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<div class="section" id="module-statistics">
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<span id="statistics-mathematical-statistics-functions"></span><h1><a class="reference internal" href="#module-statistics" title="statistics: mathematical statistics functions"><code class="xref py py-mod docutils literal notranslate"><span class="pre">statistics</span></code></a> — Mathematical statistics functions<a class="headerlink" href="#module-statistics" title="Permalink to this headline">¶</a></h1>
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<div class="versionadded">
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<p><span class="versionmodified added">New in version 3.4.</span></p>
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</div>
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<p><strong>Source code:</strong> <a class="reference external" href="https://github.com/python/cpython/tree/3.7/Lib/statistics.py">Lib/statistics.py</a></p>
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<hr class="docutils" />
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<p>This module provides functions for calculating mathematical statistics of
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numeric (<code class="xref py py-class docutils literal notranslate"><span class="pre">Real</span></code>-valued) data.</p>
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<div class="admonition note">
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<p class="admonition-title">Note</p>
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<p>Unless explicitly noted otherwise, these functions support <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a>,
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<a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a>, <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate"><span class="pre">decimal.Decimal</span></code></a> and <a class="reference internal" href="fractions.html#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">fractions.Fraction</span></code></a>.
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Behaviour with other types (whether in the numeric tower or not) is
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currently unsupported. Mixed types are also undefined and
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implementation-dependent. If your input data consists of mixed types,
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you may be able to use <a class="reference internal" href="functions.html#map" title="map"><code class="xref py py-func docutils literal notranslate"><span class="pre">map()</span></code></a> to ensure a consistent result, e.g.
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<code class="docutils literal notranslate"><span class="pre">map(float,</span> <span class="pre">input_data)</span></code>.</p>
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</div>
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<div class="section" id="averages-and-measures-of-central-location">
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<h2>Averages and measures of central location<a class="headerlink" href="#averages-and-measures-of-central-location" title="Permalink to this headline">¶</a></h2>
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<p>These functions calculate an average or typical value from a population
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or sample.</p>
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<table class="docutils align-center">
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<colgroup>
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<col style="width: 34%" />
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<col style="width: 66%" />
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</colgroup>
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<tbody>
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<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.mean" title="statistics.mean"><code class="xref py py-func docutils literal notranslate"><span class="pre">mean()</span></code></a></p></td>
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<td><p>Arithmetic mean (“average”) of data.</p></td>
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</tr>
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<tr class="row-even"><td><p><a class="reference internal" href="#statistics.harmonic_mean" title="statistics.harmonic_mean"><code class="xref py py-func docutils literal notranslate"><span class="pre">harmonic_mean()</span></code></a></p></td>
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<td><p>Harmonic mean of data.</p></td>
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</tr>
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<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.median" title="statistics.median"><code class="xref py py-func docutils literal notranslate"><span class="pre">median()</span></code></a></p></td>
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<td><p>Median (middle value) of data.</p></td>
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</tr>
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<tr class="row-even"><td><p><a class="reference internal" href="#statistics.median_low" title="statistics.median_low"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_low()</span></code></a></p></td>
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<td><p>Low median of data.</p></td>
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</tr>
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<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.median_high" title="statistics.median_high"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_high()</span></code></a></p></td>
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<td><p>High median of data.</p></td>
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</tr>
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<tr class="row-even"><td><p><a class="reference internal" href="#statistics.median_grouped" title="statistics.median_grouped"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_grouped()</span></code></a></p></td>
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<td><p>Median, or 50th percentile, of grouped data.</p></td>
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</tr>
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<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.mode" title="statistics.mode"><code class="xref py py-func docutils literal notranslate"><span class="pre">mode()</span></code></a></p></td>
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<td><p>Mode (most common value) of discrete data.</p></td>
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</tr>
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</tbody>
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</table>
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</div>
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<div class="section" id="measures-of-spread">
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<h2>Measures of spread<a class="headerlink" href="#measures-of-spread" title="Permalink to this headline">¶</a></h2>
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<p>These functions calculate a measure of how much the population or sample
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tends to deviate from the typical or average values.</p>
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<table class="docutils align-center">
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<colgroup>
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<col style="width: 34%" />
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<col style="width: 66%" />
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</colgroup>
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<tbody>
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<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.pstdev" title="statistics.pstdev"><code class="xref py py-func docutils literal notranslate"><span class="pre">pstdev()</span></code></a></p></td>
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<td><p>Population standard deviation of data.</p></td>
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</tr>
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<tr class="row-even"><td><p><a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a></p></td>
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<td><p>Population variance of data.</p></td>
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</tr>
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<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.stdev" title="statistics.stdev"><code class="xref py py-func docutils literal notranslate"><span class="pre">stdev()</span></code></a></p></td>
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<td><p>Sample standard deviation of data.</p></td>
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</tr>
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<tr class="row-even"><td><p><a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate"><span class="pre">variance()</span></code></a></p></td>
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<td><p>Sample variance of data.</p></td>
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</tr>
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</tbody>
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</table>
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</div>
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<div class="section" id="function-details">
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<h2>Function details<a class="headerlink" href="#function-details" title="Permalink to this headline">¶</a></h2>
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<p>Note: The functions do not require the data given to them to be sorted.
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However, for reading convenience, most of the examples show sorted sequences.</p>
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<dl class="function">
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<dt id="statistics.mean">
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<code class="descclassname">statistics.</code><code class="descname">mean</code><span class="sig-paren">(</span><em>data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.mean" title="Permalink to this definition">¶</a></dt>
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<dd><p>Return the sample arithmetic mean of <em>data</em> which can be a sequence or iterator.</p>
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<p>The arithmetic mean is the sum of the data divided by the number of data
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points. It is commonly called “the average”, although it is only one of many
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different mathematical averages. It is a measure of the central location of
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the data.</p>
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<p>If <em>data</em> is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> will be raised.</p>
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<p>Some examples of use:</p>
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<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">mean</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
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<span class="go">2.8</span>
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<span class="gp">>>> </span><span class="n">mean</span><span class="p">([</span><span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">3.25</span><span class="p">,</span> <span class="mf">5.75</span><span class="p">])</span>
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<span class="go">2.625</span>
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<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="k">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>
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<span class="gp">>>> </span><span class="n">mean</span><span class="p">([</span><span class="n">F</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">21</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)])</span>
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<span class="go">Fraction(13, 21)</span>
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<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="k">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>
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<span class="gp">>>> </span><span class="n">mean</span><span class="p">([</span><span class="n">D</span><span class="p">(</span><span class="s2">"0.5"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"0.75"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"0.625"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"0.375"</span><span class="p">)])</span>
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<span class="go">Decimal('0.5625')</span>
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</pre></div>
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</div>
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<div class="admonition note">
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<p class="admonition-title">Note</p>
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<p>The mean is strongly affected by outliers and is not a robust estimator
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for central location: the mean is not necessarily a typical example of the
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data points. For more robust, although less efficient, measures of
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central location, see <a class="reference internal" href="#statistics.median" title="statistics.median"><code class="xref py py-func docutils literal notranslate"><span class="pre">median()</span></code></a> and <a class="reference internal" href="#statistics.mode" title="statistics.mode"><code class="xref py py-func docutils literal notranslate"><span class="pre">mode()</span></code></a>. (In this case,
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“efficient” refers to statistical efficiency rather than computational
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efficiency.)</p>
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<p>The sample mean gives an unbiased estimate of the true population mean,
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which means that, taken on average over all the possible samples,
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<code class="docutils literal notranslate"><span class="pre">mean(sample)</span></code> converges on the true mean of the entire population. If
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||
<em>data</em> represents the entire population rather than a sample, then
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<code class="docutils literal notranslate"><span class="pre">mean(data)</span></code> is equivalent to calculating the true population mean μ.</p>
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</div>
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</dd></dl>
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<dl class="function">
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<dt id="statistics.harmonic_mean">
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<code class="descclassname">statistics.</code><code class="descname">harmonic_mean</code><span class="sig-paren">(</span><em>data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.harmonic_mean" title="Permalink to this definition">¶</a></dt>
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<dd><p>Return the harmonic mean of <em>data</em>, a sequence or iterator of
|
||
real-valued numbers.</p>
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<p>The harmonic mean, sometimes called the subcontrary mean, is the
|
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reciprocal of the arithmetic <a class="reference internal" href="#statistics.mean" title="statistics.mean"><code class="xref py py-func docutils literal notranslate"><span class="pre">mean()</span></code></a> of the reciprocals of the
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||
data. For example, the harmonic mean of three values <em>a</em>, <em>b</em> and <em>c</em>
|
||
will be equivalent to <code class="docutils literal notranslate"><span class="pre">3/(1/a</span> <span class="pre">+</span> <span class="pre">1/b</span> <span class="pre">+</span> <span class="pre">1/c)</span></code>.</p>
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<p>The harmonic mean is a type of average, a measure of the central
|
||
location of the data. It is often appropriate when averaging quantities
|
||
which are rates or ratios, for example speeds. For example:</p>
|
||
<p>Suppose an investor purchases an equal value of shares in each of
|
||
three companies, with P/E (price/earning) ratios of 2.5, 3 and 10.
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||
What is the average P/E ratio for the investor’s portfolio?</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">harmonic_mean</span><span class="p">([</span><span class="mf">2.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span> <span class="c1"># For an equal investment portfolio.</span>
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||
<span class="go">3.6</span>
|
||
</pre></div>
|
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</div>
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||
<p>Using the arithmetic mean would give an average of about 5.167, which
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is too high.</p>
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<p><a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> is raised if <em>data</em> is empty, or any element
|
||
is less than zero.</p>
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||
<div class="versionadded">
|
||
<p><span class="versionmodified added">New in version 3.6.</span></p>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.median">
|
||
<code class="descclassname">statistics.</code><code class="descname">median</code><span class="sig-paren">(</span><em>data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Return the median (middle value) of numeric data, using the common “mean of
|
||
middle two” method. If <em>data</em> is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> is raised.
|
||
<em>data</em> can be a sequence or iterator.</p>
|
||
<p>The median is a robust measure of central location, and is less affected by
|
||
the presence of outliers in your data. When the number of data points is
|
||
odd, the middle data point is returned:</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">median</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
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||
<span class="go">3</span>
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||
</pre></div>
|
||
</div>
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||
<p>When the number of data points is even, the median is interpolated by taking
|
||
the average of the two middle values:</p>
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||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">median</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
|
||
<span class="go">4.0</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>This is suited for when your data is discrete, and you don’t mind that the
|
||
median may not be an actual data point.</p>
|
||
<p>If your data is ordinal (supports order operations) but not numeric (doesn’t
|
||
support addition), you should use <a class="reference internal" href="#statistics.median_low" title="statistics.median_low"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_low()</span></code></a> or <a class="reference internal" href="#statistics.median_high" title="statistics.median_high"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_high()</span></code></a>
|
||
instead.</p>
|
||
<div class="admonition seealso">
|
||
<p class="admonition-title">See also</p>
|
||
<p><a class="reference internal" href="#statistics.median_low" title="statistics.median_low"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_low()</span></code></a>, <a class="reference internal" href="#statistics.median_high" title="statistics.median_high"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_high()</span></code></a>, <a class="reference internal" href="#statistics.median_grouped" title="statistics.median_grouped"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_grouped()</span></code></a></p>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.median_low">
|
||
<code class="descclassname">statistics.</code><code class="descname">median_low</code><span class="sig-paren">(</span><em>data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median_low" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Return the low median of numeric data. If <em>data</em> is empty,
|
||
<a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> is raised. <em>data</em> can be a sequence or iterator.</p>
|
||
<p>The low median is always a member of the data set. When the number of data
|
||
points is odd, the middle value is returned. When it is even, the smaller of
|
||
the two middle values is returned.</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">median_low</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
|
||
<span class="go">3</span>
|
||
<span class="gp">>>> </span><span class="n">median_low</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
|
||
<span class="go">3</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>Use the low median when your data are discrete and you prefer the median to
|
||
be an actual data point rather than interpolated.</p>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.median_high">
|
||
<code class="descclassname">statistics.</code><code class="descname">median_high</code><span class="sig-paren">(</span><em>data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median_high" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Return the high median of data. If <em>data</em> is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>
|
||
is raised. <em>data</em> can be a sequence or iterator.</p>
|
||
<p>The high median is always a member of the data set. When the number of data
|
||
points is odd, the middle value is returned. When it is even, the larger of
|
||
the two middle values is returned.</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">median_high</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
|
||
<span class="go">3</span>
|
||
<span class="gp">>>> </span><span class="n">median_high</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
|
||
<span class="go">5</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>Use the high median when your data are discrete and you prefer the median to
|
||
be an actual data point rather than interpolated.</p>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.median_grouped">
|
||
<code class="descclassname">statistics.</code><code class="descname">median_grouped</code><span class="sig-paren">(</span><em>data</em>, <em>interval=1</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median_grouped" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Return the median of grouped continuous data, calculated as the 50th
|
||
percentile, using interpolation. If <em>data</em> is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>
|
||
is raised. <em>data</em> can be a sequence or iterator.</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">median_grouped</span><span class="p">([</span><span class="mi">52</span><span class="p">,</span> <span class="mi">52</span><span class="p">,</span> <span class="mi">53</span><span class="p">,</span> <span class="mi">54</span><span class="p">])</span>
|
||
<span class="go">52.5</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>In the following example, the data are rounded, so that each value represents
|
||
the midpoint of data classes, e.g. 1 is the midpoint of the class 0.5–1.5, 2
|
||
is the midpoint of 1.5–2.5, 3 is the midpoint of 2.5–3.5, etc. With the data
|
||
given, the middle value falls somewhere in the class 3.5–4.5, and
|
||
interpolation is used to estimate it:</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">median_grouped</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
|
||
<span class="go">3.7</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>Optional argument <em>interval</em> represents the class interval, and defaults
|
||
to 1. Changing the class interval naturally will change the interpolation:</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">median_grouped</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="n">interval</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="go">3.25</span>
|
||
<span class="gp">>>> </span><span class="n">median_grouped</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="n">interval</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="go">3.5</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>This function does not check whether the data points are at least
|
||
<em>interval</em> apart.</p>
|
||
<div class="impl-detail compound">
|
||
<p><strong>CPython implementation detail:</strong> Under some circumstances, <a class="reference internal" href="#statistics.median_grouped" title="statistics.median_grouped"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_grouped()</span></code></a> may coerce data points to
|
||
floats. This behaviour is likely to change in the future.</p>
|
||
</div>
|
||
<div class="admonition seealso">
|
||
<p class="admonition-title">See also</p>
|
||
<ul class="simple">
|
||
<li><p>“Statistics for the Behavioral Sciences”, Frederick J Gravetter and
|
||
Larry B Wallnau (8th Edition).</p></li>
|
||
<li><p>The <a class="reference external" href="https://help.gnome.org/users/gnumeric/stable/gnumeric.html#gnumeric-function-SSMEDIAN">SSMEDIAN</a>
|
||
function in the Gnome Gnumeric spreadsheet, including <a class="reference external" href="https://mail.gnome.org/archives/gnumeric-list/2011-April/msg00018.html">this discussion</a>.</p></li>
|
||
</ul>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.mode">
|
||
<code class="descclassname">statistics.</code><code class="descname">mode</code><span class="sig-paren">(</span><em>data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.mode" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Return the most common data point from discrete or nominal <em>data</em>. The mode
|
||
(when it exists) is the most typical value, and is a robust measure of
|
||
central location.</p>
|
||
<p>If <em>data</em> is empty, or if there is not exactly one most common value,
|
||
<a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> is raised.</p>
|
||
<p><code class="docutils literal notranslate"><span class="pre">mode</span></code> assumes discrete data, and returns a single value. This is the
|
||
standard treatment of the mode as commonly taught in schools:</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">mode</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
|
||
<span class="go">3</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>The mode is unique in that it is the only statistic which also applies
|
||
to nominal (non-numeric) data:</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">mode</span><span class="p">([</span><span class="s2">"red"</span><span class="p">,</span> <span class="s2">"blue"</span><span class="p">,</span> <span class="s2">"blue"</span><span class="p">,</span> <span class="s2">"red"</span><span class="p">,</span> <span class="s2">"green"</span><span class="p">,</span> <span class="s2">"red"</span><span class="p">,</span> <span class="s2">"red"</span><span class="p">])</span>
|
||
<span class="go">'red'</span>
|
||
</pre></div>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.pstdev">
|
||
<code class="descclassname">statistics.</code><code class="descname">pstdev</code><span class="sig-paren">(</span><em>data</em>, <em>mu=None</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.pstdev" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Return the population standard deviation (the square root of the population
|
||
variance). See <a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a> for arguments and other details.</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">pstdev</span><span class="p">([</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">2.75</span><span class="p">,</span> <span class="mf">3.25</span><span class="p">,</span> <span class="mf">4.75</span><span class="p">])</span>
|
||
<span class="go">0.986893273527251</span>
|
||
</pre></div>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.pvariance">
|
||
<code class="descclassname">statistics.</code><code class="descname">pvariance</code><span class="sig-paren">(</span><em>data</em>, <em>mu=None</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.pvariance" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Return the population variance of <em>data</em>, a non-empty iterable of real-valued
|
||
numbers. Variance, or second moment about the mean, is a measure of the
|
||
variability (spread or dispersion) of data. A large variance indicates that
|
||
the data is spread out; a small variance indicates it is clustered closely
|
||
around the mean.</p>
|
||
<p>If the optional second argument <em>mu</em> is given, it should be the mean of
|
||
<em>data</em>. If it is missing or <code class="docutils literal notranslate"><span class="pre">None</span></code> (the default), the mean is
|
||
automatically calculated.</p>
|
||
<p>Use this function to calculate the variance from the entire population. To
|
||
estimate the variance from a sample, the <a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate"><span class="pre">variance()</span></code></a> function is usually
|
||
a better choice.</p>
|
||
<p>Raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> if <em>data</em> is empty.</p>
|
||
<p>Examples:</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">1.25</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">,</span> <span class="mf">2.75</span><span class="p">,</span> <span class="mf">3.25</span><span class="p">]</span>
|
||
<span class="gp">>>> </span><span class="n">pvariance</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
|
||
<span class="go">1.25</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>If you have already calculated the mean of your data, you can pass it as the
|
||
optional second argument <em>mu</em> to avoid recalculation:</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">mu</span> <span class="o">=</span> <span class="n">mean</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
|
||
<span class="gp">>>> </span><span class="n">pvariance</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">mu</span><span class="p">)</span>
|
||
<span class="go">1.25</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>This function does not attempt to verify that you have passed the actual mean
|
||
as <em>mu</em>. Using arbitrary values for <em>mu</em> may lead to invalid or impossible
|
||
results.</p>
|
||
<p>Decimals and Fractions are supported:</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="k">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>
|
||
<span class="gp">>>> </span><span class="n">pvariance</span><span class="p">([</span><span class="n">D</span><span class="p">(</span><span class="s2">"27.5"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"30.25"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"30.25"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"34.5"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"41.75"</span><span class="p">)])</span>
|
||
<span class="go">Decimal('24.815')</span>
|
||
|
||
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="k">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>
|
||
<span class="gp">>>> </span><span class="n">pvariance</span><span class="p">([</span><span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)])</span>
|
||
<span class="go">Fraction(13, 72)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="admonition note">
|
||
<p class="admonition-title">Note</p>
|
||
<p>When called with the entire population, this gives the population variance
|
||
σ². When called on a sample instead, this is the biased sample variance
|
||
s², also known as variance with N degrees of freedom.</p>
|
||
<p>If you somehow know the true population mean μ, you may use this function
|
||
to calculate the variance of a sample, giving the known population mean as
|
||
the second argument. Provided the data points are representative
|
||
(e.g. independent and identically distributed), the result will be an
|
||
unbiased estimate of the population variance.</p>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.stdev">
|
||
<code class="descclassname">statistics.</code><code class="descname">stdev</code><span class="sig-paren">(</span><em>data</em>, <em>xbar=None</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.stdev" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Return the sample standard deviation (the square root of the sample
|
||
variance). See <a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate"><span class="pre">variance()</span></code></a> for arguments and other details.</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">stdev</span><span class="p">([</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">2.75</span><span class="p">,</span> <span class="mf">3.25</span><span class="p">,</span> <span class="mf">4.75</span><span class="p">])</span>
|
||
<span class="go">1.0810874155219827</span>
|
||
</pre></div>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="statistics.variance">
|
||
<code class="descclassname">statistics.</code><code class="descname">variance</code><span class="sig-paren">(</span><em>data</em>, <em>xbar=None</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.variance" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Return the sample variance of <em>data</em>, an iterable of at least two real-valued
|
||
numbers. Variance, or second moment about the mean, is a measure of the
|
||
variability (spread or dispersion) of data. A large variance indicates that
|
||
the data is spread out; a small variance indicates it is clustered closely
|
||
around the mean.</p>
|
||
<p>If the optional second argument <em>xbar</em> is given, it should be the mean of
|
||
<em>data</em>. If it is missing or <code class="docutils literal notranslate"><span class="pre">None</span></code> (the default), the mean is
|
||
automatically calculated.</p>
|
||
<p>Use this function when your data is a sample from a population. To calculate
|
||
the variance from the entire population, see <a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a>.</p>
|
||
<p>Raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> if <em>data</em> has fewer than two values.</p>
|
||
<p>Examples:</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="mf">2.75</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">,</span> <span class="mf">1.25</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.25</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">]</span>
|
||
<span class="gp">>>> </span><span class="n">variance</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
|
||
<span class="go">1.3720238095238095</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>If you have already calculated the mean of your data, you can pass it as the
|
||
optional second argument <em>xbar</em> to avoid recalculation:</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">m</span> <span class="o">=</span> <span class="n">mean</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
|
||
<span class="gp">>>> </span><span class="n">variance</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
|
||
<span class="go">1.3720238095238095</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>This function does not attempt to verify that you have passed the actual mean
|
||
as <em>xbar</em>. Using arbitrary values for <em>xbar</em> can lead to invalid or
|
||
impossible results.</p>
|
||
<p>Decimal and Fraction values are supported:</p>
|
||
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="k">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>
|
||
<span class="gp">>>> </span><span class="n">variance</span><span class="p">([</span><span class="n">D</span><span class="p">(</span><span class="s2">"27.5"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"30.25"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"30.25"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"34.5"</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">"41.75"</span><span class="p">)])</span>
|
||
<span class="go">Decimal('31.01875')</span>
|
||
|
||
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="k">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>
|
||
<span class="gp">>>> </span><span class="n">variance</span><span class="p">([</span><span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)])</span>
|
||
<span class="go">Fraction(67, 108)</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="admonition note">
|
||
<p class="admonition-title">Note</p>
|
||
<p>This is the sample variance s² with Bessel’s correction, also known as
|
||
variance with N-1 degrees of freedom. Provided that the data points are
|
||
representative (e.g. independent and identically distributed), the result
|
||
should be an unbiased estimate of the true population variance.</p>
|
||
<p>If you somehow know the actual population mean μ you should pass it to the
|
||
<a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a> function as the <em>mu</em> parameter to get the variance of a
|
||
sample.</p>
|
||
</div>
|
||
</dd></dl>
|
||
|
||
</div>
|
||
<div class="section" id="exceptions">
|
||
<h2>Exceptions<a class="headerlink" href="#exceptions" title="Permalink to this headline">¶</a></h2>
|
||
<p>A single exception is defined:</p>
|
||
<dl class="exception">
|
||
<dt id="statistics.StatisticsError">
|
||
<em class="property">exception </em><code class="descclassname">statistics.</code><code class="descname">StatisticsError</code><a class="headerlink" href="#statistics.StatisticsError" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Subclass of <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a> for statistics-related exceptions.</p>
|
||
</dd></dl>
|
||
|
||
</div>
|
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<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">statistics</span></code> — Mathematical statistics functions</a><ul>
|
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<li><a class="reference internal" href="#averages-and-measures-of-central-location">Averages and measures of central location</a></li>
|
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<li><a class="reference internal" href="#measures-of-spread">Measures of spread</a></li>
|
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<li><a class="reference internal" href="#function-details">Function details</a></li>
|
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<li><a class="reference internal" href="#exceptions">Exceptions</a></li>
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