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  <title>simple-statistics 6.0.0 | Documentation</title>
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          <h3 class='mb0 no-anchor'>simple-statistics</h3>
          <div class='mb1'><code>6.0.0</code></div>
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            <ul class='list-reset h5 py1-ul'>
              
                
                <li><a
                  href='#basic-descriptive-statistics'
                  class="h5 bold black caps">
                  Basic Descriptive Statistics
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#min'
                  class="">
                  min
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#max'
                  class="">
                  max
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#sum'
                  class="">
                  sum
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#sumsimple'
                  class="">
                  sumSimple
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#quantile'
                  class="">
                  quantile
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#product'
                  class="">
                  product
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#sorted-basic-descriptive-statistics'
                  class="h5 bold black caps">
                  Sorted Basic Descriptive Statistics
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#minsorted'
                  class="">
                  minSorted
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#maxsorted'
                  class="">
                  maxSorted
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#quantilesorted'
                  class="">
                  quantileSorted
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#measures-of-central-tendency'
                  class="h5 bold black caps">
                  Measures of central tendency
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#mean'
                  class="">
                  mean
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#addtomean'
                  class="">
                  addToMean
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#mode'
                  class="">
                  mode
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#modesorted'
                  class="">
                  modeSorted
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#modefast'
                  class="">
                  modeFast
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#median'
                  class="">
                  median
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#mediansorted'
                  class="">
                  medianSorted
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#harmonicmean'
                  class="">
                  harmonicMean
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#geometricmean'
                  class="">
                  geometricMean
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#rootmeansquare'
                  class="">
                  rootMeanSquare
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#sampleskewness'
                  class="">
                  sampleSkewness
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#measures-of-dispersion'
                  class="h5 bold black caps">
                  Measures of dispersion
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#variance'
                  class="">
                  variance
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#samplevariance'
                  class="">
                  sampleVariance
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#standarddeviation'
                  class="">
                  standardDeviation
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#samplestandarddeviation'
                  class="">
                  sampleStandardDeviation
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#medianabsolutedeviation'
                  class="">
                  medianAbsoluteDeviation
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#interquartilerange'
                  class="">
                  interquartileRange
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#sumnthpowerdeviations'
                  class="">
                  sumNthPowerDeviations
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#zscore'
                  class="">
                  zScore
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#similarity'
                  class="h5 bold black caps">
                  Similarity
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#samplecorrelation'
                  class="">
                  sampleCorrelation
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#samplecovariance'
                  class="">
                  sampleCovariance
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#rsquared'
                  class="">
                  rSquared
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#linear-regression'
                  class="h5 bold black caps">
                  Linear Regression
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#linearregression'
                  class="">
                  linearRegression
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#linearregressionline'
                  class="">
                  linearRegressionLine
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#randomness'
                  class="h5 bold black caps">
                  Randomness
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#shuffle'
                  class="">
                  shuffle
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#shuffleinplace'
                  class="">
                  shuffleInPlace
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#samplewithreplacement'
                  class="">
                  sampleWithReplacement
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#sample'
                  class="">
                  sample
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#classifiers'
                  class="h5 bold black caps">
                  Classifiers
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#bayesianclassifier'
                  class=" toggle-sibling">
                  BayesianClassifier
                  <span class='icon'>▸</span>
                </a>
                
                <div class='toggle-target display-none'>
                  
                  
                    <ul class='list-reset py1-ul pl1'>
                      <li class='h5'><span>Instance members</span></li>
                      
                      <li><a
                        href='#bayesianclassifiertrain'
                        class='regular pre-open'>
                        #train
                      </a></li>
                      
                      <li><a
                        href='#bayesianclassifierscore'
                        class='regular pre-open'>
                        #score
                      </a></li>
                      
                    </ul>
                  
                  
                </div>
                
                </li>
              
                
                <li><a
                  href='#perceptronmodel'
                  class=" toggle-sibling">
                  PerceptronModel
                  <span class='icon'>▸</span>
                </a>
                
                <div class='toggle-target display-none'>
                  
                  
                    <ul class='list-reset py1-ul pl1'>
                      <li class='h5'><span>Instance members</span></li>
                      
                      <li><a
                        href='#perceptronmodelpredict'
                        class='regular pre-open'>
                        #predict
                      </a></li>
                      
                      <li><a
                        href='#perceptronmodeltrain'
                        class='regular pre-open'>
                        #train
                      </a></li>
                      
                    </ul>
                  
                  
                </div>
                
                </li>
              
                
                <li><a
                  href='#distributions'
                  class="h5 bold black caps">
                  Distributions
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#bernoullidistribution'
                  class="">
                  bernoulliDistribution
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#binomialdistribution'
                  class="">
                  binomialDistribution
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#poissondistribution'
                  class="">
                  poissonDistribution
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#chisquareddistributiontable'
                  class="">
                  chiSquaredDistributionTable
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#standardnormaltable'
                  class="">
                  standardNormalTable
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#ttest'
                  class="">
                  tTest
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#ttesttwosample'
                  class="">
                  tTestTwoSample
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#cumulativestdnormalprobability'
                  class="">
                  cumulativeStdNormalProbability
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#kerneldensityestimation'
                  class="">
                  kernelDensityEstimation
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#errors'
                  class="h5 bold black caps">
                  Errors
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#errorfunction'
                  class="">
                  errorFunction
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#inverseerrorfunction'
                  class="">
                  inverseErrorFunction
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#probit'
                  class="">
                  probit
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#breaks'
                  class="h5 bold black caps">
                  Breaks
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#ckmeans'
                  class="">
                  ckmeans
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#equalintervalbreaks'
                  class="">
                  equalIntervalBreaks
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#utilities'
                  class="h5 bold black caps">
                  Utilities
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#chunk'
                  class="">
                  chunk
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#chisquaredgoodnessoffit'
                  class="">
                  chiSquaredGoodnessOfFit
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#epsilon'
                  class="">
                  epsilon
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#factorial'
                  class="">
                  factorial
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#gamma'
                  class="">
                  gamma
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#uniquecountsorted'
                  class="">
                  uniqueCountSorted
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#bisect'
                  class="">
                  bisect
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#combinationsreplacement'
                  class="">
                  combinationsReplacement
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#combinations'
                  class="">
                  combinations
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#combinemeans'
                  class="">
                  combineMeans
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#combinevariances'
                  class="">
                  combineVariances
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#extentsorted'
                  class="">
                  extentSorted
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#extent'
                  class="">
                  extent
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#permutationtest'
                  class="">
                  permutationTest
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#permutationsheap'
                  class="">
                  permutationsHeap
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#quantileranksorted'
                  class="">
                  quantileRankSorted
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#quantilerank'
                  class="">
                  quantileRank
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#quickselect'
                  class="">
                  quickselect
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#samplekurtosis'
                  class="">
                  sampleKurtosis
                  
                </a>
                
                </li>
              
                
                <li><a
                  href='#subtractfrommean'
                  class="">
                  subtractFromMean
                  
                </a>
                
                </li>
              
            </ul>
          </div>
          <div class='mt1 h6 quiet'>
            <a href='http://documentation.js.org/reading-documentation.html'>Need help reading this?</a>
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            <div class='keyline-top-not py2'><section class='py2 clearfix'>

  <h2 id='basic-descriptive-statistics' class='mt0'>
    Basic Descriptive Statistics
  </h2>

  
</section>
</div>
          
        
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='min'>
      min
    </h3>
    
    
  </div>
  

  <p>The min is the lowest number in the array. This runs on <code>O(n)</code>, linear time in respect to the array</p>


  <div class='pre p1 fill-light mt0'>min(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    sample of one or more data points

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        minimum value

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if the the length of x is less than one
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>min([<span class="hljs-number">1</span>, <span class="hljs-number">5</span>, <span class="hljs-number">-10</span>, <span class="hljs-number">100</span>, <span class="hljs-number">2</span>]); <span class="hljs-comment">// =&gt; -10</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='max'>
      max
    </h3>
    
    
  </div>
  

  <p>This computes the maximum number in an array.</p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array</p>


  <div class='pre p1 fill-light mt0'>max(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    sample of one or more data points

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        maximum value

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if the the length of x is less than one
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>max([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>]);
<span class="hljs-comment">// =&gt; 4</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='sum'>
      sum
    </h3>
    
    
  </div>
  

  <p>Our default sum is the <a href="https://pdfs.semanticscholar.org/1760/7d467cda1d0277ad272deb2113533131dc09.pdf">Kahan-Babuska algorithm</a>.
This method is an improvement over the classical
<a href="https://en.wikipedia.org/wiki/Kahan_summation_algorithm">Kahan summation algorithm</a>.
It aims at computing the sum of a list of numbers while correcting for
floating-point errors. Traditionally, sums are calculated as many
successive additions, each one with its own floating-point roundoff. These
losses in precision add up as the number of numbers increases. This alternative
algorithm is more accurate than the simple way of calculating sums by simple
addition.</p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array.</p>


  <div class='pre p1 fill-light mt0'>sum(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        sum of all input numbers

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>sum([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>]); <span class="hljs-comment">// =&gt; 6</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='sumsimple'>
      sumSimple
    </h3>
    
    
  </div>
  

  <p>The simple <a href="https://en.wikipedia.org/wiki/Summation">sum</a> of an array
is the result of adding all numbers together, starting from zero.</p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array</p>


  <div class='pre p1 fill-light mt0'>sumSimple(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        sum of all input numbers

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>sumSimple([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>]); <span class="hljs-comment">// =&gt; 6</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='quantile'>
      quantile
    </h3>
    
    
  </div>
  

  <p>The <a href="https://en.wikipedia.org/wiki/Quantile">quantile</a>:
this is a population quantile, since we assume to know the entire
dataset in this library. This is an implementation of the
<a href="http://en.wikipedia.org/wiki/Quantile#Quantiles_of_a_population">Quantiles of a Population</a>
algorithm from wikipedia.</p>
<p>Sample is a one-dimensional array of numbers,
and p is either a decimal number from 0 to 1 or an array of decimal
numbers from 0 to 1.
In terms of a k/q quantile, p = k/q - it's just dealing with fractions or dealing
with decimal values.
When p is an array, the result of the function is also an array containing the appropriate
quantiles in input order</p>


  <div class='pre p1 fill-light mt0'>quantile(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, p: (<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>> | <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    sample of one or more numbers

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>p</span> <code class='quiet'>((<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>> | <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>))</code>
	    the desired quantile, as a number between 0 and 1

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        quantile

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>quantile([<span class="hljs-number">3</span>, <span class="hljs-number">6</span>, <span class="hljs-number">7</span>, <span class="hljs-number">8</span>, <span class="hljs-number">8</span>, <span class="hljs-number">9</span>, <span class="hljs-number">10</span>, <span class="hljs-number">13</span>, <span class="hljs-number">15</span>, <span class="hljs-number">16</span>, <span class="hljs-number">20</span>], <span class="hljs-number">0.5</span>); <span class="hljs-comment">// =&gt; 9</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='product'>
      product
    </h3>
    
    
  </div>
  

  <p>The <a href="https://en.wikipedia.org/wiki/Product_(mathematics)">product</a> of an array
is the result of multiplying all numbers together, starting using one as the multiplicative identity.</p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array</p>


  <div class='pre p1 fill-light mt0'>product(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        product of all input numbers

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>product([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>]); <span class="hljs-comment">// =&gt; 24</span></pre>
    
  
</section>

          
            <div class='keyline-top-not py2'><section class='py2 clearfix'>

  <h2 id='sorted-basic-descriptive-statistics' class='mt0'>
    Sorted Basic Descriptive Statistics
  </h2>

  
    <p>These are special versions of methods that assume your input
is sorted. This assumptions lets them run a lot faster, usually
in O(1).</p>

  
</section>
</div>
          
        
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='minsorted'>
      minSorted
    </h3>
    
    
  </div>
  

  <p>The minimum is the lowest number in the array. With a sorted array,
the first element in the array is always the smallest, so this calculation
can be done in one step, or constant time.</p>


  <div class='pre p1 fill-light mt0'>minSorted(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        minimum value

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>minSorted([<span class="hljs-number">-100</span>, <span class="hljs-number">-10</span>, <span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">5</span>]); <span class="hljs-comment">// =&gt; -100</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='maxsorted'>
      maxSorted
    </h3>
    
    
  </div>
  

  <p>The maximum is the highest number in the array. With a sorted array,
the last element in the array is always the largest, so this calculation
can be done in one step, or constant time.</p>


  <div class='pre p1 fill-light mt0'>maxSorted(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        maximum value

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>maxSorted([<span class="hljs-number">-100</span>, <span class="hljs-number">-10</span>, <span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">5</span>]); <span class="hljs-comment">// =&gt; 5</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='quantilesorted'>
      quantileSorted
    </h3>
    
    
  </div>
  

  <p>This is the internal implementation of quantiles: when you know
that the order is sorted, you don't need to re-sort it, and the computations
are faster.</p>


  <div class='pre p1 fill-light mt0'>quantileSorted(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, p: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    sample of one or more data points

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>p</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    desired quantile: a number between 0 to 1, inclusive

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        quantile value

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if p ix outside of the range from 0 to 1
</li>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if x is empty
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>quantileSorted([<span class="hljs-number">3</span>, <span class="hljs-number">6</span>, <span class="hljs-number">7</span>, <span class="hljs-number">8</span>, <span class="hljs-number">8</span>, <span class="hljs-number">9</span>, <span class="hljs-number">10</span>, <span class="hljs-number">13</span>, <span class="hljs-number">15</span>, <span class="hljs-number">16</span>, <span class="hljs-number">20</span>], <span class="hljs-number">0.5</span>); <span class="hljs-comment">// =&gt; 9</span></pre>
    
  
</section>

          
            <div class='keyline-top-not py2'><section class='py2 clearfix'>

  <h2 id='measures-of-central-tendency' class='mt0'>
    Measures of central tendency
  </h2>

  
    <p>These are different ways to identifying centers or locations of
a distribution.</p>

  
</section>
</div>
          
        
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='mean'>
      mean
    </h3>
    
    
  </div>
  

  <p>The mean, <em>also known as average</em>,
is the sum of all values over the number of values.
This is a <a href="https://en.wikipedia.org/wiki/Central_tendency">measure of central tendency</a>:
a method of finding a typical or central value of a set of numbers.</p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array</p>


  <div class='pre p1 fill-light mt0'>mean(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    sample of one or more data points

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        mean

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if the the length of x is less than one
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>mean([<span class="hljs-number">0</span>, <span class="hljs-number">10</span>]); <span class="hljs-comment">// =&gt; 5</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='addtomean'>
      addToMean
    </h3>
    
    
  </div>
  

  <p>When adding a new value to a list, one does not have to necessary
recompute the mean of the list in linear time. They can instead use
this function to compute the new mean by providing the current mean,
the number of elements in the list that produced it and the new
value to add.</p>


  <div class='pre p1 fill-light mt0'>addToMean(mean: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, n: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, newValue: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
  <div>Since: 2.5.0</div>

  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>mean</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    current mean

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>n</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    number of items in the list

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>newValue</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    the added value

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        the new mean

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>addToMean(<span class="hljs-number">14</span>, <span class="hljs-number">5</span>, <span class="hljs-number">53</span>); <span class="hljs-comment">// =&gt; 20.5</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='mode'>
      mode
    </h3>
    
    
  </div>
  

  <p>The <a href="http://bit.ly/W5K4Yt">mode</a> is the number that appears in a list the highest number of times.
There can be multiple modes in a list: in the event of a tie, this
algorithm will return the most recently seen mode.</p>
<p>This is a <a href="https://en.wikipedia.org/wiki/Central_tendency">measure of central tendency</a>:
a method of finding a typical or central value of a set of numbers.</p>
<p>This runs on <code>O(nlog(n))</code> because it needs to sort the array internally
before running an <code>O(n)</code> search to find the mode.</p>


  <div class='pre p1 fill-light mt0'>mode(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        mode

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>mode([<span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">1</span>]); <span class="hljs-comment">// =&gt; 0</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='modesorted'>
      modeSorted
    </h3>
    
    
  </div>
  

  <p>The <a href="http://bit.ly/W5K4Yt">mode</a> is the number that appears in a list the highest number of times.
There can be multiple modes in a list: in the event of a tie, this
algorithm will return the most recently seen mode.</p>
<p>This is a <a href="https://en.wikipedia.org/wiki/Central_tendency">measure of central tendency</a>:
a method of finding a typical or central value of a set of numbers.</p>
<p>This runs in <code>O(n)</code> because the input is sorted.</p>


  <div class='pre p1 fill-light mt0'>modeSorted(sorted: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>sorted</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    a sample of one or more data points

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        mode

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if sorted is empty
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>modeSorted([<span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">1</span>]); <span class="hljs-comment">// =&gt; 0</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='modefast'>
      modeFast
    </h3>
    
    
  </div>
  

  <p>The <a href="http://bit.ly/W5K4Yt">mode</a> is the number that appears in a list the highest number of times.
There can be multiple modes in a list: in the event of a tie, this
algorithm will return the most recently seen mode.</p>
<p>modeFast uses a Map object to keep track of the mode, instead of the approach
used with <code>mode</code>, a sorted array. As a result, it is faster
than <code>mode</code> and supports any data type that can be compared with <code>==</code>.
It also requires a
<a href="https://kangax.github.io/compat-table/es6/#test-Map">JavaScript environment with support for Map</a>,
and will throw an error if Map is not available.</p>
<p>This is a <a href="https://en.wikipedia.org/wiki/Central_tendency">measure of central tendency</a>:
a method of finding a typical or central value of a set of numbers.</p>


  <div class='pre p1 fill-light mt0'>modeFast(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;any>): any?</div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;any>)</code>
	    a sample of one or more data points

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code>any?</code>:
        mode

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/ReferenceError">ReferenceError</a>: if the JavaScript environment doesn't support Map
</li>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if x is empty
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>modeFast([<span class="hljs-string">'rabbits'</span>, <span class="hljs-string">'rabbits'</span>, <span class="hljs-string">'squirrels'</span>]); <span class="hljs-comment">// =&gt; 'rabbits'</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='median'>
      median
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Median">median</a> is
the middle number of a list. This is often a good indicator of 'the middle'
when there are outliers that skew the <code>mean()</code> value.
This is a <a href="https://en.wikipedia.org/wiki/Central_tendency">measure of central tendency</a>:
a method of finding a typical or central value of a set of numbers.</p>
<p>The median isn't necessarily one of the elements in the list: the value
can be the average of two elements if the list has an even length
and the two central values are different.</p>


  <div class='pre p1 fill-light mt0'>median(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        median value

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>median([<span class="hljs-number">10</span>, <span class="hljs-number">2</span>, <span class="hljs-number">5</span>, <span class="hljs-number">100</span>, <span class="hljs-number">2</span>, <span class="hljs-number">1</span>]); <span class="hljs-comment">// =&gt; 3.5</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='mediansorted'>
      medianSorted
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Median">median</a> is
the middle number of a list. This is often a good indicator of 'the middle'
when there are outliers that skew the <code>mean()</code> value.
This is a <a href="https://en.wikipedia.org/wiki/Central_tendency">measure of central tendency</a>:
a method of finding a typical or central value of a set of numbers.</p>
<p>The median isn't necessarily one of the elements in the list: the value
can be the average of two elements if the list has an even length
and the two central values are different.</p>


  <div class='pre p1 fill-light mt0'>medianSorted(sorted: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>sorted</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        median value

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>medianSorted([<span class="hljs-number">10</span>, <span class="hljs-number">2</span>, <span class="hljs-number">5</span>, <span class="hljs-number">100</span>, <span class="hljs-number">2</span>, <span class="hljs-number">1</span>]); <span class="hljs-comment">// =&gt; 52.5</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='harmonicmean'>
      harmonicMean
    </h3>
    
    
  </div>
  

  <p>The <a href="https://en.wikipedia.org/wiki/Harmonic_mean">Harmonic Mean</a> is
a mean function typically used to find the average of rates.
This mean is calculated by taking the reciprocal of the arithmetic mean
of the reciprocals of the input numbers.</p>
<p>This is a <a href="https://en.wikipedia.org/wiki/Central_tendency">measure of central tendency</a>:
a method of finding a typical or central value of a set of numbers.</p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array.</p>


  <div class='pre p1 fill-light mt0'>harmonicMean(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    sample of one or more data points

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        harmonic mean

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if x is empty
</li>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if x contains a negative number
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>harmonicMean([<span class="hljs-number">2</span>, <span class="hljs-number">3</span>]).toFixed(<span class="hljs-number">2</span>) <span class="hljs-comment">// =&gt; '2.40'</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='geometricmean'>
      geometricMean
    </h3>
    
    
  </div>
  

  <p>The <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric Mean</a> is
a mean function that is more useful for numbers in different
ranges.</p>
<p>This is the nth root of the input numbers multiplied by each other.</p>
<p>The geometric mean is often useful for
<strong><a href="https://en.wikipedia.org/wiki/Geometric_mean#Proportional_growth">proportional growth</a></strong>: given
growth rates for multiple years, like <em>80%, 16.66% and 42.85%</em>, a simple
mean will incorrectly estimate an average growth rate, whereas a geometric
mean will correctly estimate a growth rate that, over those years,
will yield the same end value.</p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array</p>


  <div class='pre p1 fill-light mt0'>geometricMean(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    sample of one or more data points

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        geometric mean

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if x is empty
</li>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if x contains a negative number
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'><span class="hljs-keyword">var</span> growthRates = [<span class="hljs-number">1.80</span>, <span class="hljs-number">1.166666</span>, <span class="hljs-number">1.428571</span>];
<span class="hljs-keyword">var</span> averageGrowth = ss.geometricMean(growthRates);
<span class="hljs-keyword">var</span> averageGrowthRates = [averageGrowth, averageGrowth, averageGrowth];
<span class="hljs-keyword">var</span> startingValue = <span class="hljs-number">10</span>;
<span class="hljs-keyword">var</span> startingValueMean = <span class="hljs-number">10</span>;
growthRates.forEach(<span class="hljs-function"><span class="hljs-keyword">function</span>(<span class="hljs-params">rate</span>) </span>{
  startingValue *= rate;
});
averageGrowthRates.forEach(<span class="hljs-function"><span class="hljs-keyword">function</span>(<span class="hljs-params">rate</span>) </span>{
  startingValueMean *= rate;
});
startingValueMean === startingValue;</pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='rootmeansquare'>
      rootMeanSquare
    </h3>
    
    
  </div>
  

  <p>The Root Mean Square (RMS) is
a mean function used as a measure of the magnitude of a set
of numbers, regardless of their sign.
This is the square root of the mean of the squares of the
input numbers.
This runs on <code>O(n)</code>, linear time in respect to the array</p>


  <div class='pre p1 fill-light mt0'>rootMeanSquare(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    a sample of one or more data points

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        root mean square

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if x is empty
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>rootMeanSquare([<span class="hljs-number">-1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">-1</span>, <span class="hljs-number">1</span>]); <span class="hljs-comment">// =&gt; 1</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='sampleskewness'>
      sampleSkewness
    </h3>
    
    
  </div>
  

  <p><a href="http://en.wikipedia.org/wiki/Skewness">Skewness</a> is
a measure of the extent to which a probability distribution of a
real-valued random variable "leans" to one side of the mean.
The skewness value can be positive or negative, or even undefined.</p>
<p>Implementation is based on the adjusted Fisher-Pearson standardized
moment coefficient, which is the version found in Excel and several
statistical packages including Minitab, SAS and SPSS.</p>


  <div class='pre p1 fill-light mt0'>sampleSkewness(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
  <div>Since: 4.1.0</div>

  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    a sample of 3 or more data points

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        sample skewness

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if x has length less than 3
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>sampleSkewness([<span class="hljs-number">2</span>, <span class="hljs-number">4</span>, <span class="hljs-number">6</span>, <span class="hljs-number">3</span>, <span class="hljs-number">1</span>]); <span class="hljs-comment">// =&gt; 0.590128656384365</span></pre>
    
  
</section>

          
            <div class='keyline-top-not py2'><section class='py2 clearfix'>

  <h2 id='measures-of-dispersion' class='mt0'>
    Measures of dispersion
  </h2>

  
    <p>These are different ways of determining how spread out a distribution is.</p>

  
</section>
</div>
          
        
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='variance'>
      variance
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Variance">variance</a>
is the sum of squared deviations from the mean.</p>
<p>This is an implementation of variance, not sample variance:
see the <code>sampleVariance</code> method if you want a sample measure.</p>


  <div class='pre p1 fill-light mt0'>variance(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    a population of one or more data points

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        variance: a value greater than or equal to zero.
zero indicates that all values are identical.

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if x's length is 0
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>variance([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>, <span class="hljs-number">5</span>, <span class="hljs-number">6</span>]); <span class="hljs-comment">// =&gt; 2.9166666666666665</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='samplevariance'>
      sampleVariance
    </h3>
    
    
  </div>
  

  <p>The <a href="https://en.wikipedia.org/wiki/Variance#Sample_variance">sample variance</a>
is the sum of squared deviations from the mean. The sample variance
is distinguished from the variance by the usage of <a href="https://en.wikipedia.org/wiki/Bessel&#x27;s_correction">Bessel's Correction</a>:
instead of dividing the sum of squared deviations by the length of the input,
it is divided by the length minus one. This corrects the bias in estimating
a value from a set that you don't know if full.</p>
<p>References:</p>
<ul>
<li><a href="http://mathworld.wolfram.com/SampleVariance.html">Wolfram MathWorld on Sample Variance</a></li>
</ul>


  <div class='pre p1 fill-light mt0'>sampleVariance(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    a sample of two or more data points

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        sample variance

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if the length of x is less than 2
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>sampleVariance([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>, <span class="hljs-number">5</span>]); <span class="hljs-comment">// =&gt; 2.5</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='standarddeviation'>
      standardDeviation
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Standard_deviation">standard deviation</a>
is the square root of the variance. This is also known as the population
standard deviation. It's useful for measuring the amount
of variation or dispersion in a set of values.</p>
<p>Standard deviation is only appropriate for full-population knowledge: for
samples of a population, <a href="#samplestandarddeviation">sampleStandardDeviation</a> is
more appropriate.</p>


  <div class='pre p1 fill-light mt0'>standardDeviation(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        standard deviation

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>variance([<span class="hljs-number">2</span>, <span class="hljs-number">4</span>, <span class="hljs-number">4</span>, <span class="hljs-number">4</span>, <span class="hljs-number">5</span>, <span class="hljs-number">5</span>, <span class="hljs-number">7</span>, <span class="hljs-number">9</span>]); <span class="hljs-comment">// =&gt; 4</span>
standardDeviation([<span class="hljs-number">2</span>, <span class="hljs-number">4</span>, <span class="hljs-number">4</span>, <span class="hljs-number">4</span>, <span class="hljs-number">5</span>, <span class="hljs-number">5</span>, <span class="hljs-number">7</span>, <span class="hljs-number">9</span>]); <span class="hljs-comment">// =&gt; 2</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='samplestandarddeviation'>
      sampleStandardDeviation
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Standard_deviation#Sample_standard_deviation">sample standard deviation</a>
is the square root of the sample variance.</p>


  <div class='pre p1 fill-light mt0'>sampleStandardDeviation(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input array

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        sample standard deviation

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>sampleStandardDeviation([<span class="hljs-number">2</span>, <span class="hljs-number">4</span>, <span class="hljs-number">4</span>, <span class="hljs-number">4</span>, <span class="hljs-number">5</span>, <span class="hljs-number">5</span>, <span class="hljs-number">7</span>, <span class="hljs-number">9</span>]).toFixed(<span class="hljs-number">2</span>);
<span class="hljs-comment">// =&gt; '2.14'</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='medianabsolutedeviation'>
      medianAbsoluteDeviation
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Median_absolute_deviation">Median Absolute Deviation</a> is
a robust measure of statistical
dispersion. It is more resilient to outliers than the standard deviation.</p>


  <div class='pre p1 fill-light mt0'>medianAbsoluteDeviation(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input array

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        median absolute deviation

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>medianAbsoluteDeviation([<span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>, <span class="hljs-number">4</span>, <span class="hljs-number">6</span>, <span class="hljs-number">9</span>]); <span class="hljs-comment">// =&gt; 1</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='interquartilerange'>
      interquartileRange
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Interquartile_range">Interquartile range</a> is
a measure of statistical dispersion, or how scattered, spread, or
concentrated a distribution is. It's computed as the difference between
the third quartile and first quartile.</p>


  <div class='pre p1 fill-light mt0'>interquartileRange(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    sample of one or more numbers

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        interquartile range: the span between lower and upper quartile,
0.25 and 0.75

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>interquartileRange([<span class="hljs-number">0</span>, <span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>]); <span class="hljs-comment">// =&gt; 2</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='sumnthpowerdeviations'>
      sumNthPowerDeviations
    </h3>
    
    
  </div>
  

  <p>The sum of deviations to the Nth power.
When n=2 it's the sum of squared deviations.
When n=3 it's the sum of cubed deviations.</p>


  <div class='pre p1 fill-light mt0'>sumNthPowerDeviations(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, n: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    
          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>n</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    power

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        sum of nth power deviations

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'><span class="hljs-keyword">var</span> input = [<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>];
<span class="hljs-comment">// since the variance of a set is the mean squared</span>
<span class="hljs-comment">// deviations, we can calculate that with sumNthPowerDeviations:</span>
sumNthPowerDeviations(input, <span class="hljs-number">2</span>) / input.length;</pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='zscore'>
      zScore
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Standard_score">Z-Score, or Standard Score</a>.</p>
<p>The standard score is the number of standard deviations an observation
or datum is above or below the mean. Thus, a positive standard score
represents a datum above the mean, while a negative standard score
represents a datum below the mean. It is a dimensionless quantity
obtained by subtracting the population mean from an individual raw
score and then dividing the difference by the population standard
deviation.</p>
<p>The z-score is only defined if one knows the population parameters;
if one only has a sample set, then the analogous computation with
sample mean and sample standard deviation yields the
Student's t-statistic.</p>


  <div class='pre p1 fill-light mt0'>zScore(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, mean: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, standardDeviation: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    
          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>mean</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    
          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>standardDeviation</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    
          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        z score

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>zScore(<span class="hljs-number">78</span>, <span class="hljs-number">80</span>, <span class="hljs-number">5</span>); <span class="hljs-comment">// =&gt; -0.4</span></pre>
    
  
</section>

          
            <div class='keyline-top-not py2'><section class='py2 clearfix'>

  <h2 id='similarity' class='mt0'>
    Similarity
  </h2>

  
</section>
</div>
          
        
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='samplecorrelation'>
      sampleCorrelation
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Correlation_and_dependence">correlation</a> is
a measure of how correlated two datasets are, between -1 and 1</p>


  <div class='pre p1 fill-light mt0'>sampleCorrelation(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, y: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    first input

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>y</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    second input

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        sample correlation

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>sampleCorrelation([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>, <span class="hljs-number">5</span>, <span class="hljs-number">6</span>], [<span class="hljs-number">2</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>, <span class="hljs-number">5</span>, <span class="hljs-number">60</span>]).toFixed(<span class="hljs-number">2</span>);
<span class="hljs-comment">// =&gt; '0.69'</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='samplecovariance'>
      sampleCovariance
    </h3>
    
    
  </div>
  

  <p><a href="https://en.wikipedia.org/wiki/Sample_mean_and_sampleCovariance">Sample covariance</a> of two datasets:
how much do the two datasets move together?
x and y are two datasets, represented as arrays of numbers.</p>


  <div class='pre p1 fill-light mt0'>sampleCovariance(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, y: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    a sample of two or more data points

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>y</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    a sample of two or more data points

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        sample covariance

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if x and y do not have equal lengths
</li>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if x or y have length of one or less
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>sampleCovariance([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>, <span class="hljs-number">5</span>, <span class="hljs-number">6</span>], [<span class="hljs-number">6</span>, <span class="hljs-number">5</span>, <span class="hljs-number">4</span>, <span class="hljs-number">3</span>, <span class="hljs-number">2</span>, <span class="hljs-number">1</span>]); <span class="hljs-comment">// =&gt; -3.5</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='rsquared'>
      rSquared
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Coefficient_of_determination">R Squared</a>
value of data compared with a function <code>f</code>
is the sum of the squared differences between the prediction
and the actual value.</p>


  <div class='pre p1 fill-light mt0'>rSquared(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>>, func: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>>)</code>
	    input data: this should be doubly-nested

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>func</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a>)</code>
	    function called on 
<code>[i][0]</code>
 values within the dataset

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        r-squared value

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'><span class="hljs-keyword">var</span> samples = [[<span class="hljs-number">0</span>, <span class="hljs-number">0</span>], [<span class="hljs-number">1</span>, <span class="hljs-number">1</span>]];
<span class="hljs-keyword">var</span> regressionLine = linearRegressionLine(linearRegression(samples));
rSquared(samples, regressionLine); <span class="hljs-comment">// = 1 this line is a perfect fit</span></pre>
    
  
</section>

          
            <div class='keyline-top-not py2'><section class='py2 clearfix'>

  <h2 id='linear-regression' class='mt0'>
    Linear Regression
  </h2>

  
</section>
</div>
          
        
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='linearregression'>
      linearRegression
    </h3>
    
    
  </div>
  

  <p><a href="http://en.wikipedia.org/wiki/Simple_linear_regression">Simple linear regression</a>
is a simple way to find a fitted line
between a set of coordinates. This algorithm finds the slope and y-intercept of a regression line
using the least sum of squares.</p>


  <div class='pre p1 fill-light mt0'>linearRegression(data: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object">Object</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>data</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>>)</code>
	    an array of two-element of arrays,
like 
<code>[[0, 1], [2, 3]]</code>

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object">Object</a></code>:
        object containing slope and intersect of regression line

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>linearRegression([[<span class="hljs-number">0</span>, <span class="hljs-number">0</span>], [<span class="hljs-number">1</span>, <span class="hljs-number">1</span>]]); <span class="hljs-comment">// =&gt; { m: 1, b: 0 }</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='linearregressionline'>
      linearRegressionLine
    </h3>
    
    
  </div>
  

  <p>Given the output of <code>linearRegression</code>: an object
with <code>m</code> and <code>b</code> values indicating slope and intercept,
respectively, generate a line function that translates
x values into y values.</p>


  <div class='pre p1 fill-light mt0'>linearRegressionLine(mb: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object">Object</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>mb</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object">Object</a>)</code>
	    object with 
<code>m</code>
 and 
<code>b</code>
 members, representing
slope and intersect of desired line

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a></code>:
        method that computes y-value at any given
x-value on the line.

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'><span class="hljs-keyword">var</span> l = linearRegressionLine(linearRegression([[<span class="hljs-number">0</span>, <span class="hljs-number">0</span>], [<span class="hljs-number">1</span>, <span class="hljs-number">1</span>]]));
l(<span class="hljs-number">0</span>) <span class="hljs-comment">// = 0</span>
l(<span class="hljs-number">2</span>) <span class="hljs-comment">// = 2</span>
linearRegressionLine({ <span class="hljs-attr">b</span>: <span class="hljs-number">0</span>, <span class="hljs-attr">m</span>: <span class="hljs-number">1</span> })(<span class="hljs-number">1</span>); <span class="hljs-comment">// =&gt; 1</span>
linearRegressionLine({ <span class="hljs-attr">b</span>: <span class="hljs-number">1</span>, <span class="hljs-attr">m</span>: <span class="hljs-number">1</span> })(<span class="hljs-number">1</span>); <span class="hljs-comment">// =&gt; 2</span></pre>
    
  
</section>

          
            <div class='keyline-top-not py2'><section class='py2 clearfix'>

  <h2 id='randomness' class='mt0'>
    Randomness
  </h2>

  
</section>
</div>
          
        
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='shuffle'>
      shuffle
    </h3>
    
    
  </div>
  

  <p>A <a href="http://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle">Fisher-Yates shuffle</a>
is a fast way to create a random permutation of a finite set. This is
a function around <code>shuffle_in_place</code> that adds the guarantee that
it will not modify its input.</p>


  <div class='pre p1 fill-light mt0'>shuffle(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>, randomSource: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>)</code>
	    sample of 0 or more numbers

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>randomSource</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a>
            = <code>Math.random</code>)</code>
	    an optional entropy source that
returns numbers between 0 inclusive and 1 exclusive: the range [0, 1)

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a></code>:
        shuffled version of input

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'><span class="hljs-keyword">var</span> shuffled = shuffle([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>]);
shuffled; <span class="hljs-comment">// = [2, 3, 1, 4] or any other random permutation</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='shuffleinplace'>
      shuffleInPlace
    </h3>
    
    
  </div>
  

  <p>A <a href="http://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle">Fisher-Yates shuffle</a>
in-place - which means that it <strong>will change the order of the original
array by reference</strong>.</p>
<p>This is an algorithm that generates a random <a href="https://en.wikipedia.org/wiki/Permutation">permutation</a>
of a set.</p>


  <div class='pre p1 fill-light mt0'>shuffleInPlace(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>, randomSource: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>)</code>
	    sample of one or more numbers

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>randomSource</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a>
            = <code>Math.random</code>)</code>
	    an optional entropy source that
returns numbers between 0 inclusive and 1 exclusive: the range [0, 1)

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a></code>:
        x

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'><span class="hljs-keyword">var</span> x = [<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>];
shuffleInPlace(x);
<span class="hljs-comment">// x is shuffled to a value like [2, 1, 4, 3]</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='samplewithreplacement'>
      sampleWithReplacement
    </h3>
    
    
  </div>
  

  <p>Sampling with replacement is a type of sampling that allows the same
item to be picked out of a population more than once.</p>


  <div class='pre p1 fill-light mt0'>sampleWithReplacement(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;any>, n: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, randomSource: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;any>)</code>
	    an array of any kind of value

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>n</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    count of how many elements to take

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>randomSource</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a>
            = <code>Math.random</code>)</code>
	    an optional entropy source that
returns numbers between 0 inclusive and 1 exclusive: the range [0, 1)

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a></code>:
        n sampled items from the population

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'><span class="hljs-keyword">var</span> values = [<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>];
sampleWithReplacement(values, <span class="hljs-number">2</span>); <span class="hljs-comment">// returns 2 random values, like [2, 4];</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='sample'>
      sample
    </h3>
    
    
  </div>
  

  <p>Create a <a href="http://en.wikipedia.org/wiki/Simple_random_sample">simple random sample</a>
from a given array of <code>n</code> elements.</p>
<p>The sampled values will be in any order, not necessarily the order
they appear in the input.</p>


  <div class='pre p1 fill-light mt0'>sample(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;any>, n: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, randomSource: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;any>)</code>
	    input array. can contain any type

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>n</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    count of how many elements to take

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>randomSource</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a>
            = <code>Math.random</code>)</code>
	    an optional entropy source that
returns numbers between 0 inclusive and 1 exclusive: the range [0, 1)

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a></code>:
        subset of n elements in original array

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'><span class="hljs-keyword">var</span> values = [<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">4</span>, <span class="hljs-number">5</span>, <span class="hljs-number">6</span>, <span class="hljs-number">7</span>, <span class="hljs-number">8</span>, <span class="hljs-number">9</span>];
sample(values, <span class="hljs-number">3</span>); <span class="hljs-comment">// returns 3 random values, like [2, 5, 8];</span></pre>
    
  
</section>

          
            <div class='keyline-top-not py2'><section class='py2 clearfix'>

  <h2 id='classifiers' class='mt0'>
    Classifiers
  </h2>

  
</section>
</div>
          
        
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='bayesianclassifier'>
      BayesianClassifier
    </h3>
    
    
  </div>
  

  <p><a href="http://en.wikipedia.org/wiki/Naive_Bayes_classifier">Bayesian Classifier</a></p>
<p>This is a naïve bayesian classifier that takes
singly-nested objects.</p>


  <div class='pre p1 fill-light mt0'>new BayesianClassifier()</div>
  
  
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'><span class="hljs-keyword">var</span> bayes = <span class="hljs-keyword">new</span> BayesianClassifier();
bayes.train({
  <span class="hljs-attr">species</span>: <span class="hljs-string">'Cat'</span>
}, <span class="hljs-string">'animal'</span>);
<span class="hljs-keyword">var</span> result = bayes.score({
  <span class="hljs-attr">species</span>: <span class="hljs-string">'Cat'</span>
})
<span class="hljs-comment">// result</span>
<span class="hljs-comment">// {</span>
<span class="hljs-comment">//   animal: 1</span>
<span class="hljs-comment">// }</span></pre>
    
  
    <div class='py1 quiet mt1 prose-big'>Instance Members</div>
    <div class="clearfix">
  
    <div class='border-bottom' id='bayesianclassifiertrain'>
      <div class="clearfix small pointer toggle-sibling">
        <div class="py1 contain">
            <a class='icon pin-right py1 dark-link caret-right'>▸</a>
            <span class='code strong strong truncate'>train(item, category)</span>
        </div>
      </div>
      <div class="clearfix display-none toggle-target">
        <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <p>Train the classifier with a new item, which has a single
dimension of Javascript literal keys and values.</p>


  <div class='pre p1 fill-light mt0'>train(item: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object">Object</a>, category: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/String">string</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/undefined">undefined</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>item</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object">Object</a>)</code>
	    an object with singly-deep properties

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>category</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/String">string</a>)</code>
	    the category this item belongs to

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/undefined">undefined</a></code>:
        adds the item to the classifier

      
</section>

      </div>
    </div>
  
    <div class='border-bottom' id='bayesianclassifierscore'>
      <div class="clearfix small pointer toggle-sibling">
        <div class="py1 contain">
            <a class='icon pin-right py1 dark-link caret-right'>▸</a>
            <span class='code strong strong truncate'>score(item)</span>
        </div>
      </div>
      <div class="clearfix display-none toggle-target">
        <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <p>Generate a score of how well this item matches all
possible categories based on its attributes</p>


  <div class='pre p1 fill-light mt0'>score(item: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object">Object</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object">Object</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>item</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object">Object</a>)</code>
	    an item in the same format as with train

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object">Object</a></code>:
        of probabilities that this item belongs to a
given category.

      
</section>

      </div>
    </div>
  
</div>

  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='perceptronmodel'>
      PerceptronModel
    </h3>
    
    
  </div>
  

  <p>This is a single-layer <a href="http://en.wikipedia.org/wiki/Perceptron">Perceptron Classifier</a> that takes
arrays of numbers and predicts whether they should be classified
as either 0 or 1 (negative or positive examples).</p>


  <div class='pre p1 fill-light mt0'>new PerceptronModel()</div>
  
  
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'><span class="hljs-comment">// Create the model</span>
<span class="hljs-keyword">var</span> p = <span class="hljs-keyword">new</span> PerceptronModel();
<span class="hljs-comment">// Train the model with input with a diagonal boundary.</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">var</span> i = <span class="hljs-number">0</span>; i &lt; <span class="hljs-number">5</span>; i++) {
    p.train([<span class="hljs-number">1</span>, <span class="hljs-number">1</span>], <span class="hljs-number">1</span>);
    p.train([<span class="hljs-number">0</span>, <span class="hljs-number">1</span>], <span class="hljs-number">0</span>);
    p.train([<span class="hljs-number">1</span>, <span class="hljs-number">0</span>], <span class="hljs-number">0</span>);
    p.train([<span class="hljs-number">0</span>, <span class="hljs-number">0</span>], <span class="hljs-number">0</span>);
}
p.predict([<span class="hljs-number">0</span>, <span class="hljs-number">0</span>]); <span class="hljs-comment">// 0</span>
p.predict([<span class="hljs-number">0</span>, <span class="hljs-number">1</span>]); <span class="hljs-comment">// 0</span>
p.predict([<span class="hljs-number">1</span>, <span class="hljs-number">0</span>]); <span class="hljs-comment">// 0</span>
p.predict([<span class="hljs-number">1</span>, <span class="hljs-number">1</span>]); <span class="hljs-comment">// 1</span></pre>
    
  
    <div class='py1 quiet mt1 prose-big'>Instance Members</div>
    <div class="clearfix">
  
    <div class='border-bottom' id='perceptronmodelpredict'>
      <div class="clearfix small pointer toggle-sibling">
        <div class="py1 contain">
            <a class='icon pin-right py1 dark-link caret-right'>▸</a>
            <span class='code strong strong truncate'>predict(features)</span>
        </div>
      </div>
      <div class="clearfix display-none toggle-target">
        <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <p><strong>Predict</strong>: Use an array of features with the weight array and bias
to predict whether an example is labeled 0 or 1.</p>


  <div class='pre p1 fill-light mt0'>predict(features: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>features</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    an array of features as numbers

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        1 if the score is over 0, otherwise 0

      
</section>

      </div>
    </div>
  
    <div class='border-bottom' id='perceptronmodeltrain'>
      <div class="clearfix small pointer toggle-sibling">
        <div class="py1 contain">
            <a class='icon pin-right py1 dark-link caret-right'>▸</a>
            <span class='code strong strong truncate'>train(features, label)</span>
        </div>
      </div>
      <div class="clearfix display-none toggle-target">
        <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <p><strong>Train</strong> the classifier with a new example, which is
a numeric array of features and a 0 or 1 label.</p>


  <div class='pre p1 fill-light mt0'>train(features: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, label: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="#perceptronmodel">PerceptronModel</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>features</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    an array of features as numbers

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>label</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    either 0 or 1

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="#perceptronmodel">PerceptronModel</a></code>:
        this

      
</section>

      </div>
    </div>
  
</div>

  
</section>

          
            <div class='keyline-top-not py2'><section class='py2 clearfix'>

  <h2 id='distributions' class='mt0'>
    Distributions
  </h2>

  
</section>
</div>
          
        
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='bernoullidistribution'>
      bernoulliDistribution
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution">Bernoulli distribution</a>
is the probability discrete
distribution of a random variable which takes value 1 with success
probability <code>p</code> and value 0 with failure
probability <code>q</code> = 1 - <code>p</code>. It can be used, for example, to represent the
toss of a coin, where "1" is defined to mean "heads" and "0" is defined
to mean "tails" (or vice versa). It is
a special case of a Binomial Distribution
where <code>n</code> = 1.</p>


  <div class='pre p1 fill-light mt0'>bernoulliDistribution(p: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>p</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    input value, between 0 and 1 inclusive

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>></code>:
        values of bernoulli distribution at this point

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if p is outside 0 and 1
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>bernoulliDistribution(<span class="hljs-number">0.3</span>); <span class="hljs-comment">// =&gt; [0.7, 0.3]</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='binomialdistribution'>
      binomialDistribution
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Binomial_distribution">Binomial Distribution</a> is the discrete probability
distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields
success with probability <code>probability</code>. Such a success/failure experiment is also called a Bernoulli experiment or
Bernoulli trial; when trials = 1, the Binomial Distribution is a Bernoulli Distribution.</p>


  <div class='pre p1 fill-light mt0'>binomialDistribution(trials: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, probability: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>trials</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    number of trials to simulate

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>probability</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    
          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>></code>:
        output

      
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='poissondistribution'>
      poissonDistribution
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Poisson_distribution">Poisson Distribution</a>
is a discrete probability distribution that expresses the probability
of a given number of events occurring in a fixed interval of time
and/or space if these events occur with a known average rate and
independently of the time since the last event.</p>
<p>The Poisson Distribution is characterized by the strictly positive
mean arrival or occurrence rate, <code>λ</code>.</p>


  <div class='pre p1 fill-light mt0'>poissonDistribution(lambda: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>lambda</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    location poisson distribution

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>></code>:
        values of poisson distribution at that point

      
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='chisquareddistributiontable'>
      chiSquaredDistributionTable
    </h3>
    
    
  </div>
  

  <p><strong>Percentage Points of the χ2 (Chi-Squared) Distribution</strong></p>
<p>The <a href="http://en.wikipedia.org/wiki/Chi-squared_distribution">χ2 (Chi-Squared) Distribution</a> is used in the common
chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two
criteria of classification of qualitative data, and in confidence interval estimation for a population standard
deviation of a normal distribution from a sample standard deviation.</p>
<p>Values from Appendix 1, Table III of William W. Hines &#x26; Douglas C. Montgomery, "Probability and Statistics in
Engineering and Management Science", Wiley (1980).</p>


  <div class='pre p1 fill-light mt0'>chiSquaredDistributionTable</div>
  
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='standardnormaltable'>
      standardNormalTable
    </h3>
    
    
  </div>
  

  <p>A standard normal table, also called the unit normal table or Z table,
is a mathematical table for the values of Φ (phi), which are the values of
the cumulative distribution function of the normal distribution.
It is used to find the probability that a statistic is observed below,
above, or between values on the standard normal distribution, and by
extension, any normal distribution.</p>
<p>The probabilities are calculated using the
<a href="https://en.wikipedia.org/wiki/Normal_distribution#Cumulative_distribution_function">Cumulative distribution function</a>.
The table used is the cumulative, and not cumulative from 0 to mean
(even though the latter has 5 digits precision, instead of 4).</p>


  <div class='pre p1 fill-light mt0'>standardNormalTable</div>
  
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='ttest'>
      tTest
    </h3>
    
    
  </div>
  

  <p>This is to compute <a href="https://en.wikipedia.org/wiki/Student%27s_t-test#One-sample_t-test">a one-sample t-test</a>, comparing the mean
of a sample to a known value, x.</p>
<p>in this case, we're trying to determine whether the
population mean is equal to the value that we know, which is <code>x</code>
here. usually the results here are used to look up a
<a href="http://en.wikipedia.org/wiki/P-value">p-value</a>, which, for
a certain level of significance, will let you determine that the
null hypothesis can or cannot be rejected.</p>


  <div class='pre p1 fill-light mt0'>tTest(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, expectedValue: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    sample of one or more numbers

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>expectedValue</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    expected value of the population mean

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        value

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>tTest([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>, <span class="hljs-number">5</span>, <span class="hljs-number">6</span>], <span class="hljs-number">3.385</span>).toFixed(<span class="hljs-number">2</span>); <span class="hljs-comment">// =&gt; '0.16'</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='ttesttwosample'>
      tTestTwoSample
    </h3>
    
    
  </div>
  

  <p>This is to compute <a href="http://en.wikipedia.org/wiki/Student&#x27;s_t-test">two sample t-test</a>.
Tests whether "mean(X)-mean(Y) = difference", (
in the most common case, we often have <code>difference == 0</code> to test if two samples
are likely to be taken from populations with the same mean value) with
no prior knowledge on standard deviations of both samples
other than the fact that they have the same standard deviation.</p>
<p>Usually the results here are used to look up a
<a href="http://en.wikipedia.org/wiki/P-value">p-value</a>, which, for
a certain level of significance, will let you determine that the
null hypothesis can or cannot be rejected.</p>
<p><code>diff</code> can be omitted if it equals 0.</p>
<p><a href="http://www.monarchlab.org/Lab/Research/Stats/2SampleT.aspx">This is used to confirm or deny</a>
a null hypothesis that the two populations that have been sampled into
<code>sampleX</code> and <code>sampleY</code> are equal to each other.</p>


  <div class='pre p1 fill-light mt0'>tTestTwoSample(sampleX: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, sampleY: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, difference: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): (<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a> | null)</div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>sampleX</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    a sample as an array of numbers

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>sampleY</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    a sample as an array of numbers

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>difference</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>
            = <code>0</code>)</code>
	    
          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a> | null)</code>:
        test result

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>tTestTwoSample([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>], [<span class="hljs-number">3</span>, <span class="hljs-number">4</span>, <span class="hljs-number">5</span>, <span class="hljs-number">6</span>], <span class="hljs-number">0</span>); <span class="hljs-comment">// =&gt; -2.1908902300206643</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='cumulativestdnormalprobability'>
      cumulativeStdNormalProbability
    </h3>
    
    
  </div>
  

  <p><strong><a href="http://en.wikipedia.org/wiki/Standard_normal_table">Cumulative Standard Normal Probability</a></strong></p>
<p>Since probability tables cannot be
printed for every normal distribution, as there are an infinite variety
of normal distributions, it is common practice to convert a normal to a
standard normal and then use the standard normal table to find probabilities.</p>
<p>You can use <code>.5 + .5 * errorFunction(x / Math.sqrt(2))</code> to calculate the probability
instead of looking it up in a table.</p>


  <div class='pre p1 fill-light mt0'>cumulativeStdNormalProbability(z: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>z</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    
          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        cumulative standard normal probability

      
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='kerneldensityestimation'>
      kernelDensityEstimation
    </h3>
    
    
  </div>
  

  <p><a href="https://en.wikipedia.org/wiki/Kernel_density_estimation">Kernel density estimation</a>
is a useful tool for, among other things, estimating the shape of the
underlying probability distribution from a sample.</p>


  <div class='pre p1 fill-light mt0'>kernelDensityEstimation</div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>X</span> <code class='quiet'>(any)</code>
	    sample values

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>kernel</span> <code class='quiet'>(any)</code>
	    The kernel function to use. If a function is provided, it should return non-negative values and integrate to 1. Defaults to 'gaussian'.

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>bandwidthMethod</span> <code class='quiet'>(any)</code>
	    The "bandwidth selection" method to use, or a fixed bandwidth value. Defaults to "nrd", the commonly-used 
<a href="https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/bandwidth.nrd.html">"normal reference distribution" rule-of-thumb</a>
.

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a></code>:
        An estimated 
<a href="https://en.wikipedia.org/wiki/Probability_density_function">probability density function</a>
 for the given sample. The returned function runs in 
<code>O(X.length)</code>
.

      
</section>

          
            <div class='keyline-top-not py2'><section class='py2 clearfix'>

  <h2 id='errors' class='mt0'>
    Errors
  </h2>

  
</section>
</div>
          
        
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='errorfunction'>
      errorFunction
    </h3>
    
    
  </div>
  

  <p><strong><a href="http://en.wikipedia.org/wiki/Error_function">Gaussian error function</a></strong></p>
<p>The <code>errorFunction(x/(sd * Math.sqrt(2)))</code> is the probability that a value in a
normal distribution with standard deviation sd is within x of the mean.</p>
<p>This function returns a numerical approximation to the exact value.</p>


  <div class='pre p1 fill-light mt0'>errorFunction(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    input

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        error estimation

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>errorFunction(<span class="hljs-number">1</span>).toFixed(<span class="hljs-number">2</span>); <span class="hljs-comment">// =&gt; '0.84'</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='inverseerrorfunction'>
      inverseErrorFunction
    </h3>
    
    
  </div>
  

  <p>The Inverse <a href="http://en.wikipedia.org/wiki/Error_function">Gaussian error function</a>
returns a numerical approximation to the value that would have caused
<code>errorFunction()</code> to return x.</p>


  <div class='pre p1 fill-light mt0'>inverseErrorFunction(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    value of error function

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        estimated inverted value

      
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='probit'>
      probit
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Probit">Probit</a>
is the inverse of cumulativeStdNormalProbability(),
and is also known as the normal quantile function.</p>
<p>It returns the number of standard deviations from the mean
where the p'th quantile of values can be found in a normal distribution.
So, for example, probit(0.5 + 0.6827/2) ≈ 1 because 68.27% of values are
normally found within 1 standard deviation above or below the mean.</p>


  <div class='pre p1 fill-light mt0'>probit(p: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>p</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    
          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        probit

      
</section>

          
            <div class='keyline-top-not py2'><section class='py2 clearfix'>

  <h2 id='breaks' class='mt0'>
    Breaks
  </h2>

  
    <p>Breaks methods split datasets into chunks. Often these are used
for segmentation or visualization of a dataset. A method of computing
breaks that splits data evenly can make for a better choropleth map,
for instance, because each color will be represented equally.</p>

  
</section>
</div>
          
        
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='ckmeans'>
      ckmeans
    </h3>
    
    
  </div>
  

  <p>Ckmeans clustering is an improvement on heuristic-based clustering
approaches like Jenks. The algorithm was developed in
<a href="http://journal.r-project.org/archive/2011-2/RJournal_2011-2_Wang+Song.pdf">Haizhou Wang and Mingzhou Song</a>
as a <a href="https://en.wikipedia.org/wiki/Dynamic_programming">dynamic programming</a> approach
to the problem of clustering numeric data into groups with the least
within-group sum-of-squared-deviations.</p>
<p>Minimizing the difference within groups - what Wang &#x26; Song refer to as
<code>withinss</code>, or within sum-of-squares, means that groups are optimally
homogenous within and the data is split into representative groups.
This is very useful for visualization, where you may want to represent
a continuous variable in discrete color or style groups. This function
can provide groups that emphasize differences between data.</p>
<p>Being a dynamic approach, this algorithm is based on two matrices that
store incrementally-computed values for squared deviations and backtracking
indexes.</p>
<p>This implementation is based on Ckmeans 3.4.6, which introduced a new divide
and conquer approach that improved runtime from O(kn^2) to O(kn log(n)).</p>
<p>Unlike the <a href="https://cran.r-project.org/web/packages/Ckmeans.1d.dp/index.html">original implementation</a>,
this implementation does not include any code to automatically determine
the optimal number of clusters: this information needs to be explicitly
provided.</p>
<h3>References</h3>
<p><em>Ckmeans.1d.dp: Optimal k-means Clustering in One Dimension by Dynamic
Programming</em> Haizhou Wang and Mingzhou Song ISSN 2073-4859</p>
<p>from The R Journal Vol. 3/2, December 2011</p>


  <div class='pre p1 fill-light mt0'>ckmeans(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, nClusters: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input data, as an array of number values

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>nClusters</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    number of desired classes. This cannot be
greater than the number of values in the data array.

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>></code>:
        clustered input

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if the number of requested clusters is higher than the size of the data
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>ckmeans([<span class="hljs-number">-1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">-1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">4</span>, <span class="hljs-number">5</span>, <span class="hljs-number">6</span>, <span class="hljs-number">-1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">-1</span>], <span class="hljs-number">3</span>);
<span class="hljs-comment">// The input, clustered into groups of similar numbers.</span>
<span class="hljs-comment">//= [[-1, -1, -1, -1], [2, 2, 2], [4, 5, 6]]);</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='equalintervalbreaks'>
      equalIntervalBreaks
    </h3>
    
    
  </div>
  

  <p>Given an array of x, this will find the extent of the
x and return an array of breaks that can be used
to categorize the x into a number of classes. The
returned array will always be 1 longer than the number of
classes because it includes the minimum value.</p>


  <div class='pre p1 fill-light mt0'>equalIntervalBreaks(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, nClasses: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    an array of number values

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>nClasses</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    number of desired classes

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>></code>:
        array of class break positions

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>equalIntervalBreaks([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>, <span class="hljs-number">5</span>, <span class="hljs-number">6</span>], <span class="hljs-number">4</span>); <span class="hljs-comment">// =&gt; [1, 2.25, 3.5, 4.75, 6]</span></pre>
    
  
</section>

          
            <div class='keyline-top-not py2'><section class='py2 clearfix'>

  <h2 id='utilities' class='mt0'>
    Utilities
  </h2>

  
</section>
</div>
          
        
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='chunk'>
      chunk
    </h3>
    
    
  </div>
  

  <p>Split an array into chunks of a specified size. This function
has the same behavior as <a href="http://php.net/manual/en/function.array-chunk.php">PHP's array_chunk</a>
function, and thus will insert smaller-sized chunks at the end if
the input size is not divisible by the chunk size.</p>
<p><code>x</code> is expected to be an array, and <code>chunkSize</code> a number.
The <code>x</code> array can contain any kind of data.</p>


  <div class='pre p1 fill-light mt0'>chunk(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>, chunkSize: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>)</code>
	    a sample

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>chunkSize</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    size of each output array. must be a positive integer

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>></code>:
        a chunked array

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if chunk size is less than 1 or not an integer
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>chunk([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>, <span class="hljs-number">5</span>, <span class="hljs-number">6</span>], <span class="hljs-number">2</span>);
<span class="hljs-comment">// =&gt; [[1, 2], [3, 4], [5, 6]]</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='chisquaredgoodnessoffit'>
      chiSquaredGoodnessOfFit
    </h3>
    
    
  </div>
  

  <p>The <a href="http://en.wikipedia.org/wiki/Goodness_of_fit#Pearson.27s_chi-squared_test">χ2 (Chi-Squared) Goodness-of-Fit Test</a>
uses a measure of goodness of fit which is the sum of differences between observed and expected outcome frequencies
(that is, counts of observations), each squared and divided by the number of observations expected given the
hypothesized distribution. The resulting χ2 statistic, <code>chiSquared</code>, can be compared to the chi-squared distribution
to determine the goodness of fit. In order to determine the degrees of freedom of the chi-squared distribution, one
takes the total number of observed frequencies and subtracts the number of estimated parameters. The test statistic
follows, approximately, a chi-square distribution with (k − c) degrees of freedom where <code>k</code> is the number of non-empty
cells and <code>c</code> is the number of estimated parameters for the distribution.</p>


  <div class='pre p1 fill-light mt0'>chiSquaredGoodnessOfFit(data: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, distributionType: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a>, significance: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>data</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    
          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>distributionType</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a>)</code>
	    a function that returns a point in a distribution:
for instance, binomial, bernoulli, or poisson

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>significance</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    
          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        chi squared goodness of fit

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'><span class="hljs-comment">// Data from Poisson goodness-of-fit example 10-19 in William W. Hines &amp; Douglas C. Montgomery,</span>
<span class="hljs-comment">// "Probability and Statistics in Engineering and Management Science", Wiley (1980).</span>
<span class="hljs-keyword">var</span> data1019 = [
    <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>,
    <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span>,
    <span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">1</span>,
    <span class="hljs-number">2</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>,
    <span class="hljs-number">3</span>, <span class="hljs-number">3</span>, <span class="hljs-number">3</span>, <span class="hljs-number">3</span>
];
ss.chiSquaredGoodnessOfFit(data1019, ss.poissonDistribution, <span class="hljs-number">0.05</span>); <span class="hljs-comment">//= false</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='epsilon'>
      epsilon
    </h3>
    
    
  </div>
  

  <p>We use <code>ε</code>, epsilon, as a stopping criterion when we want to iterate
until we're "close enough". Epsilon is a very small number: for
simple statistics, that number is <strong>0.0001</strong></p>
<p>This is used in calculations like the binomialDistribution, in which
the process of finding a value is <a href="https://en.wikipedia.org/wiki/Iterative_method">iterative</a>:
it progresses until it is close enough.</p>
<p>Below is an example of using epsilon in <a href="https://en.wikipedia.org/wiki/Gradient_descent">gradient descent</a>,
where we're trying to find a local minimum of a function's derivative,
given by the <code>fDerivative</code> method.</p>


  <div class='pre p1 fill-light mt0'>epsilon</div>
  
  
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'><span class="hljs-comment">// From calculation, we expect that the local minimum occurs at x=9/4</span>
<span class="hljs-keyword">var</span> x_old = <span class="hljs-number">0</span>;
<span class="hljs-comment">// The algorithm starts at x=6</span>
<span class="hljs-keyword">var</span> x_new = <span class="hljs-number">6</span>;
<span class="hljs-keyword">var</span> stepSize = <span class="hljs-number">0.01</span>;

<span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">fDerivative</span>(<span class="hljs-params">x</span>) </span>{
  <span class="hljs-keyword">return</span> <span class="hljs-number">4</span> * <span class="hljs-built_in">Math</span>.pow(x, <span class="hljs-number">3</span>) - <span class="hljs-number">9</span> * <span class="hljs-built_in">Math</span>.pow(x, <span class="hljs-number">2</span>);
}

<span class="hljs-comment">// The loop runs until the difference between the previous</span>
<span class="hljs-comment">// value and the current value is smaller than epsilon - a rough</span>
<span class="hljs-comment">// meaure of 'close enough'</span>
<span class="hljs-keyword">while</span> (<span class="hljs-built_in">Math</span>.abs(x_new - x_old) &gt; ss.epsilon) {
  x_old = x_new;
  x_new = x_old - stepSize * fDerivative(x_old);
}

<span class="hljs-built_in">console</span>.log(<span class="hljs-string">'Local minimum occurs at'</span>, x_new);</pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='factorial'>
      factorial
    </h3>
    
    
  </div>
  

  <p>A <a href="https://en.wikipedia.org/wiki/Factorial">Factorial</a>, usually written n!, is the product of all positive
integers less than or equal to n. Often factorial is implemented
recursively, but this iterative approach is significantly faster
and simpler.</p>


  <div class='pre p1 fill-light mt0'>factorial(n: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>n</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    input, must be an integer number 1 or greater

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        factorial: n!

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if n is less than 0 or not an integer
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>factorial(<span class="hljs-number">5</span>); <span class="hljs-comment">// =&gt; 120</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='gamma'>
      gamma
    </h3>
    
    
  </div>
  

  <p>Compute the <a href="https://en.wikipedia.org/wiki/Gamma_function">gamma function</a> of a value using Nemes' approximation.
The gamma of n is equivalent to (n-1)!, but unlike the factorial function, gamma is defined for all real n except zero
and negative integers (where NaN is returned). Note, the gamma function is also well-defined for complex numbers,
though this implementation currently does not handle complex numbers as input values.
Nemes' approximation is defined <a href="https://arxiv.org/abs/1003.6020">here</a> as Theorem 2.2.
Negative values use <a href="https://en.wikipedia.org/wiki/Gamma_function#Properties">Euler's reflection formula</a> for computation.</p>


  <div class='pre p1 fill-light mt0'>gamma(n: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>n</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    Any real number except for zero and negative integers.

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        The gamma of the input value.

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>gamma(<span class="hljs-number">11.5</span>); <span class="hljs-comment">// 11899423.084037038</span>
gamma(<span class="hljs-number">-11.5</span>); <span class="hljs-comment">// 2.29575810481609e-8 </span>
gamma(<span class="hljs-number">5</span>); <span class="hljs-comment">// 24</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='uniquecountsorted'>
      uniqueCountSorted
    </h3>
    
    
  </div>
  

  <p>For a sorted input, counting the number of unique values
is possible in constant time and constant memory. This is
a simple implementation of the algorithm.</p>
<p>Values are compared with <code>===</code>, so objects and non-primitive objects
are not handled in any special way.</p>


  <div class='pre p1 fill-light mt0'>uniqueCountSorted(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;any>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;any>)</code>
	    an array of any kind of value

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        count of unique values

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>uniqueCountSorted([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>]); <span class="hljs-comment">// =&gt; 3</span>
uniqueCountSorted([<span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">1</span>]); <span class="hljs-comment">// =&gt; 1</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='bisect'>
      bisect
    </h3>
    
    
  </div>
  

  <p><a href="https://en.wikipedia.org/wiki/Bisection_method">Bisection method</a> is a root-finding
method that repeatedly bisects an interval to find the root.</p>
<p>This function returns a numerical approximation to the exact value.</p>


  <div class='pre p1 fill-light mt0'>bisect(func: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a>, start: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, end: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, maxIterations: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, errorTolerance: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>func</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Statements/function">Function</a>)</code>
	    input function

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>start</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    start of interval

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>end</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    end of interval

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>maxIterations</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    the maximum number of iterations

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>errorTolerance</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    the error tolerance

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        estimated root value

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/TypeError">TypeError</a>: Argument func must be a function
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>bisect(<span class="hljs-built_in">Math</span>.cos,<span class="hljs-number">0</span>,<span class="hljs-number">4</span>,<span class="hljs-number">100</span>,<span class="hljs-number">0.003</span>); <span class="hljs-comment">// =&gt; 1.572265625</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='combinationsreplacement'>
      combinationsReplacement
    </h3>
    
    
  </div>
  

  <p>Implementation of <a href="https://en.wikipedia.org/wiki/Combination">Combinations</a> with replacement
Combinations are unique subsets of a collection - in this case, k x from a collection at a time.
'With replacement' means that a given element can be chosen multiple times.
Unlike permutation, order doesn't matter for combinations.</p>


  <div class='pre p1 fill-light mt0'>combinationsReplacement(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>, k: int): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>)</code>
	    any type of data

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>k</span> <code class='quiet'>(int)</code>
	    the number of objects in each group (without replacement)

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>></code>:
        array of permutations

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>combinationsReplacement([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>], <span class="hljs-number">2</span>); <span class="hljs-comment">// =&gt; [[1, 1], [1, 2], [2, 2]]</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='combinations'>
      combinations
    </h3>
    
    
  </div>
  

  <p>Implementation of Combinations
Combinations are unique subsets of a collection - in this case, k x from a collection at a time.
<a href="https://en.wikipedia.org/wiki/Combination">https://en.wikipedia.org/wiki/Combination</a></p>


  <div class='pre p1 fill-light mt0'>combinations(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>, k: int): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>)</code>
	    any type of data

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>k</span> <code class='quiet'>(int)</code>
	    the number of objects in each group (without replacement)

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>></code>:
        array of permutations

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>combinations([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>], <span class="hljs-number">2</span>); <span class="hljs-comment">// =&gt; [[1,2], [1,3], [2,3]]</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='combinemeans'>
      combineMeans
    </h3>
    
    
  </div>
  

  <p>When combining two lists of values for which one already knows the means,
one does not have to necessary recompute the mean of the combined lists in
linear time. They can instead use this function to compute the combined
mean by providing the mean &#x26; number of values of the first list and the mean
&#x26; number of values of the second list.</p>


  <div class='pre p1 fill-light mt0'>combineMeans(mean1: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, n1: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, mean2: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, n2: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
  <div>Since: 3.0.0</div>

  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>mean1</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    mean of the first list

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>n1</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    number of items in the first list

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>mean2</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    mean of the second list

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>n2</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    number of items in the second list

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        the combined mean

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>combineMeans(<span class="hljs-number">5</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>, <span class="hljs-number">3</span>); <span class="hljs-comment">// =&gt; 4.5</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='combinevariances'>
      combineVariances
    </h3>
    
    
  </div>
  

  <p>When combining two lists of values for which one already knows the variances,
one does not have to necessary recompute the variance of the combined lists
in linear time. They can instead use this function to compute the combined
variance by providing the variance, mean &#x26; number of values of the first list
and the variance, mean &#x26; number of values of the second list.</p>


  <div class='pre p1 fill-light mt0'>combineVariances(variance1: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, mean1: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, n1: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, variance2: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, mean2: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, n2: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
  <div>Since: 3.0.0</div>

  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>variance1</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    variance of the first list

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>mean1</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    mean of the first list

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>n1</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    number of items in the first list

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>variance2</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    variance of the second list

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>mean2</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    mean of the second list

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>n2</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    number of items in the second list

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        the combined mean

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>combineVariances(<span class="hljs-number">14</span> / <span class="hljs-number">3</span>, <span class="hljs-number">5</span>, <span class="hljs-number">3</span>, <span class="hljs-number">8</span> / <span class="hljs-number">3</span>, <span class="hljs-number">4</span>, <span class="hljs-number">3</span>); <span class="hljs-comment">// =&gt; 47 / 12</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='extentsorted'>
      extentSorted
    </h3>
    
    
  </div>
  

  <p>The extent is the lowest &#x26; highest number in the array. With a sorted array,
the first element in the array is always the lowest while the last element is always the largest, so this calculation
can be done in one step, or constant time.</p>


  <div class='pre p1 fill-light mt0'>extentSorted(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>></code>:
        minimum &#x26; maximum value

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>extentSorted([<span class="hljs-number">-100</span>, <span class="hljs-number">-10</span>, <span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">5</span>]); <span class="hljs-comment">// =&gt; [-100, 5]</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='extent'>
      extent
    </h3>
    
    
  </div>
  

  <p>This computes the minimum &#x26; maximum number in an array.</p>
<p>This runs on <code>O(n)</code>, linear time in respect to the array</p>


  <div class='pre p1 fill-light mt0'>extent(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    sample of one or more data points

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>></code>:
        minimum &#x26; maximum value

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if the the length of x is less than one
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>extent([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>]);
<span class="hljs-comment">// =&gt; [1, 4]</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='permutationtest'>
      permutationTest
    </h3>
    
    
  </div>
  

  <p>Conducts a <a href="https://en.wikipedia.org/wiki/Resampling_(statistics)#Permutation_tests">permutation test</a>
to determine if two data sets are <em>significantly</em> different from each other, using
the difference of means between the groups as the test statistic.
The function allows for the following hypotheses:</p>
<ul>
<li>two_tail = Null hypothesis: the two distributions are equal.</li>
<li>greater = Null hypothesis: observations from sampleX tend to be smaller than those from sampleY.</li>
<li>less = Null hypothesis: observations from sampleX tend to be greater than those from sampleY.
<a href="https://en.wikipedia.org/wiki/One-_and_two-tailed_tests">Learn more about one-tail vs two-tail tests.</a></li>
</ul>


  <div class='pre p1 fill-light mt0'>permutationTest(sampleX: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, sampleY: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, alternative: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/String">string</a>, k: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>sampleX</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    first dataset (e.g. treatment data)

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>sampleY</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    second dataset (e.g. control data)

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>alternative</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/String">string</a>)</code>
	    alternative hypothesis, either 'two_sided' (default), 'greater', or 'less'

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>k</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    number of values in permutation distribution.

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        p-value The probability of observing the difference between groups (as or more extreme than what we did), assuming the null hypothesis.

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'><span class="hljs-keyword">var</span> control = [<span class="hljs-number">2</span>, <span class="hljs-number">5</span>, <span class="hljs-number">3</span>, <span class="hljs-number">6</span>, <span class="hljs-number">7</span>, <span class="hljs-number">2</span>, <span class="hljs-number">5</span>];
<span class="hljs-keyword">var</span> treatment = [<span class="hljs-number">20</span>, <span class="hljs-number">5</span>, <span class="hljs-number">13</span>, <span class="hljs-number">12</span>, <span class="hljs-number">7</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>];
permutationTest(control, treatment); <span class="hljs-comment">// ~0.1324</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='permutationsheap'>
      permutationsHeap
    </h3>
    
    
  </div>
  

  <p>Implementation of <a href="https://en.wikipedia.org/wiki/Heap%27s_algorithm">Heap's Algorithm</a>
for generating permutations.</p>


  <div class='pre p1 fill-light mt0'>permutationsHeap(elements: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>elements</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>)</code>
	    any type of data

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>></code>:
        array of permutations

      
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='quantileranksorted'>
      quantileRankSorted
    </h3>
    
    
  </div>
  

  <p>This function returns the quantile in which one would find the given value in
the given array. With a sorted array, leveraging binary search, we can find
this information in logarithmic time.</p>


  <div class='pre p1 fill-light mt0'>quantileRankSorted(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, value: any): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>value</span> <code class='quiet'>(any)</code>
	    
          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        value value

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>quantileRankSorted([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>], <span class="hljs-number">3</span>); <span class="hljs-comment">// =&gt; 0.75</span>
quantileRankSorted([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>], <span class="hljs-number">3</span>); <span class="hljs-comment">// =&gt; 0.7</span>
quantileRankSorted([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>], <span class="hljs-number">6</span>); <span class="hljs-comment">// =&gt; 1</span>
quantileRankSorted([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">3</span>, <span class="hljs-number">5</span>], <span class="hljs-number">4</span>); <span class="hljs-comment">// =&gt; 0.8</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='quantilerank'>
      quantileRank
    </h3>
    
    
  </div>
  

  <p>This function returns the quantile in which one would find the given value in
the given array. It will require to copy and sort your array beforehand, so
if you know your array is already sorted, you would rather use
<code>quantileRankSorted</code>.</p>


  <div class='pre p1 fill-light mt0'>quantileRank(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, value: any): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>value</span> <code class='quiet'>(any)</code>
	    
          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        value value

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>quantileRank([<span class="hljs-number">4</span>, <span class="hljs-number">3</span>, <span class="hljs-number">1</span>, <span class="hljs-number">2</span>], <span class="hljs-number">3</span>); <span class="hljs-comment">// =&gt; 0.75</span>
quantileRank([<span class="hljs-number">4</span>, <span class="hljs-number">3</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">1</span>], <span class="hljs-number">3</span>); <span class="hljs-comment">// =&gt; 0.7</span>
quantileRank([<span class="hljs-number">2</span>, <span class="hljs-number">4</span>, <span class="hljs-number">1</span>, <span class="hljs-number">3</span>], <span class="hljs-number">6</span>); <span class="hljs-comment">// =&gt; 1</span>
quantileRank([<span class="hljs-number">5</span>, <span class="hljs-number">3</span>, <span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>], <span class="hljs-number">4</span>); <span class="hljs-comment">// =&gt; 0.8</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='quickselect'>
      quickselect
    </h3>
    
    
  </div>
  

  <p>Rearrange items in <code>arr</code> so that all items in <code>[left, k]</code> range are the smallest.
The <code>k</code>-th element will have the <code>(k - left + 1)</code>-th smallest value in <code>[left, right]</code>.</p>
<p>Implements Floyd-Rivest selection algorithm <a href="https://en.wikipedia.org/wiki/Floyd-Rivest_algorithm">https://en.wikipedia.org/wiki/Floyd-Rivest_algorithm</a></p>


  <div class='pre p1 fill-light mt0'>quickselect(arr: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>, k: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, left: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>?, right: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>?): void</div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>arr</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    input array

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>k</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    pivot index

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>left</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>?)</code>
	    left index

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>right</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>?)</code>
	    right index

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code>void</code>:
        mutates input array

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'><span class="hljs-keyword">var</span> arr = [<span class="hljs-number">65</span>, <span class="hljs-number">28</span>, <span class="hljs-number">59</span>, <span class="hljs-number">33</span>, <span class="hljs-number">21</span>, <span class="hljs-number">56</span>, <span class="hljs-number">22</span>, <span class="hljs-number">95</span>, <span class="hljs-number">50</span>, <span class="hljs-number">12</span>, <span class="hljs-number">90</span>, <span class="hljs-number">53</span>, <span class="hljs-number">28</span>, <span class="hljs-number">77</span>, <span class="hljs-number">39</span>];
quickselect(arr, <span class="hljs-number">8</span>);
<span class="hljs-comment">// = [39, 28, 28, 33, 21, 12, 22, 50, 53, 56, 59, 65, 90, 77, 95]</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='samplekurtosis'>
      sampleKurtosis
    </h3>
    
    
  </div>
  

  <p><a href="http://en.wikipedia.org/wiki/Kurtosis">Kurtosis</a> is
a measure of the heaviness of a distribution's tails relative to its
variance. The kurtosis value can be positive or negative, or even undefined.</p>
<p>Implementation is based on Fisher's excess kurtosis definition and uses
unbiased moment estimators. This is the version found in Excel and available
in several statistical packages, including SAS and SciPy.</p>


  <div class='pre p1 fill-light mt0'>sampleKurtosis(x: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>x</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Array">Array</a>&#x3C;<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>>)</code>
	    a sample of 4 or more data points

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        sample kurtosis

      
    <div class='py1 quiet mt1 prose-big'>Throws</div>
    <ul>
      
        <li><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Error">Error</a>: if x has length less than 4
</li>
      
    </ul>
  

    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>sampleKurtosis([<span class="hljs-number">1</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">5</span>]); <span class="hljs-comment">// =&gt; 1.4555765595463122</span></pre>
    
  
</section>

          
            <section class='p2 mb2 clearfix bg-white minishadow'>

  
  <div class='clearfix'>
    
    <h3 class='fl m0' id='subtractfrommean'>
      subtractFromMean
    </h3>
    
    
  </div>
  

  <p>When removing a value from a list, one does not have to necessary
recompute the mean of the list in linear time. They can instead use
this function to compute the new mean by providing the current mean,
the number of elements in the list that produced it and the value to remove.</p>


  <div class='pre p1 fill-light mt0'>subtractFromMean(mean: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, n: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>, value: <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>): <a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></div>
  
  
  <div>Since: 3.0.0</div>

  
    <div class='py1 quiet mt1 prose-big'>Parameters</div>
    <div class='prose'>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>mean</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    current mean

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>n</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    number of items in the list

          </div>
          
        </div>
      
        <div class='space-bottom0'>
          <div>
            <span class='code bold'>value</span> <code class='quiet'>(<a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a>)</code>
	    the value to remove

          </div>
          
        </div>
      
    </div>
  

      <div class='py1 quiet mt1 prose-big'>Returns</div>
      <code><a href="https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Number">number</a></code>:
        the new mean

      
    <div class='py1 quiet mt1 prose-big'>Example</div>
    
      
      <pre class='p1 overflow-auto round fill-light'>subtractFromMean(<span class="hljs-number">20.5</span>, <span class="hljs-number">6</span>, <span class="hljs-number">53</span>); <span class="hljs-comment">// =&gt; 14</span></pre>
    
  
</section>

          
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