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href="../contents.html">Table of Contents</a></h3>106    <ul>107<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>108<li><a class="reference internal" href="#averages-and-measures-of-central-location">Averages and measures of central location</a></li>109<li><a class="reference internal" href="#measures-of-spread">Measures of spread</a></li>110<li><a class="reference internal" href="#statistics-for-relations-between-two-inputs">Statistics for relations between two inputs</a></li>111<li><a class="reference internal" href="#function-details">Function details</a></li>112<li><a class="reference internal" href="#exceptions">Exceptions</a></li>113<li><a class="reference internal" href="#normaldist-objects"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code> objects</a></li>114<li><a class="reference internal" href="#examples-and-recipes">Examples and Recipes</a><ul>115<li><a class="reference internal" href="#classic-probability-problems">Classic probability problems</a></li>116<li><a class="reference internal" href="#monte-carlo-inputs-for-simulations">Monte Carlo inputs for simulations</a></li>117<li><a class="reference internal" href="#approximating-binomial-distributions">Approximating binomial distributions</a></li>118<li><a class="reference internal" href="#naive-bayesian-classifier">Naive bayesian classifier</a></li>119</ul>120</li>121</ul>122</li>123</ul>124 125  </div>126  <div>127    <h4>Previous topic</h4>128    <p class="topless"><a href="random.html"129                          title="previous chapter"><code class="xref py py-mod docutils literal notranslate"><span class="pre">random</span></code> — Generate pseudo-random numbers</a></p>130  </div>131  <div>132    <h4>Next topic</h4>133    <p class="topless"><a href="functional.html"134                          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     <li class="right">201                    202 203    <div class="inline-search" role="search">204        <form class="inline-search" action="../search.html" method="get">205          <input placeholder="Quick search" aria-label="Quick search" type="search" name="q" id="search-box">206          <input type="submit" value="Go">207        </form>208    </div>209                     |210                </li>211            <li class="right">212<label class="theme-selector-label">213    Theme214    <select class="theme-selector" oninput="activateTheme(this.value)">215        <option value="auto" selected>Auto</option>216        <option value="light">Light</option>217        <option value="dark">Dark</option>218    </select>219</label> |</li>220            221      </ul>222    </div>    223 224    <div class="document">225      <div class="documentwrapper">226        <div class="bodywrapper">227          <div class="body" role="main">228            229  <section id="module-statistics">230<span id="statistics-mathematical-statistics-functions"></span><h1><code class="xref py py-mod docutils literal notranslate"><span class="pre">statistics</span></code> — Mathematical statistics functions<a class="headerlink" href="#module-statistics" title="Link to this heading">¶</a></h1>231<div class="versionadded">232<p><span class="versionmodified added">Added in version 3.4.</span></p>233</div>234<p><strong>Source code:</strong> <a class="extlink-source reference external" href="https://github.com/python/cpython/tree/main/Lib/statistics.py">Lib/statistics.py</a></p>235<hr class="docutils" />236<p>This module provides functions for calculating mathematical statistics of237numeric (<a class="reference internal" href="numbers.html#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate"><span class="pre">Real</span></code></a>-valued) data.</p>238<p>The module is not intended to be a competitor to third-party libraries such239as <a class="reference external" href="https://numpy.org">NumPy</a>, <a class="reference external" href="https://scipy.org/">SciPy</a>, or240proprietary full-featured statistics packages aimed at professional241statisticians such as Minitab, SAS and Matlab. It is aimed at the level of242graphing and scientific calculators.</p>243<p>Unless explicitly noted, 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>,244<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</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">Fraction</span></code></a>.245Behaviour with other types (whether in the numeric tower or not) is246currently unsupported.  Collections with a mix of types are also undefined247and implementation-dependent.  If your input data consists of mixed types,248you 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, for249example: <code class="docutils literal notranslate"><span class="pre">map(float,</span> <span class="pre">input_data)</span></code>.</p>250<p>Some datasets use <code class="docutils literal notranslate"><span class="pre">NaN</span></code> (not a number) values to represent missing data.251Since NaNs have unusual comparison semantics, they cause surprising or252undefined behaviors in the statistics functions that sort data or that count253occurrences.  The functions affected are <code class="docutils literal notranslate"><span class="pre">median()</span></code>, <code class="docutils literal notranslate"><span class="pre">median_low()</span></code>,254<code class="docutils literal notranslate"><span class="pre">median_high()</span></code>, <code class="docutils literal notranslate"><span class="pre">median_grouped()</span></code>, <code class="docutils literal notranslate"><span class="pre">mode()</span></code>, <code class="docutils literal notranslate"><span class="pre">multimode()</span></code>, and255<code class="docutils literal notranslate"><span class="pre">quantiles()</span></code>.  The <code class="docutils literal notranslate"><span class="pre">NaN</span></code> values should be stripped before calling these256functions:</p>257<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span><span class="w"> </span><span class="nn">statistics</span><span class="w"> </span><span class="kn">import</span> <span class="n">median</span>258<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span><span class="w"> </span><span class="nn">math</span><span class="w"> </span><span class="kn">import</span> <span class="n">isnan</span>259<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span><span class="w"> </span><span class="nn">itertools</span><span class="w"> </span><span class="kn">import</span> <span class="n">filterfalse</span>260 261<span class="gp">&gt;&gt;&gt; </span><span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="mf">20.7</span><span class="p">,</span> <span class="nb">float</span><span class="p">(</span><span class="s1">&#39;NaN&#39;</span><span class="p">),</span><span class="mf">19.2</span><span class="p">,</span> <span class="mf">18.3</span><span class="p">,</span> <span class="nb">float</span><span class="p">(</span><span class="s1">&#39;NaN&#39;</span><span class="p">),</span> <span class="mf">14.4</span><span class="p">]</span>262<span class="gp">&gt;&gt;&gt; </span><span class="nb">sorted</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>  <span class="c1"># This has surprising behavior</span>263<span class="go">[20.7, nan, 14.4, 18.3, 19.2, nan]</span>264<span class="gp">&gt;&gt;&gt; </span><span class="n">median</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>  <span class="c1"># This result is unexpected</span>265<span class="go">16.35</span>266 267<span class="gp">&gt;&gt;&gt; </span><span class="nb">sum</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">isnan</span><span class="p">,</span> <span class="n">data</span><span class="p">))</span>    <span class="c1"># Number of missing values</span>268<span class="go">2</span>269<span class="gp">&gt;&gt;&gt; </span><span class="n">clean</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">filterfalse</span><span class="p">(</span><span class="n">isnan</span><span class="p">,</span> <span class="n">data</span><span class="p">))</span>  <span class="c1"># Strip NaN values</span>270<span class="gp">&gt;&gt;&gt; </span><span class="n">clean</span>271<span class="go">[20.7, 19.2, 18.3, 14.4]</span>272<span class="gp">&gt;&gt;&gt; </span><span class="nb">sorted</span><span class="p">(</span><span class="n">clean</span><span class="p">)</span>  <span class="c1"># Sorting now works as expected</span>273<span class="go">[14.4, 18.3, 19.2, 20.7]</span>274<span class="gp">&gt;&gt;&gt; </span><span class="n">median</span><span class="p">(</span><span class="n">clean</span><span class="p">)</span>       <span class="c1"># This result is now well defined</span>275<span class="go">18.75</span>276</pre></div>277</div>278<section id="averages-and-measures-of-central-location">279<h2>Averages and measures of central location<a class="headerlink" href="#averages-and-measures-of-central-location" title="Link to this heading">¶</a></h2>280<p>These functions calculate an average or typical value from a population281or sample.</p>282<table class="docutils align-default">283<tbody>284<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>285<td><p>Arithmetic mean (“average”) of data.</p></td>286</tr>287<tr class="row-even"><td><p><a class="reference internal" href="#statistics.fmean" title="statistics.fmean"><code class="xref py py-func docutils literal notranslate"><span class="pre">fmean()</span></code></a></p></td>288<td><p>Fast, floating-point arithmetic mean, with optional weighting.</p></td>289</tr>290<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.geometric_mean" title="statistics.geometric_mean"><code class="xref py py-func docutils literal notranslate"><span class="pre">geometric_mean()</span></code></a></p></td>291<td><p>Geometric mean of data.</p></td>292</tr>293<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>294<td><p>Harmonic mean of data.</p></td>295</tr>296<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.kde" title="statistics.kde"><code class="xref py py-func docutils literal notranslate"><span class="pre">kde()</span></code></a></p></td>297<td><p>Estimate the probability density distribution of the data.</p></td>298</tr>299<tr class="row-even"><td><p><a class="reference internal" href="#statistics.kde_random" title="statistics.kde_random"><code class="xref py py-func docutils literal notranslate"><span class="pre">kde_random()</span></code></a></p></td>300<td><p>Random sampling from the PDF generated by kde().</p></td>301</tr>302<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>303<td><p>Median (middle value) of data.</p></td>304</tr>305<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>306<td><p>Low median of data.</p></td>307</tr>308<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>309<td><p>High median of data.</p></td>310</tr>311<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>312<td><p>Median (50th percentile) of grouped data.</p></td>313</tr>314<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>315<td><p>Single mode (most common value) of discrete or nominal data.</p></td>316</tr>317<tr class="row-even"><td><p><a class="reference internal" href="#statistics.multimode" title="statistics.multimode"><code class="xref py py-func docutils literal notranslate"><span class="pre">multimode()</span></code></a></p></td>318<td><p>List of modes (most common values) of discrete or nominal data.</p></td>319</tr>320<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.quantiles" title="statistics.quantiles"><code class="xref py py-func docutils literal notranslate"><span class="pre">quantiles()</span></code></a></p></td>321<td><p>Divide data into intervals with equal probability.</p></td>322</tr>323</tbody>324</table>325</section>326<section id="measures-of-spread">327<h2>Measures of spread<a class="headerlink" href="#measures-of-spread" title="Link to this heading">¶</a></h2>328<p>These functions calculate a measure of how much the population or sample329tends to deviate from the typical or average values.</p>330<table class="docutils align-default">331<tbody>332<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>333<td><p>Population standard deviation of data.</p></td>334</tr>335<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>336<td><p>Population variance of data.</p></td>337</tr>338<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>339<td><p>Sample standard deviation of data.</p></td>340</tr>341<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>342<td><p>Sample variance of data.</p></td>343</tr>344</tbody>345</table>346</section>347<section id="statistics-for-relations-between-two-inputs">348<h2>Statistics for relations between two inputs<a class="headerlink" href="#statistics-for-relations-between-two-inputs" title="Link to this heading">¶</a></h2>349<p>These functions calculate statistics regarding relations between two inputs.</p>350<table class="docutils align-default">351<tbody>352<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.covariance" title="statistics.covariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">covariance()</span></code></a></p></td>353<td><p>Sample covariance for two variables.</p></td>354</tr>355<tr class="row-even"><td><p><a class="reference internal" href="#statistics.correlation" title="statistics.correlation"><code class="xref py py-func docutils literal notranslate"><span class="pre">correlation()</span></code></a></p></td>356<td><p>Pearson and Spearman’s correlation coefficients.</p></td>357</tr>358<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.linear_regression" title="statistics.linear_regression"><code class="xref py py-func docutils literal notranslate"><span class="pre">linear_regression()</span></code></a></p></td>359<td><p>Slope and intercept for simple linear regression.</p></td>360</tr>361</tbody>362</table>363</section>364<section id="function-details">365<h2>Function details<a class="headerlink" href="#function-details" title="Link to this heading">¶</a></h2>366<p>Note: The functions do not require the data given to them to be sorted.367However, for reading convenience, most of the examples show sorted sequences.</p>368<dl class="py function">369<dt class="sig sig-object py" id="statistics.mean">370<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">mean</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.mean" title="Link to this definition">¶</a></dt>371<dd><p>Return the sample arithmetic mean of <em>data</em> which can be a sequence or iterable.</p>372<p>The arithmetic mean is the sum of the data divided by the number of data373points.  It is commonly called “the average”, although it is only one of many374different mathematical averages.  It is a measure of the central location of375the data.</p>376<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>377<p>Some examples of use:</p>378<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </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>379<span class="go">2.8</span>380<span class="gp">&gt;&gt;&gt; </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>381<span class="go">2.625</span>382 383<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span><span class="w"> </span><span class="nn">fractions</span><span class="w"> </span><span class="kn">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>384<span class="gp">&gt;&gt;&gt; </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>385<span class="go">Fraction(13, 21)</span>386 387<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span><span class="w"> </span><span class="nn">decimal</span><span class="w"> </span><span class="kn">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>388<span class="gp">&gt;&gt;&gt; </span><span class="n">mean</span><span class="p">([</span><span class="n">D</span><span class="p">(</span><span class="s2">&quot;0.5&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;0.75&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;0.625&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;0.375&quot;</span><span class="p">)])</span>389<span class="go">Decimal(&#39;0.5625&#39;)</span>390</pre></div>391</div>392<div class="admonition note">393<p class="admonition-title">Note</p>394<p>The mean is strongly affected by <a class="reference external" href="https://en.wikipedia.org/wiki/Outlier">outliers</a> and is not necessarily a395typical example of the data points. For a more robust, although less396efficient, measure of <a class="reference external" href="https://en.wikipedia.org/wiki/Central_tendency">central tendency</a>, 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>.</p>397<p>The sample mean gives an unbiased estimate of the true population mean,398so that when taken on average over all the possible samples,399<code class="docutils literal notranslate"><span class="pre">mean(sample)</span></code> converges on the true mean of the entire population.  If400<em>data</em> represents the entire population rather than a sample, then401<code class="docutils literal notranslate"><span class="pre">mean(data)</span></code> is equivalent to calculating the true population mean μ.</p>402</div>403</dd></dl>404 405<dl class="py function">406<dt class="sig sig-object py" id="statistics.fmean">407<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">fmean</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">weights</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.fmean" title="Link to this definition">¶</a></dt>408<dd><p>Convert <em>data</em> to floats and compute the arithmetic mean.</p>409<p>This runs faster than the <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> function and it always returns a410<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>.  The <em>data</em> may be a sequence or iterable.  If the input411dataset is empty, raises a <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>.</p>412<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">fmean</span><span class="p">([</span><span class="mf">3.5</span><span class="p">,</span> <span class="mf">4.0</span><span class="p">,</span> <span class="mf">5.25</span><span class="p">])</span>413<span class="go">4.25</span>414</pre></div>415</div>416<p>Optional weighting is supported.  For example, a professor assigns a417grade for a course by weighting quizzes at 20%, homework at 20%, a418midterm exam at 30%, and a final exam at 30%:</p>419<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">grades</span> <span class="o">=</span> <span class="p">[</span><span class="mi">85</span><span class="p">,</span> <span class="mi">92</span><span class="p">,</span> <span class="mi">83</span><span class="p">,</span> <span class="mi">91</span><span class="p">]</span>420<span class="gp">&gt;&gt;&gt; </span><span class="n">weights</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.20</span><span class="p">,</span> <span class="mf">0.20</span><span class="p">,</span> <span class="mf">0.30</span><span class="p">,</span> <span class="mf">0.30</span><span class="p">]</span>421<span class="gp">&gt;&gt;&gt; </span><span class="n">fmean</span><span class="p">(</span><span class="n">grades</span><span class="p">,</span> <span class="n">weights</span><span class="p">)</span>422<span class="go">87.6</span>423</pre></div>424</div>425<p>If <em>weights</em> is supplied, it must be the same length as the <em>data</em> or426a <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> will be raised.</p>427<div class="versionadded">428<p><span class="versionmodified added">Added in version 3.8.</span></p>429</div>430<div class="versionchanged">431<p><span class="versionmodified changed">Changed in version 3.11: </span>Added support for <em>weights</em>.</p>432</div>433</dd></dl>434 435<dl class="py function">436<dt class="sig sig-object py" id="statistics.geometric_mean">437<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">geometric_mean</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.geometric_mean" title="Link to this definition">¶</a></dt>438<dd><p>Convert <em>data</em> to floats and compute the geometric mean.</p>439<p>The geometric mean indicates the central tendency or typical value of the440<em>data</em> using the product of the values (as opposed to the arithmetic mean441which uses their sum).</p>442<p>Raises a <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 the input dataset is empty,443if it contains a zero, or if it contains a negative value.444The <em>data</em> may be a sequence or iterable.</p>445<p>No special efforts are made to achieve exact results.446(However, this may change in the future.)</p>447<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">geometric_mean</span><span class="p">([</span><span class="mi">54</span><span class="p">,</span> <span class="mi">24</span><span class="p">,</span> <span class="mi">36</span><span class="p">]),</span> <span class="mi">1</span><span class="p">)</span>448<span class="go">36.0</span>449</pre></div>450</div>451<div class="versionadded">452<p><span class="versionmodified added">Added in version 3.8.</span></p>453</div>454</dd></dl>455 456<dl class="py function">457<dt class="sig sig-object py" id="statistics.harmonic_mean">458<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">harmonic_mean</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">weights</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.harmonic_mean" title="Link to this definition">¶</a></dt>459<dd><p>Return the harmonic mean of <em>data</em>, a sequence or iterable of460real-valued numbers.  If <em>weights</em> is omitted or <code class="docutils literal notranslate"><span class="pre">None</span></code>, then461equal weighting is assumed.</p>462<p>The harmonic mean is the 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 the463reciprocals of the data. For example, the harmonic mean of three values <em>a</em>,464<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>.  If one of the465values is zero, the result will be zero.</p>466<p>The harmonic mean is a type of average, a measure of the central467location of the data.  It is often appropriate when averaging468ratios or rates, for example speeds.</p>469<p>Suppose a car travels 10 km at 40 km/hr, then another 10 km at 60 km/hr.470What is the average speed?</p>471<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">harmonic_mean</span><span class="p">([</span><span class="mi">40</span><span class="p">,</span> <span class="mi">60</span><span class="p">])</span>472<span class="go">48.0</span>473</pre></div>474</div>475<p>Suppose a car travels 40 km/hr for 5 km, and when traffic clears,476speeds-up to 60 km/hr for the remaining 30 km of the journey. What477is the average speed?</p>478<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">harmonic_mean</span><span class="p">([</span><span class="mi">40</span><span class="p">,</span> <span class="mi">60</span><span class="p">],</span> <span class="n">weights</span><span class="o">=</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">30</span><span class="p">])</span>479<span class="go">56.0</span>480</pre></div>481</div>482<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, any element483is less than zero, or if the weighted sum isn’t positive.</p>484<p>The current algorithm has an early-out when it encounters a zero485in the input.  This means that the subsequent inputs are not tested486for validity.  (This behavior may change in the future.)</p>487<div class="versionadded">488<p><span class="versionmodified added">Added in version 3.6.</span></p>489</div>490<div class="versionchanged">491<p><span class="versionmodified changed">Changed in version 3.10: </span>Added support for <em>weights</em>.</p>492</div>493</dd></dl>494 495<dl class="py function">496<dt class="sig sig-object py" id="statistics.kde">497<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">kde</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">h</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">kernel</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'normal'</span></span></em>, <em class="sig-param"><span class="keyword-only-separator o"><abbr title="Keyword-only parameters separator (PEP 3102)"><span class="pre">*</span></abbr></span></em>, <em class="sig-param"><span class="n"><span class="pre">cumulative</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.kde" title="Link to this definition">¶</a></dt>498<dd><p><a class="reference external" href="https://www.itm-conferences.org/articles/itmconf/pdf/2018/08/itmconf_sam2018_00037.pdf">Kernel Density Estimation (KDE)</a>:499Create a continuous probability density function or cumulative500distribution function from discrete samples.</p>501<p>The basic idea is to smooth the data using <a class="reference external" href="https://en.wikipedia.org/wiki/Kernel_(statistics)">a kernel function</a>.502to help draw inferences about a population from a sample.</p>503<p>The degree of smoothing is controlled by the scaling parameter <em>h</em>504which is called the bandwidth.  Smaller values emphasize local505features while larger values give smoother results.</p>506<p>The <em>kernel</em> determines the relative weights of the sample data507points.  Generally, the choice of kernel shape does not matter508as much as the more influential bandwidth smoothing parameter.</p>509<p>Kernels that give some weight to every sample point include510<em>normal</em> (<em>gauss</em>), <em>logistic</em>, and <em>sigmoid</em>.</p>511<p>Kernels that only give weight to sample points within the bandwidth512include <em>rectangular</em> (<em>uniform</em>), <em>triangular</em>, <em>parabolic</em>513(<em>epanechnikov</em>), <em>quartic</em> (<em>biweight</em>), <em>triweight</em>, and <em>cosine</em>.</p>514<p>If <em>cumulative</em> is true, will return a cumulative distribution function.</p>515<p>A <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 if the <em>data</em> sequence is empty.</p>516<p><a class="reference external" href="https://en.wikipedia.org/wiki/Kernel_density_estimation#Example">Wikipedia has an example</a>517where we can use <code class="xref py py-func docutils literal notranslate"><span class="pre">kde()</span></code> to generate and plot a probability518density function estimated from a small sample:</p>519<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">sample</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mf">2.1</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.3</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.4</span><span class="p">,</span> <span class="mf">1.9</span><span class="p">,</span> <span class="mf">5.1</span><span class="p">,</span> <span class="mf">6.2</span><span class="p">]</span>520<span class="gp">&gt;&gt;&gt; </span><span class="n">f_hat</span> <span class="o">=</span> <span class="n">kde</span><span class="p">(</span><span class="n">sample</span><span class="p">,</span> <span class="n">h</span><span class="o">=</span><span class="mf">1.5</span><span class="p">)</span>521<span class="gp">&gt;&gt;&gt; </span><span class="n">xarr</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span><span class="o">/</span><span class="mi">100</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">750</span><span class="p">,</span> <span class="mi">1100</span><span class="p">)]</span>522<span class="gp">&gt;&gt;&gt; </span><span class="n">yarr</span> <span class="o">=</span> <span class="p">[</span><span class="n">f_hat</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xarr</span><span class="p">]</span>523</pre></div>524</div>525<p>The points in <code class="docutils literal notranslate"><span class="pre">xarr</span></code> and <code class="docutils literal notranslate"><span class="pre">yarr</span></code> can be used to make a PDF plot:</p>526<img alt="Scatter plot of the estimated probability density function." src="../_images/kde_example.png" />527<div class="versionadded">528<p><span class="versionmodified added">Added in version 3.13.</span></p>529</div>530</dd></dl>531 532<dl class="py function">533<dt class="sig sig-object py" id="statistics.kde_random">534<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">kde_random</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">h</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">kernel</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'normal'</span></span></em>, <em class="sig-param"><span class="keyword-only-separator o"><abbr title="Keyword-only parameters separator (PEP 3102)"><span class="pre">*</span></abbr></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.kde_random" title="Link to this definition">¶</a></dt>535<dd><p>Return a function that makes a random selection from the estimated536probability density function produced by <code class="docutils literal notranslate"><span class="pre">kde(data,</span> <span class="pre">h,</span> <span class="pre">kernel)</span></code>.</p>537<p>Providing a <em>seed</em> allows reproducible selections. In the future, the538values may change slightly as more accurate kernel inverse CDF estimates539are implemented.  The seed may be an integer, float, str, or bytes.</p>540<p>A <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 if the <em>data</em> sequence is empty.</p>541<p>Continuing the example for <a class="reference internal" href="#statistics.kde" title="statistics.kde"><code class="xref py py-func docutils literal notranslate"><span class="pre">kde()</span></code></a>, we can use542<code class="xref py py-func docutils literal notranslate"><span class="pre">kde_random()</span></code> to generate new random selections from an543estimated probability density function:</p>544<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mf">2.1</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.3</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.4</span><span class="p">,</span> <span class="mf">1.9</span><span class="p">,</span> <span class="mf">5.1</span><span class="p">,</span> <span class="mf">6.2</span><span class="p">]</span>545<span class="gp">&gt;&gt;&gt; </span><span class="n">rand</span> <span class="o">=</span> <span class="n">kde_random</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">h</span><span class="o">=</span><span class="mf">1.5</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">8675309</span><span class="p">)</span>546<span class="gp">&gt;&gt;&gt; </span><span class="n">new_selections</span> <span class="o">=</span> <span class="p">[</span><span class="n">rand</span><span class="p">()</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)]</span>547<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="nb">round</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">new_selections</span><span class="p">]</span>548<span class="go">[0.7, 6.2, 1.2, 6.9, 7.0, 1.8, 2.5, -0.5, -1.8, 5.6]</span>549</pre></div>550</div>551<div class="versionadded">552<p><span class="versionmodified added">Added in version 3.13.</span></p>553</div>554</dd></dl>555 556<dl class="py function">557<dt class="sig sig-object py" id="statistics.median">558<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">median</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median" title="Link to this definition">¶</a></dt>559<dd><p>Return the median (middle value) of numeric data, using the common “mean of560middle 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.561<em>data</em> can be a sequence or iterable.</p>562<p>The median is a robust measure of central location and is less affected by563the presence of outliers.  When the number of data points is odd, the564middle data point is returned:</p>565<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </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>566<span class="go">3</span>567</pre></div>568</div>569<p>When the number of data points is even, the median is interpolated by taking570the average of the two middle values:</p>571<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </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>572<span class="go">4.0</span>573</pre></div>574</div>575<p>This is suited for when your data is discrete, and you don’t mind that the576median may not be an actual data point.</p>577<p>If the data is ordinal (supports order operations) but not numeric (doesn’t578support addition), consider using <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>579instead.</p>580</dd></dl>581 582<dl class="py function">583<dt class="sig sig-object py" id="statistics.median_low">584<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">median_low</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median_low" title="Link to this definition">¶</a></dt>585<dd><p>Return the low median of numeric data.  If <em>data</em> is empty,586<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 iterable.</p>587<p>The low median is always a member of the data set.  When the number of data588points is odd, the middle value is returned.  When it is even, the smaller of589the two middle values is returned.</p>590<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </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>591<span class="go">3</span>592<span class="gp">&gt;&gt;&gt; </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>593<span class="go">3</span>594</pre></div>595</div>596<p>Use the low median when your data are discrete and you prefer the median to597be an actual data point rather than interpolated.</p>598</dd></dl>599 600<dl class="py function">601<dt class="sig sig-object py" id="statistics.median_high">602<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">median_high</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median_high" title="Link to this definition">¶</a></dt>603<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>604is raised.  <em>data</em> can be a sequence or iterable.</p>605<p>The high median is always a member of the data set.  When the number of data606points is odd, the middle value is returned.  When it is even, the larger of607the two middle values is returned.</p>608<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </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>609<span class="go">3</span>610<span class="gp">&gt;&gt;&gt; </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>611<span class="go">5</span>612</pre></div>613</div>614<p>Use the high median when your data are discrete and you prefer the median to615be an actual data point rather than interpolated.</p>616</dd></dl>617 618<dl class="py function">619<dt class="sig sig-object py" id="statistics.median_grouped">620<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">median_grouped</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">interval</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median_grouped" title="Link to this definition">¶</a></dt>621<dd><p>Estimates the median for numeric data that has been <a class="reference external" href="https://en.wikipedia.org/wiki/Data_binning">grouped or binned</a> around the midpoints622of consecutive, fixed-width intervals.</p>623<p>The <em>data</em> can be any iterable of numeric data with each value being624exactly the midpoint of a bin.  At least one value must be present.</p>625<p>The <em>interval</em> is the width of each bin.</p>626<p>For example, demographic information may have been summarized into627consecutive ten-year age groups with each group being represented628by the 5-year midpoints of the intervals:</p>629<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span><span class="w"> </span><span class="nn">collections</span><span class="w"> </span><span class="kn">import</span> <span class="n">Counter</span>630<span class="gp">&gt;&gt;&gt; </span><span class="n">demographics</span> <span class="o">=</span> <span class="n">Counter</span><span class="p">({</span>631<span class="gp">... </span>   <span class="mi">25</span><span class="p">:</span> <span class="mi">172</span><span class="p">,</span>   <span class="c1"># 20 to 30 years old</span>632<span class="gp">... </span>   <span class="mi">35</span><span class="p">:</span> <span class="mi">484</span><span class="p">,</span>   <span class="c1"># 30 to 40 years old</span>633<span class="gp">... </span>   <span class="mi">45</span><span class="p">:</span> <span class="mi">387</span><span class="p">,</span>   <span class="c1"># 40 to 50 years old</span>634<span class="gp">... </span>   <span class="mi">55</span><span class="p">:</span>  <span class="mi">22</span><span class="p">,</span>   <span class="c1"># 50 to 60 years old</span>635<span class="gp">... </span>   <span class="mi">65</span><span class="p">:</span>   <span class="mi">6</span><span class="p">,</span>   <span class="c1"># 60 to 70 years old</span>636<span class="gp">... </span><span class="p">})</span>637<span class="gp">...</span>638</pre></div>639</div>640<p>The 50th percentile (median) is the 536th person out of the 1071641member cohort.  That person is in the 30 to 40 year old age group.</p>642<p>The regular <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> function would assume that everyone in the643tricenarian age group was exactly 35 years old.  A more tenable644assumption is that the 484 members of that age group are evenly645distributed between 30 and 40.  For that, we use646<code class="xref py py-func docutils literal notranslate"><span class="pre">median_grouped()</span></code>:</p>647<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">data</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">demographics</span><span class="o">.</span><span class="n">elements</span><span class="p">())</span>648<span class="gp">&gt;&gt;&gt; </span><span class="n">median</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>649<span class="go">35</span>650<span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">median_grouped</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">interval</span><span class="o">=</span><span class="mi">10</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>651<span class="go">37.5</span>652</pre></div>653</div>654<p>The caller is responsible for making sure the data points are separated655by exact multiples of <em>interval</em>.  This is essential for getting a656correct result.  The function does not check this precondition.</p>657<p>Inputs may be any numeric type that can be coerced to a float during658the interpolation step.</p>659</dd></dl>660 661<dl class="py function">662<dt class="sig sig-object py" id="statistics.mode">663<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">mode</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.mode" title="Link to this definition">¶</a></dt>664<dd><p>Return the single most common data point from discrete or nominal <em>data</em>.665The mode (when it exists) is the most typical value and serves as a666measure of central location.</p>667<p>If there are multiple modes with the same frequency, returns the first one668encountered in the <em>data</em>.  If the smallest or largest of those is669desired instead, use <code class="docutils literal notranslate"><span class="pre">min(multimode(data))</span></code> or <code class="docutils literal notranslate"><span class="pre">max(multimode(data))</span></code>.670If the input <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.</p>671<p><code class="docutils literal notranslate"><span class="pre">mode</span></code> assumes discrete data and returns a single value. This is the672standard treatment of the mode as commonly taught in schools:</p>673<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </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>674<span class="go">3</span>675</pre></div>676</div>677<p>The mode is unique in that it is the only statistic in this package that678also applies to nominal (non-numeric) data:</p>679<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">mode</span><span class="p">([</span><span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="s2">&quot;blue&quot;</span><span class="p">,</span> <span class="s2">&quot;blue&quot;</span><span class="p">,</span> <span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="s2">&quot;green&quot;</span><span class="p">,</span> <span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="s2">&quot;red&quot;</span><span class="p">])</span>680<span class="go">&#39;red&#39;</span>681</pre></div>682</div>683<p>Only hashable inputs are supported.  To handle type <a class="reference internal" href="stdtypes.html#set" title="set"><code class="xref py py-class docutils literal notranslate"><span class="pre">set</span></code></a>,684consider casting to <a class="reference internal" href="stdtypes.html#frozenset" title="frozenset"><code class="xref py py-class docutils literal notranslate"><span class="pre">frozenset</span></code></a>.  To handle type <a class="reference internal" href="stdtypes.html#list" title="list"><code class="xref py py-class docutils literal notranslate"><span class="pre">list</span></code></a>,685consider casting to <a class="reference internal" href="stdtypes.html#tuple" title="tuple"><code class="xref py py-class docutils literal notranslate"><span class="pre">tuple</span></code></a>.  For mixed or nested inputs, consider686using this slower quadratic algorithm that only depends on equality tests:687<code class="docutils literal notranslate"><span class="pre">max(data,</span> <span class="pre">key=data.count)</span></code>.</p>688<div class="versionchanged">689<p><span class="versionmodified changed">Changed in version 3.8: </span>Now handles multimodal datasets by returning the first mode encountered.690Formerly, it raised <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> when more than one mode was691found.</p>692</div>693</dd></dl>694 695<dl class="py function">696<dt class="sig sig-object py" id="statistics.multimode">697<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">multimode</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.multimode" title="Link to this definition">¶</a></dt>698<dd><p>Return a list of the most frequently occurring values in the order they699were first encountered in the <em>data</em>.  Will return more than one result if700there are multiple modes or an empty list if the <em>data</em> is empty:</p>701<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">multimode</span><span class="p">(</span><span class="s1">&#39;aabbbbccddddeeffffgg&#39;</span><span class="p">)</span>702<span class="go">[&#39;b&#39;, &#39;d&#39;, &#39;f&#39;]</span>703<span class="gp">&gt;&gt;&gt; </span><span class="n">multimode</span><span class="p">(</span><span class="s1">&#39;&#39;</span><span class="p">)</span>704<span class="go">[]</span>705</pre></div>706</div>707<div class="versionadded">708<p><span class="versionmodified added">Added in version 3.8.</span></p>709</div>710</dd></dl>711 712<dl class="py function">713<dt class="sig sig-object py" id="statistics.pstdev">714<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">pstdev</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">mu</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.pstdev" title="Link to this definition">¶</a></dt>715<dd><p>Return the population standard deviation (the square root of the population716variance).  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>717<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </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>718<span class="go">0.986893273527251</span>719</pre></div>720</div>721</dd></dl>722 723<dl class="py function">724<dt class="sig sig-object py" id="statistics.pvariance">725<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">pvariance</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">mu</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.pvariance" title="Link to this definition">¶</a></dt>726<dd><p>Return the population variance of <em>data</em>, a non-empty sequence or iterable727of real-valued numbers.  Variance, or second moment about the mean, is a728measure of the variability (spread or dispersion) of data.  A large729variance indicates that the data is spread out; a small variance indicates730it is clustered closely around the mean.</p>731<p>If the optional second argument <em>mu</em> is given, it should be the <em>population</em>732mean of the <em>data</em>.  It can also be used to compute the second moment around733a point that is not the mean.  If it is missing or <code class="docutils literal notranslate"><span class="pre">None</span></code> (the default),734the arithmetic mean is automatically calculated.</p>735<p>Use this function to calculate the variance from the entire population.  To736estimate 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 usually737a better choice.</p>738<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>739<p>Examples:</p>740<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </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>741<span class="gp">&gt;&gt;&gt; </span><span class="n">pvariance</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>742<span class="go">1.25</span>743</pre></div>744</div>745<p>If you have already calculated the mean of your data, you can pass it as the746optional second argument <em>mu</em> to avoid recalculation:</p>747<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </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>748<span class="gp">&gt;&gt;&gt; </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>749<span class="go">1.25</span>750</pre></div>751</div>752<p>Decimals and Fractions are supported:</p>753<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span><span class="w"> </span><span class="nn">decimal</span><span class="w"> </span><span class="kn">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>754<span class="gp">&gt;&gt;&gt; </span><span class="n">pvariance</span><span class="p">([</span><span class="n">D</span><span class="p">(</span><span class="s2">&quot;27.5&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;30.25&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;30.25&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;34.5&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;41.75&quot;</span><span class="p">)])</span>755<span class="go">Decimal(&#39;24.815&#39;)</span>756 757<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span><span class="w"> </span><span class="nn">fractions</span><span class="w"> </span><span class="kn">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>758<span class="gp">&gt;&gt;&gt; </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>759<span class="go">Fraction(13, 72)</span>760</pre></div>761</div>762<div class="admonition note">763<p class="admonition-title">Note</p>764<p>When called with the entire population, this gives the population variance765σ².  When called on a sample instead, this is the biased sample variance766s², also known as variance with N degrees of freedom.</p>767<p>If you somehow know the true population mean μ, you may use this768function to calculate the variance of a sample, giving the known769population mean as the second argument.  Provided the data points are a770random sample of the population, the result will be an unbiased estimate771of the population variance.</p>772</div>773</dd></dl>774 775<dl class="py function">776<dt class="sig sig-object py" id="statistics.stdev">777<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">stdev</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">xbar</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.stdev" title="Link to this definition">¶</a></dt>778<dd><p>Return the sample standard deviation (the square root of the sample779variance).  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>780<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </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>781<span class="go">1.0810874155219827</span>782</pre></div>783</div>784</dd></dl>785 786<dl class="py function">787<dt class="sig sig-object py" id="statistics.variance">788<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">variance</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">xbar</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.variance" title="Link to this definition">¶</a></dt>789<dd><p>Return the sample variance of <em>data</em>, an iterable of at least two real-valued790numbers.  Variance, or second moment about the mean, is a measure of the791variability (spread or dispersion) of data.  A large variance indicates that792the data is spread out; a small variance indicates it is clustered closely793around the mean.</p>794<p>If the optional second argument <em>xbar</em> is given, it should be the <em>sample</em>795mean of <em>data</em>.  If it is missing or <code class="docutils literal notranslate"><span class="pre">None</span></code> (the default), the mean is796automatically calculated.</p>797<p>Use this function when your data is a sample from a population. To calculate798the 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>799<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>800<p>Examples:</p>801<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </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>802<span class="gp">&gt;&gt;&gt; </span><span class="n">variance</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>803<span class="go">1.3720238095238095</span>804</pre></div>805</div>806<p>If you have already calculated the sample mean of your data, you can pass it807as the optional second argument <em>xbar</em> to avoid recalculation:</p>808<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </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>809<span class="gp">&gt;&gt;&gt; </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>810<span class="go">1.3720238095238095</span>811</pre></div>812</div>813<p>This function does not attempt to verify that you have passed the actual mean814as <em>xbar</em>.  Using arbitrary values for <em>xbar</em> can lead to invalid or815impossible results.</p>816<p>Decimal and Fraction values are supported:</p>817<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span><span class="w"> </span><span class="nn">decimal</span><span class="w"> </span><span class="kn">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>818<span class="gp">&gt;&gt;&gt; </span><span class="n">variance</span><span class="p">([</span><span class="n">D</span><span class="p">(</span><span class="s2">&quot;27.5&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;30.25&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;30.25&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;34.5&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;41.75&quot;</span><span class="p">)])</span>819<span class="go">Decimal(&#39;31.01875&#39;)</span>820 821<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span><span class="w"> </span><span class="nn">fractions</span><span class="w"> </span><span class="kn">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>822<span class="gp">&gt;&gt;&gt; </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>823<span class="go">Fraction(67, 108)</span>824</pre></div>825</div>826<div class="admonition note">827<p class="admonition-title">Note</p>828<p>This is the sample variance s² with Bessel’s correction, also known as829variance with N-1 degrees of freedom.  Provided that the data points are830representative (e.g. independent and identically distributed), the result831should be an unbiased estimate of the true population variance.</p>832<p>If you somehow know the actual population mean μ you should pass it to the833<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 a834sample.</p>835</div>836</dd></dl>837 838<dl class="py function">839<dt class="sig sig-object py" id="statistics.quantiles">840<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">quantiles</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em>, <em class="sig-param"><span class="keyword-only-separator o"><abbr title="Keyword-only parameters separator (PEP 3102)"><span class="pre">*</span></abbr></span></em>, <em class="sig-param"><span class="n"><span class="pre">n</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">4</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">method</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'exclusive'</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.quantiles" title="Link to this definition">¶</a></dt>841<dd><p>Divide <em>data</em> into <em>n</em> continuous intervals with equal probability.842Returns a list of <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">-</span> <span class="pre">1</span></code> cut points separating the intervals.</p>843<p>Set <em>n</em> to 4 for quartiles (the default).  Set <em>n</em> to 10 for deciles.  Set844<em>n</em> to 100 for percentiles which gives the 99 cuts points that separate845<em>data</em> into 100 equal sized groups.  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>n</em>846is not least 1.</p>847<p>The <em>data</em> can be any iterable containing sample data.  For meaningful848results, the number of data points in <em>data</em> should be larger than <em>n</em>.849Raises <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 there is not at least one data point.</p>850<p>The cut points are linearly interpolated from the851two nearest data points.  For example, if a cut point falls one-third852of the distance between two sample values, <code class="docutils literal notranslate"><span class="pre">100</span></code> and <code class="docutils literal notranslate"><span class="pre">112</span></code>, the853cut-point will evaluate to <code class="docutils literal notranslate"><span class="pre">104</span></code>.</p>854<p>The <em>method</em> for computing quantiles can be varied depending on855whether the <em>data</em> includes or excludes the lowest and856highest possible values from the population.</p>857<p>The default <em>method</em> is “exclusive” and is used for data sampled from858a population that can have more extreme values than found in the859samples.  The portion of the population falling below the <em>i-th</em> of860<em>m</em> sorted data points is computed as <code class="docutils literal notranslate"><span class="pre">i</span> <span class="pre">/</span> <span class="pre">(m</span> <span class="pre">+</span> <span class="pre">1)</span></code>.  Given nine861sample values, the method sorts them and assigns the following862percentiles: 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%.</p>863<p>Setting the <em>method</em> to “inclusive” is used for describing population864data or for samples that are known to include the most extreme values865from the population.  The minimum value in <em>data</em> is treated as the 0th866percentile and the maximum value is treated as the 100th percentile.867The portion of the population falling below the <em>i-th</em> of <em>m</em> sorted868data points is computed as <code class="docutils literal notranslate"><span class="pre">(i</span> <span class="pre">-</span> <span class="pre">1)</span> <span class="pre">/</span> <span class="pre">(m</span> <span class="pre">-</span> <span class="pre">1)</span></code>.  Given 11 sample869values, the method sorts them and assigns the following percentiles:8700%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 100%.</p>871<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="go"># Decile cut points for empirically sampled data</span>872<span class="gp">&gt;&gt;&gt; </span><span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="mi">105</span><span class="p">,</span> <span class="mi">129</span><span class="p">,</span> <span class="mi">87</span><span class="p">,</span> <span class="mi">86</span><span class="p">,</span> <span class="mi">111</span><span class="p">,</span> <span class="mi">111</span><span class="p">,</span> <span class="mi">89</span><span class="p">,</span> <span class="mi">81</span><span class="p">,</span> <span class="mi">108</span><span class="p">,</span> <span class="mi">92</span><span class="p">,</span> <span class="mi">110</span><span class="p">,</span>873<span class="gp">... </span>        <span class="mi">100</span><span class="p">,</span> <span class="mi">75</span><span class="p">,</span> <span class="mi">105</span><span class="p">,</span> <span class="mi">103</span><span class="p">,</span> <span class="mi">109</span><span class="p">,</span> <span class="mi">76</span><span class="p">,</span> <span class="mi">119</span><span class="p">,</span> <span class="mi">99</span><span class="p">,</span> <span class="mi">91</span><span class="p">,</span> <span class="mi">103</span><span class="p">,</span> <span class="mi">129</span><span class="p">,</span>874<span class="gp">... </span>        <span class="mi">106</span><span class="p">,</span> <span class="mi">101</span><span class="p">,</span> <span class="mi">84</span><span class="p">,</span> <span class="mi">111</span><span class="p">,</span> <span class="mi">74</span><span class="p">,</span> <span class="mi">87</span><span class="p">,</span> <span class="mi">86</span><span class="p">,</span> <span class="mi">103</span><span class="p">,</span> <span class="mi">103</span><span class="p">,</span> <span class="mi">106</span><span class="p">,</span> <span class="mi">86</span><span class="p">,</span>875<span class="gp">... </span>        <span class="mi">111</span><span class="p">,</span> <span class="mi">75</span><span class="p">,</span> <span class="mi">87</span><span class="p">,</span> <span class="mi">102</span><span class="p">,</span> <span class="mi">121</span><span class="p">,</span> <span class="mi">111</span><span class="p">,</span> <span class="mi">88</span><span class="p">,</span> <span class="mi">89</span><span class="p">,</span> <span class="mi">101</span><span class="p">,</span> <span class="mi">106</span><span class="p">,</span> <span class="mi">95</span><span class="p">,</span>876<span class="gp">... </span>        <span class="mi">103</span><span class="p">,</span> <span class="mi">107</span><span class="p">,</span> <span class="mi">101</span><span class="p">,</span> <span class="mi">81</span><span class="p">,</span> <span class="mi">109</span><span class="p">,</span> <span class="mi">104</span><span class="p">]</span>877<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="nb">round</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">quantiles</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">10</span><span class="p">)]</span>878<span class="go">[81.0, 86.2, 89.0, 99.4, 102.5, 103.6, 106.0, 109.8, 111.0]</span>879</pre></div>880</div>881<div class="versionadded">882<p><span class="versionmodified added">Added in version 3.8.</span></p>883</div>884<div class="versionchanged">885<p><span class="versionmodified changed">Changed in version 3.13: </span>No longer raises an exception for an input with only a single data point.886This allows quantile estimates to be built up one sample point887at a time becoming gradually more refined with each new data point.</p>888</div>889</dd></dl>890 891<dl class="py function">892<dt class="sig sig-object py" id="statistics.covariance">893<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">covariance</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span></em>, <em class="sig-param"><span class="positional-only-separator o"><abbr title="Positional-only parameter separator (PEP 570)"><span class="pre">/</span></abbr></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.covariance" title="Link to this definition">¶</a></dt>894<dd><p>Return the sample covariance of two inputs <em>x</em> and <em>y</em>. Covariance895is a measure of the joint variability of two inputs.</p>896<p>Both inputs must be of the same length (no less than two), otherwise897<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>898<p>Examples:</p>899<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</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">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">]</span>900<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</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">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">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>901<span class="gp">&gt;&gt;&gt; </span><span class="n">covariance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>902<span class="go">0.75</span>903<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="p">[</span><span class="mi">9</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">6</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="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>904<span class="gp">&gt;&gt;&gt; </span><span class="n">covariance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>905<span class="go">-7.5</span>906<span class="gp">&gt;&gt;&gt; </span><span class="n">covariance</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>907<span class="go">-7.5</span>908</pre></div>909</div>910<div class="versionadded">911<p><span class="versionmodified added">Added in version 3.10.</span></p>912</div>913</dd></dl>914 915<dl class="py function">916<dt class="sig sig-object py" id="statistics.correlation">917<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">correlation</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span></em>, <em class="sig-param"><span class="positional-only-separator o"><abbr title="Positional-only parameter separator (PEP 570)"><span class="pre">/</span></abbr></span></em>, <em class="sig-param"><span class="keyword-only-separator o"><abbr title="Keyword-only parameters separator (PEP 3102)"><span class="pre">*</span></abbr></span></em>, <em class="sig-param"><span class="n"><span class="pre">method</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'linear'</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.correlation" title="Link to this definition">¶</a></dt>918<dd><p>Return the <a class="reference external" href="https://en.wikipedia.org/wiki/Pearson_correlation_coefficient">Pearson’s correlation coefficient</a>919for two inputs. Pearson’s correlation coefficient <em>r</em> takes values920between -1 and +1. It measures the strength and direction of a linear921relationship.</p>922<p>If <em>method</em> is “ranked”, computes <a class="reference external" href="https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient">Spearman’s rank correlation coefficient</a>923for two inputs. The data is replaced by ranks.  Ties are averaged so that924equal values receive the same rank.  The resulting coefficient measures the925strength of a monotonic relationship.</p>926<p>Spearman’s correlation coefficient is appropriate for ordinal data or for927continuous data that doesn’t meet the linear proportion requirement for928Pearson’s correlation coefficient.</p>929<p>Both inputs must be of the same length (no less than two), and need930not to be constant, otherwise <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>931<p>Example with <a class="reference external" href="https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion">Kepler’s laws of planetary motion</a>:</p>932<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="c1"># Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and  Neptune</span>933<span class="gp">&gt;&gt;&gt; </span><span class="n">orbital_period</span> <span class="o">=</span> <span class="p">[</span><span class="mi">88</span><span class="p">,</span> <span class="mi">225</span><span class="p">,</span> <span class="mi">365</span><span class="p">,</span> <span class="mi">687</span><span class="p">,</span> <span class="mi">4331</span><span class="p">,</span> <span class="mi">10_756</span><span class="p">,</span> <span class="mi">30_687</span><span class="p">,</span> <span class="mi">60_190</span><span class="p">]</span>    <span class="c1"># days</span>934<span class="gp">&gt;&gt;&gt; </span><span class="n">dist_from_sun</span> <span class="o">=</span> <span class="p">[</span><span class="mi">58</span><span class="p">,</span> <span class="mi">108</span><span class="p">,</span> <span class="mi">150</span><span class="p">,</span> <span class="mi">228</span><span class="p">,</span> <span class="mi">778</span><span class="p">,</span> <span class="mi">1_400</span><span class="p">,</span> <span class="mi">2_900</span><span class="p">,</span> <span class="mi">4_500</span><span class="p">]</span> <span class="c1"># million km</span>935 936<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Show that a perfect monotonic relationship exists</span>937<span class="gp">&gt;&gt;&gt; </span><span class="n">correlation</span><span class="p">(</span><span class="n">orbital_period</span><span class="p">,</span> <span class="n">dist_from_sun</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s1">&#39;ranked&#39;</span><span class="p">)</span>938<span class="go">1.0</span>939 940<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Observe that a linear relationship is imperfect</span>941<span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">correlation</span><span class="p">(</span><span class="n">orbital_period</span><span class="p">,</span> <span class="n">dist_from_sun</span><span class="p">),</span> <span class="mi">4</span><span class="p">)</span>942<span class="go">0.9882</span>943 944<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Demonstrate Kepler&#39;s third law: There is a linear correlation</span>945<span class="gp">&gt;&gt;&gt; </span><span class="c1"># between the square of the orbital period and the cube of the</span>946<span class="gp">&gt;&gt;&gt; </span><span class="c1"># distance from the sun.</span>947<span class="gp">&gt;&gt;&gt; </span><span class="n">period_squared</span> <span class="o">=</span> <span class="p">[</span><span class="n">p</span> <span class="o">*</span> <span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">orbital_period</span><span class="p">]</span>948<span class="gp">&gt;&gt;&gt; </span><span class="n">dist_cubed</span> <span class="o">=</span> <span class="p">[</span><span class="n">d</span> <span class="o">*</span> <span class="n">d</span> <span class="o">*</span> <span class="n">d</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">dist_from_sun</span><span class="p">]</span>949<span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">correlation</span><span class="p">(</span><span class="n">period_squared</span><span class="p">,</span> <span class="n">dist_cubed</span><span class="p">),</span> <span class="mi">4</span><span class="p">)</span>950<span class="go">1.0</span>951</pre></div>952</div>953<div class="versionadded">954<p><span class="versionmodified added">Added in version 3.10.</span></p>955</div>956<div class="versionchanged">957<p><span class="versionmodified changed">Changed in version 3.12: </span>Added support for Spearman’s rank correlation coefficient.</p>958</div>959</dd></dl>960 961<dl class="py function">962<dt class="sig sig-object py" id="statistics.linear_regression">963<span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">linear_regression</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span></em>, <em class="sig-param"><span class="positional-only-separator o"><abbr title="Positional-only parameter separator (PEP 570)"><span class="pre">/</span></abbr></span></em>, <em class="sig-param"><span class="keyword-only-separator o"><abbr title="Keyword-only parameters separator (PEP 3102)"><span class="pre">*</span></abbr></span></em>, <em class="sig-param"><span class="n"><span class="pre">proportional</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.linear_regression" title="Link to this definition">¶</a></dt>964<dd><p>Return the slope and intercept of <a class="reference external" href="https://en.wikipedia.org/wiki/Simple_linear_regression">simple linear regression</a>965parameters estimated using ordinary least squares. Simple linear966regression describes the relationship between an independent variable <em>x</em> and967a dependent variable <em>y</em> in terms of this linear function:</p>968<blockquote>969<div><p><em>y = slope * x + intercept + noise</em></p>970</div></blockquote>971<p>where <code class="docutils literal notranslate"><span class="pre">slope</span></code> and <code class="docutils literal notranslate"><span class="pre">intercept</span></code> are the regression parameters that are972estimated, and <code class="docutils literal notranslate"><span class="pre">noise</span></code> represents the973variability of the data that was not explained by the linear regression974(it is equal to the difference between predicted and actual values975of the dependent variable).</p>976<p>Both inputs must be of the same length (no less than two), and977the independent variable <em>x</em> cannot be constant;978otherwise a <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>979<p>For example, we can use the <a class="reference external" href="https://en.wikipedia.org/wiki/Monty_Python#Films">release dates of the Monty980Python films</a>981to predict the cumulative number of Monty Python films982that would have been produced by 2019983assuming that they had kept the pace.</p>984<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">year</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1971</span><span class="p">,</span> <span class="mi">1975</span><span class="p">,</span> <span class="mi">1979</span><span class="p">,</span> <span class="mi">1982</span><span class="p">,</span> <span class="mi">1983</span><span class="p">]</span>985<span class="gp">&gt;&gt;&gt; </span><span class="n">films_total</span> <span class="o">=</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">5</span><span class="p">]</span>986<span class="gp">&gt;&gt;&gt; </span><span class="n">slope</span><span class="p">,</span> <span class="n">intercept</span> <span class="o">=</span> <span class="n">linear_regression</span><span class="p">(</span><span class="n">year</span><span class="p">,</span> <span class="n">films_total</span><span class="p">)</span>987<span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">slope</span> <span class="o">*</span> <span class="mi">2019</span> <span class="o">+</span> <span class="n">intercept</span><span class="p">)</span>988<span class="go">16</span>989</pre></div>990</div>991<p>If <em>proportional</em> is true, the independent variable <em>x</em> and the992dependent variable <em>y</em> are assumed to be directly proportional.993The data is fit to a line passing through the origin.994Since the <em>intercept</em> will always be 0.0, the underlying linear995function simplifies to:</p>996<blockquote>997<div><p><em>y = slope * x + noise</em></p>998</div></blockquote>999<p>Continuing the example from <a class="reference internal" href="#statistics.correlation" title="statistics.correlation"><code class="xref py py-func docutils literal notranslate"><span class="pre">correlation()</span></code></a>, we look to see1000how well a model based on major planets can predict the orbital1001distances for dwarf planets:</p>1002<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">model</span> <span class="o">=</span> <span class="n">linear_regression</span><span class="p">(</span><span class="n">period_squared</span><span class="p">,</span> <span class="n">dist_cubed</span><span class="p">,</span> <span class="n">proportional</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>1003<span class="gp">&gt;&gt;&gt; </span><span class="n">slope</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">slope</span>1004 1005<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Dwarf planets:   Pluto,  Eris,    Makemake, Haumea, Ceres</span>1006<span class="gp">&gt;&gt;&gt; </span><span class="n">orbital_periods</span> <span class="o">=</span> <span class="p">[</span><span class="mi">90_560</span><span class="p">,</span> <span class="mi">204_199</span><span class="p">,</span> <span class="mi">111_845</span><span class="p">,</span> <span class="mi">103_410</span><span class="p">,</span> <span class="mi">1_680</span><span class="p">]</span>  <span class="c1"># days</span>1007<span class="gp">&gt;&gt;&gt; </span><span class="n">predicted_dist</span> <span class="o">=</span> <span class="p">[</span><span class="n">math</span><span class="o">.</span><span class="n">cbrt</span><span class="p">(</span><span class="n">slope</span> <span class="o">*</span> <span class="p">(</span><span class="n">p</span> <span class="o">*</span> <span class="n">p</span><span class="p">))</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">orbital_periods</span><span class="p">]</span>1008<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">round</span><span class="p">,</span> <span class="n">predicted_dist</span><span class="p">))</span>1009<span class="go">[5912, 10166, 6806, 6459, 414]</span>1010 1011<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="mi">5_906</span><span class="p">,</span> <span class="mi">10_152</span><span class="p">,</span> <span class="mi">6_796</span><span class="p">,</span> <span class="mi">6_450</span><span class="p">,</span> <span class="mi">414</span><span class="p">]</span>  <span class="c1"># actual distance in million km</span>1012<span class="go">[5906, 10152, 6796, 6450, 414]</span>1013</pre></div>1014</div>1015<div class="versionadded">1016<p><span class="versionmodified added">Added in version 3.10.</span></p>1017</div>1018<div class="versionchanged">1019<p><span class="versionmodified changed">Changed in version 3.11: </span>Added support for <em>proportional</em>.</p>1020</div>1021</dd></dl>1022 1023</section>1024<section id="exceptions">1025<h2>Exceptions<a class="headerlink" href="#exceptions" title="Link to this heading">¶</a></h2>1026<p>A single exception is defined:</p>1027<dl class="py exception">1028<dt class="sig sig-object py" id="statistics.StatisticsError">1029<em class="property"><span class="k"><span class="pre">exception</span></span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">StatisticsError</span></span><a class="headerlink" href="#statistics.StatisticsError" title="Link to this definition">¶</a></dt>1030<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>1031</dd></dl>1032 1033</section>1034<section id="normaldist-objects">1035<h2><a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> objects<a class="headerlink" href="#normaldist-objects" title="Link to this heading">¶</a></h2>1036<p><a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> is a tool for creating and manipulating normal1037distributions of a <a class="reference external" href="http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm">random variable</a>.  It is a1038class that treats the mean and standard deviation of data1039measurements as a single entity.</p>1040<p>Normal distributions arise from the <a class="reference external" href="https://en.wikipedia.org/wiki/Central_limit_theorem">Central Limit Theorem</a> and have a wide range1041of applications in statistics.</p>1042<dl class="py class">1043<dt class="sig sig-object py" id="statistics.NormalDist">1044<em class="property"><span class="k"><span class="pre">class</span></span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">statistics.</span></span><span class="sig-name descname"><span class="pre">NormalDist</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">mu</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sigma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist" title="Link to this definition">¶</a></dt>1045<dd><p>Returns a new <em>NormalDist</em> object where <em>mu</em> represents the <a class="reference external" href="https://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic1046mean</a> and <em>sigma</em>1047represents the <a class="reference external" href="https://en.wikipedia.org/wiki/Standard_deviation">standard deviation</a>.</p>1048<p>If <em>sigma</em> is negative, 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>.</p>1049<dl class="py attribute">1050<dt class="sig sig-object py" id="statistics.NormalDist.mean">1051<span class="sig-name descname"><span class="pre">mean</span></span><a class="headerlink" href="#statistics.NormalDist.mean" title="Link to this definition">¶</a></dt>1052<dd><p>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of a normal1053distribution.</p>1054</dd></dl>1055 1056<dl class="py attribute">1057<dt class="sig sig-object py" id="statistics.NormalDist.median">1058<span class="sig-name descname"><span class="pre">median</span></span><a class="headerlink" href="#statistics.NormalDist.median" title="Link to this definition">¶</a></dt>1059<dd><p>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Median">median</a> of a normal1060distribution.</p>1061</dd></dl>1062 1063<dl class="py attribute">1064<dt class="sig sig-object py" id="statistics.NormalDist.mode">1065<span class="sig-name descname"><span class="pre">mode</span></span><a class="headerlink" href="#statistics.NormalDist.mode" title="Link to this definition">¶</a></dt>1066<dd><p>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Mode_(statistics)">mode</a> of a normal1067distribution.</p>1068</dd></dl>1069 1070<dl class="py attribute">1071<dt class="sig sig-object py" id="statistics.NormalDist.stdev">1072<span class="sig-name descname"><span class="pre">stdev</span></span><a class="headerlink" href="#statistics.NormalDist.stdev" title="Link to this definition">¶</a></dt>1073<dd><p>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Standard_deviation">standard deviation</a> of a normal1074distribution.</p>1075</dd></dl>1076 1077<dl class="py attribute">1078<dt class="sig sig-object py" id="statistics.NormalDist.variance">1079<span class="sig-name descname"><span class="pre">variance</span></span><a class="headerlink" href="#statistics.NormalDist.variance" title="Link to this definition">¶</a></dt>1080<dd><p>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Variance">variance</a> of a normal1081distribution. Equal to the square of the standard deviation.</p>1082</dd></dl>1083 1084<dl class="py method">1085<dt class="sig sig-object py" id="statistics.NormalDist.from_samples">1086<em class="property"><span class="k"><span class="pre">classmethod</span></span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">from_samples</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">data</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.from_samples" title="Link to this definition">¶</a></dt>1087<dd><p>Makes a normal distribution instance with <em>mu</em> and <em>sigma</em> parameters1088estimated from the <em>data</em> using <a class="reference internal" href="#statistics.fmean" title="statistics.fmean"><code class="xref py py-func docutils literal notranslate"><span class="pre">fmean()</span></code></a> and <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>1089<p>The <em>data</em> can be any <a class="reference internal" href="../glossary.html#term-iterable"><span class="xref std std-term">iterable</span></a> and should consist of values1090that can be converted to type <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>.  If <em>data</em> does not1091contain at least two elements, 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> because it1092takes at least one point to estimate a central value and at least two1093points to estimate dispersion.</p>1094</dd></dl>1095 1096<dl class="py method">1097<dt class="sig sig-object py" id="statistics.NormalDist.samples">1098<span class="sig-name descname"><span class="pre">samples</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em>, <em class="sig-param"><span class="keyword-only-separator o"><abbr title="Keyword-only parameters separator (PEP 3102)"><span class="pre">*</span></abbr></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.samples" title="Link to this definition">¶</a></dt>1099<dd><p>Generates <em>n</em> random samples for a given mean and standard deviation.1100Returns a <a class="reference internal" href="stdtypes.html#list" title="list"><code class="xref py py-class docutils literal notranslate"><span class="pre">list</span></code></a> of <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> values.</p>1101<p>If <em>seed</em> is given, creates a new instance of the underlying random1102number generator.  This is useful for creating reproducible results,1103even in a multi-threading context.</p>1104<div class="versionchanged">1105<p><span class="versionmodified changed">Changed in version 3.13.</span></p>1106</div>1107<p>Switched to a faster algorithm.  To reproduce samples from previous1108versions, use <a class="reference internal" href="random.html#random.seed" title="random.seed"><code class="xref py py-func docutils literal notranslate"><span class="pre">random.seed()</span></code></a> and <a class="reference internal" href="random.html#random.gauss" title="random.gauss"><code class="xref py py-func docutils literal notranslate"><span class="pre">random.gauss()</span></code></a>.</p>1109</dd></dl>1110 1111<dl class="py method">1112<dt class="sig sig-object py" id="statistics.NormalDist.pdf">1113<span class="sig-name descname"><span class="pre">pdf</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.pdf" title="Link to this definition">¶</a></dt>1114<dd><p>Using a <a class="reference external" href="https://en.wikipedia.org/wiki/Probability_density_function">probability density function (pdf)</a>, compute1115the relative likelihood that a random variable <em>X</em> will be near the1116given value <em>x</em>.  Mathematically, it is the limit of the ratio <code class="docutils literal notranslate"><span class="pre">P(x</span> <span class="pre">&lt;=</span>1117<span class="pre">X</span> <span class="pre">&lt;</span> <span class="pre">x+dx)</span> <span class="pre">/</span> <span class="pre">dx</span></code> as <em>dx</em> approaches zero.</p>1118<p>The relative likelihood is computed as the probability of a sample1119occurring in a narrow range divided by the width of the range (hence1120the word “density”).  Since the likelihood is relative to other points,1121its value can be greater than <code class="docutils literal notranslate"><span class="pre">1.0</span></code>.</p>1122</dd></dl>1123 1124<dl class="py method">1125<dt class="sig sig-object py" id="statistics.NormalDist.cdf">1126<span class="sig-name descname"><span class="pre">cdf</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.cdf" title="Link to this definition">¶</a></dt>1127<dd><p>Using a <a class="reference external" href="https://en.wikipedia.org/wiki/Cumulative_distribution_function">cumulative distribution function (cdf)</a>,1128compute the probability that a random variable <em>X</em> will be less than or1129equal to <em>x</em>.  Mathematically, it is written <code class="docutils literal notranslate"><span class="pre">P(X</span> <span class="pre">&lt;=</span> <span class="pre">x)</span></code>.</p>1130</dd></dl>1131 1132<dl class="py method">1133<dt class="sig sig-object py" id="statistics.NormalDist.inv_cdf">1134<span class="sig-name descname"><span class="pre">inv_cdf</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">p</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.inv_cdf" title="Link to this definition">¶</a></dt>1135<dd><p>Compute the inverse cumulative distribution function, also known as the1136<a class="reference external" href="https://en.wikipedia.org/wiki/Quantile_function">quantile function</a>1137or the <a class="reference external" href="https://web.archive.org/web/20190203145224/https://www.statisticshowto.datasciencecentral.com/inverse-distribution-function/">percent-point</a>1138function.  Mathematically, it is written <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">:</span> <span class="pre">P(X</span> <span class="pre">&lt;=</span> <span class="pre">x)</span> <span class="pre">=</span> <span class="pre">p</span></code>.</p>1139<p>Finds the value <em>x</em> of the random variable <em>X</em> such that the1140probability of the variable being less than or equal to that value1141equals the given probability <em>p</em>.</p>1142</dd></dl>1143 1144<dl class="py method">1145<dt class="sig sig-object py" id="statistics.NormalDist.overlap">1146<span class="sig-name descname"><span class="pre">overlap</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">other</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.overlap" title="Link to this definition">¶</a></dt>1147<dd><p>Measures the agreement between two normal probability distributions.1148Returns a value between 0.0 and 1.0 giving <a class="reference external" href="https://www.rasch.org/rmt/rmt101r.htm">the overlapping area for1149the two probability density functions</a>.</p>1150</dd></dl>1151 1152<dl class="py method">1153<dt class="sig sig-object py" id="statistics.NormalDist.quantiles">1154<span class="sig-name descname"><span class="pre">quantiles</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">4</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.quantiles" title="Link to this definition">¶</a></dt>1155<dd><p>Divide the normal distribution into <em>n</em> continuous intervals with1156equal probability.  Returns a list of (n - 1) cut points separating1157the intervals.</p>1158<p>Set <em>n</em> to 4 for quartiles (the default).  Set <em>n</em> to 10 for deciles.1159Set <em>n</em> to 100 for percentiles which gives the 99 cuts points that1160separate the normal distribution into 100 equal sized groups.</p>1161</dd></dl>1162 1163<dl class="py method">1164<dt class="sig sig-object py" id="statistics.NormalDist.zscore">1165<span class="sig-name descname"><span class="pre">zscore</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.zscore" title="Link to this definition">¶</a></dt>1166<dd><p>Compute the1167<a class="reference external" href="https://www.statisticshowto.com/probability-and-statistics/z-score/">Standard Score</a>1168describing <em>x</em> in terms of the number of standard deviations1169above or below the mean of the normal distribution:1170<code class="docutils literal notranslate"><span class="pre">(x</span> <span class="pre">-</span> <span class="pre">mean)</span> <span class="pre">/</span> <span class="pre">stdev</span></code>.</p>1171<div class="versionadded">1172<p><span class="versionmodified added">Added in version 3.9.</span></p>1173</div>1174</dd></dl>1175 1176<p>Instances of <code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code> support addition, subtraction,1177multiplication and division by a constant.  These operations1178are used for translation and scaling.  For example:</p>1179<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">temperature_february</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span>             <span class="c1"># Celsius</span>1180<span class="gp">&gt;&gt;&gt; </span><span class="n">temperature_february</span> <span class="o">*</span> <span class="p">(</span><span class="mi">9</span><span class="o">/</span><span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="mi">32</span>                     <span class="c1"># Fahrenheit</span>1181<span class="go">NormalDist(mu=41.0, sigma=4.5)</span>1182</pre></div>1183</div>1184<p>Dividing a constant by an instance of <code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code> is not supported1185because the result wouldn’t be normally distributed.</p>1186<p>Since normal distributions arise from additive effects of independent1187variables, it is possible to <a class="reference external" href="https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables">add and subtract two independent normally1188distributed random variables</a>1189represented as instances of <code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code>.  For example:</p>1190<div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">birth_weights</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mf">2.5</span><span class="p">,</span> <span class="mf">3.1</span><span class="p">,</span> <span class="mf">2.1</span><span class="p">,</span> <span class="mf">2.4</span><span class="p">,</span> <span class="mf">2.7</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">])</span>1191<span class="gp">&gt;&gt;&gt; </span><span class="n">drug_effects</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.15</span><span class="p">)</span>1192<span class="gp">&gt;&gt;&gt; </span><span class="n">combined</span> <span class="o">=</span> <span class="n">birth_weights</span> <span class="o">+</span> <span class="n">drug_effects</span>1193<span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">combined</span><span class="o">.</span><span class="n">mean</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>1194<span class="go">3.1</span>1195<span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">combined</span><span class="o">.</span><span class="n">stdev</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>1196<span class="go">0.5</span>1197</pre></div>1198</div>1199<div class="versionadded">1200<p><span class="versionmodified added">Added in version 3.8.</span></p>

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