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  99. <li><a class="reference internal" href="#averages-and-measures-of-central-location">Averages and measures of central location</a></li>

  100. <li><a class="reference internal" href="#measures-of-spread">Measures of spread</a></li>

  101. <li><a class="reference internal" href="#statistics-for-relations-between-two-inputs">Statistics for relations between two inputs</a></li>

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  205.   <section id="module-statistics">

  206. <span id="statistics-mathematical-statistics-functions"></span><h1><a class="reference internal" href="#module-statistics" title="statistics: Mathematical statistics functions"><code class="xref py py-mod docutils literal notranslate"><span class="pre">statistics</span></code></a> — Mathematical statistics functions<a class="headerlink" href="#module-statistics" title="Link to this heading">¶</a></h1>

  207. <div class="versionadded">

  208. <p><span class="versionmodified added">New in version 3.4.</span></p>

  209. </div>

  210. <p><strong>Source code:</strong> <a class="reference external" href="https://github.com/python/cpython/tree/3.12/Lib/statistics.py">Lib/statistics.py</a></p>

  211. <hr class="docutils" />

  212. <p>This module provides functions for calculating mathematical statistics of

  213. numeric (<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>

  214. <p>The module is not intended to be a competitor to third-party libraries such

  215. as <a class="reference external" href="https://numpy.org">NumPy</a>, <a class="reference external" href="https://scipy.org/">SciPy</a>, or

  216. proprietary full-featured statistics packages aimed at professional

  217. statisticians such as Minitab, SAS and Matlab. It is aimed at the level of

  218. graphing and scientific calculators.</p>

  219. <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>,

  220. <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>.

  221. Behaviour with other types (whether in the numeric tower or not) is

  222. currently unsupported.  Collections with a mix of types are also undefined

  223. and implementation-dependent.  If your input data consists of mixed types,

  224. you may be able to use <a class="reference internal" href="functions.html#map" title="map"><code class="xref py py-func docutils literal notranslate"><span class="pre">map()</span></code></a> to ensure a consistent result, for

  225. example: <code class="docutils literal notranslate"><span class="pre">map(float,</span> <span class="pre">input_data)</span></code>.</p>

  226. <p>Some datasets use <code class="docutils literal notranslate"><span class="pre">NaN</span></code> (not a number) values to represent missing data.

  227. Since NaNs have unusual comparison semantics, they cause surprising or

  228. undefined behaviors in the statistics functions that sort data or that count

  229. occurrences.  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>,

  230. <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>, and

  231. <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 these

  232. functions:</p>

  233. <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="nn">statistics</span> <span class="kn">import</span> <span class="n">median</span>

  234. <span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">isnan</span>

  235. <span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">filterfalse</span>

  236. 

  237. <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>

  238. <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>

  239. <span class="go">[20.7, nan, 14.4, 18.3, 19.2, nan]</span>

  240. <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>

  241. <span class="go">16.35</span>

  242. 

  243. <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>

  244. <span class="go">2</span>

  245. <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>

  246. <span class="gp">&gt;&gt;&gt; </span><span class="n">clean</span>

  247. <span class="go">[20.7, 19.2, 18.3, 14.4]</span>

  248. <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>

  249. <span class="go">[14.4, 18.3, 19.2, 20.7]</span>

  250. <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>

  251. <span class="go">18.75</span>

  252. </pre></div>

  253. </div>

  254. <section id="averages-and-measures-of-central-location">

  255. <h2>Averages and measures of central location<a class="headerlink" href="#averages-and-measures-of-central-location" title="Link to this heading">¶</a></h2>

  256. <p>These functions calculate an average or typical value from a population

  257. or sample.</p>

  258. <table class="docutils align-default">

  259. <tbody>

  260. <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>

  261. <td><p>Arithmetic mean (“average”) of data.</p></td>

  262. </tr>

  263. <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>

  264. <td><p>Fast, floating point arithmetic mean, with optional weighting.</p></td>

  265. </tr>

  266. <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>

  267. <td><p>Geometric mean of data.</p></td>

  268. </tr>

  269. <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>

  270. <td><p>Harmonic mean of data.</p></td>

  271. </tr>

  272. <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>

  273. <td><p>Median (middle value) of data.</p></td>

  274. </tr>

  275. <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>

  276. <td><p>Low median of data.</p></td>

  277. </tr>

  278. <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>

  279. <td><p>High median of data.</p></td>

  280. </tr>

  281. <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>

  282. <td><p>Median (50th percentile) of grouped data.</p></td>

  283. </tr>

  284. <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>

  285. <td><p>Single mode (most common value) of discrete or nominal data.</p></td>

  286. </tr>

  287. <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>

  288. <td><p>List of modes (most common values) of discrete or nominal data.</p></td>

  289. </tr>

  290. <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>

  291. <td><p>Divide data into intervals with equal probability.</p></td>

  292. </tr>

  293. </tbody>

  294. </table>

  295. </section>

  296. <section id="measures-of-spread">

  297. <h2>Measures of spread<a class="headerlink" href="#measures-of-spread" title="Link to this heading">¶</a></h2>

  298. <p>These functions calculate a measure of how much the population or sample

  299. tends to deviate from the typical or average values.</p>

  300. <table class="docutils align-default">

  301. <tbody>

  302. <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>

  303. <td><p>Population standard deviation of data.</p></td>

  304. </tr>

  305. <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>

  306. <td><p>Population variance of data.</p></td>

  307. </tr>

  308. <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>

  309. <td><p>Sample standard deviation of data.</p></td>

  310. </tr>

  311. <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>

  312. <td><p>Sample variance of data.</p></td>

  313. </tr>

  314. </tbody>

  315. </table>

  316. </section>

  317. <section id="statistics-for-relations-between-two-inputs">

  318. <h2>Statistics for relations between two inputs<a class="headerlink" href="#statistics-for-relations-between-two-inputs" title="Link to this heading">¶</a></h2>

  319. <p>These functions calculate statistics regarding relations between two inputs.</p>

  320. <table class="docutils align-default">

  321. <tbody>

  322. <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>

  323. <td><p>Sample covariance for two variables.</p></td>

  324. </tr>

  325. <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>

  326. <td><p>Pearson and Spearman’s correlation coefficients.</p></td>

  327. </tr>

  328. <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>

  329. <td><p>Slope and intercept for simple linear regression.</p></td>

  330. </tr>

  331. </tbody>

  332. </table>

  333. </section>

  334. <section id="function-details">

  335. <h2>Function details<a class="headerlink" href="#function-details" title="Link to this heading">¶</a></h2>

  336. <p>Note: The functions do not require the data given to them to be sorted.

  337. However, for reading convenience, most of the examples show sorted sequences.</p>

  338. <dl class="py function">

  339. <dt class="sig sig-object py" id="statistics.mean">

  340. <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>

  341. <dd><p>Return the sample arithmetic mean of <em>data</em> which can be a sequence or iterable.</p>

  342. <p>The arithmetic mean is the sum of the data divided by the number of data

  343. points.  It is commonly called “the average”, although it is only one of many

  344. different mathematical averages.  It is a measure of the central location of

  345. the data.</p>

  346. <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>

  347. <p>Some examples of use:</p>

  348. <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>

  349. <span class="go">2.8</span>

  350. <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>

  351. <span class="go">2.625</span>

  352. 

  353. <span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="kn">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>

  354. <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>

  355. <span class="go">Fraction(13, 21)</span>

  356. 

  357. <span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="kn">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>

  358. <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>

  359. <span class="go">Decimal(&#39;0.5625&#39;)</span>

  360. </pre></div>

  361. </div>

  362. <div class="admonition note">

  363. <p class="admonition-title">Note</p>

  364. <p>The mean is strongly affected by <a class="reference external" href="https://en.wikipedia.org/wiki/Outlier">outliers</a> and is not necessarily a

  365. typical example of the data points. For a more robust, although less

  366. efficient, 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>

  367. <p>The sample mean gives an unbiased estimate of the true population mean,

  368. so that when taken on average over all the possible samples,

  369. <code class="docutils literal notranslate"><span class="pre">mean(sample)</span></code> converges on the true mean of the entire population.  If

  370. <em>data</em> represents the entire population rather than a sample, then

  371. <code class="docutils literal notranslate"><span class="pre">mean(data)</span></code> is equivalent to calculating the true population mean μ.</p>

  372. </div>

  373. </dd></dl>

  374. 

  375. <dl class="py function">

  376. <dt class="sig sig-object py" id="statistics.fmean">

  377. <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>

  378. <dd><p>Convert <em>data</em> to floats and compute the arithmetic mean.</p>

  379. <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 a

  380. <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 input

  381. dataset 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>

  382. <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>

  383. <span class="go">4.25</span>

  384. </pre></div>

  385. </div>

  386. <p>Optional weighting is supported.  For example, a professor assigns a

  387. grade for a course by weighting quizzes at 20%, homework at 20%, a

  388. midterm exam at 30%, and a final exam at 30%:</p>

  389. <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>

  390. <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>

  391. <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>

  392. <span class="go">87.6</span>

  393. </pre></div>

  394. </div>

  395. <p>If <em>weights</em> is supplied, it must be the same length as the <em>data</em> or

  396. a <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>

  397. <div class="versionadded">

  398. <p><span class="versionmodified added">New in version 3.8.</span></p>

  399. </div>

  400. <div class="versionchanged">

  401. <p><span class="versionmodified changed">Changed in version 3.11: </span>Added support for <em>weights</em>.</p>

  402. </div>

  403. </dd></dl>

  404. 

  405. <dl class="py function">

  406. <dt class="sig sig-object py" id="statistics.geometric_mean">

  407. <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>

  408. <dd><p>Convert <em>data</em> to floats and compute the geometric mean.</p>

  409. <p>The geometric mean indicates the central tendency or typical value of the

  410. <em>data</em> using the product of the values (as opposed to the arithmetic mean

  411. which uses their sum).</p>

  412. <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,

  413. if it contains a zero, or if it contains a negative value.

  414. The <em>data</em> may be a sequence or iterable.</p>

  415. <p>No special efforts are made to achieve exact results.

  416. (However, this may change in the future.)</p>

  417. <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>

  418. <span class="go">36.0</span>

  419. </pre></div>

  420. </div>

  421. <div class="versionadded">

  422. <p><span class="versionmodified added">New in version 3.8.</span></p>

  423. </div>

  424. </dd></dl>

  425. 

  426. <dl class="py function">

  427. <dt class="sig sig-object py" id="statistics.harmonic_mean">

  428. <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>

  429. <dd><p>Return the harmonic mean of <em>data</em>, a sequence or iterable of

  430. real-valued numbers.  If <em>weights</em> is omitted or <em>None</em>, then

  431. equal weighting is assumed.</p>

  432. <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 the

  433. reciprocals of the data. For example, the harmonic mean of three values <em>a</em>,

  434. <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 the

  435. values is zero, the result will be zero.</p>

  436. <p>The harmonic mean is a type of average, a measure of the central

  437. location of the data.  It is often appropriate when averaging

  438. ratios or rates, for example speeds.</p>

  439. <p>Suppose a car travels 10 km at 40 km/hr, then another 10 km at 60 km/hr.

  440. What is the average speed?</p>

  441. <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>

  442. <span class="go">48.0</span>

  443. </pre></div>

  444. </div>

  445. <p>Suppose a car travels 40 km/hr for 5 km, and when traffic clears,

  446. speeds-up to 60 km/hr for the remaining 30 km of the journey. What

  447. is the average speed?</p>

  448. <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>

  449. <span class="go">56.0</span>

  450. </pre></div>

  451. </div>

  452. <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 element

  453. is less than zero, or if the weighted sum isn’t positive.</p>

  454. <p>The current algorithm has an early-out when it encounters a zero

  455. in the input.  This means that the subsequent inputs are not tested

  456. for validity.  (This behavior may change in the future.)</p>

  457. <div class="versionadded">

  458. <p><span class="versionmodified added">New in version 3.6.</span></p>

  459. </div>

  460. <div class="versionchanged">

  461. <p><span class="versionmodified changed">Changed in version 3.10: </span>Added support for <em>weights</em>.</p>

  462. </div>

  463. </dd></dl>

  464. 

  465. <dl class="py function">

  466. <dt class="sig sig-object py" id="statistics.median">

  467. <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>

  468. <dd><p>Return the median (middle value) of numeric data, using the common “mean of

  469. middle two” method.  If <em>data</em> is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> is raised.

  470. <em>data</em> can be a sequence or iterable.</p>

  471. <p>The median is a robust measure of central location and is less affected by

  472. the presence of outliers.  When the number of data points is odd, the

  473. middle data point is returned:</p>

  474. <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>

  475. <span class="go">3</span>

  476. </pre></div>

  477. </div>

  478. <p>When the number of data points is even, the median is interpolated by taking

  479. the average of the two middle values:</p>

  480. <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>

  481. <span class="go">4.0</span>

  482. </pre></div>

  483. </div>

  484. <p>This is suited for when your data is discrete, and you don’t mind that the

  485. median may not be an actual data point.</p>

  486. <p>If the data is ordinal (supports order operations) but not numeric (doesn’t

  487. support 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>

  488. instead.</p>

  489. </dd></dl>

  490. 

  491. <dl class="py function">

  492. <dt class="sig sig-object py" id="statistics.median_low">

  493. <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>

  494. <dd><p>Return the low median of numeric data.  If <em>data</em> is empty,

  495. <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>

  496. <p>The low median is always a member of the data set.  When the number of data

  497. points is odd, the middle value is returned.  When it is even, the smaller of

  498. the two middle values is returned.</p>

  499. <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>

  500. <span class="go">3</span>

  501. <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>

  502. <span class="go">3</span>

  503. </pre></div>

  504. </div>

  505. <p>Use the low median when your data are discrete and you prefer the median to

  506. be an actual data point rather than interpolated.</p>

  507. </dd></dl>

  508. 

  509. <dl class="py function">

  510. <dt class="sig sig-object py" id="statistics.median_high">

  511. <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>

  512. <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>

  513. is raised.  <em>data</em> can be a sequence or iterable.</p>

  514. <p>The high median is always a member of the data set.  When the number of data

  515. points is odd, the middle value is returned.  When it is even, the larger of

  516. the two middle values is returned.</p>

  517. <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>

  518. <span class="go">3</span>

  519. <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>

  520. <span class="go">5</span>

  521. </pre></div>

  522. </div>

  523. <p>Use the high median when your data are discrete and you prefer the median to

  524. be an actual data point rather than interpolated.</p>

  525. </dd></dl>

  526. 

  527. <dl class="py function">

  528. <dt class="sig sig-object py" id="statistics.median_grouped">

  529. <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>

  530. <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 midpoints

  531. of consecutive, fixed-width intervals.</p>

  532. <p>The <em>data</em> can be any iterable of numeric data with each value being

  533. exactly the midpoint of a bin.  At least one value must be present.</p>

  534. <p>The <em>interval</em> is the width of each bin.</p>

  535. <p>For example, demographic information may have been summarized into

  536. consecutive ten-year age groups with each group being represented

  537. by the 5-year midpoints of the intervals:</p>

  538. <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="nn">collections</span> <span class="kn">import</span> <span class="n">Counter</span>

  539. <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>

  540. <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>

  541. <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>

  542. <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>

  543. <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>

  544. <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>

  545. <span class="gp">... </span><span class="p">})</span>

  546. <span class="gp">...</span>

  547. </pre></div>

  548. </div>

  549. <p>The 50th percentile (median) is the 536th person out of the 1071

  550. member cohort.  That person is in the 30 to 40 year old age group.</p>

  551. <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 the

  552. tricenarian age group was exactly 35 years old.  A more tenable

  553. assumption is that the 484 members of that age group are evenly

  554. distributed between 30 and 40.  For that, we use

  555. <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>

  556. <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>

  557. <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>

  558. <span class="go">35</span>

  559. <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>

  560. <span class="go">37.5</span>

  561. </pre></div>

  562. </div>

  563. <p>The caller is responsible for making sure the data points are separated

  564. by exact multiples of <em>interval</em>.  This is essential for getting a

  565. correct result.  The function does not check this precondition.</p>

  566. <p>Inputs may be any numeric type that can be coerced to a float during

  567. the interpolation step.</p>

  568. </dd></dl>

  569. 

  570. <dl class="py function">

  571. <dt class="sig sig-object py" id="statistics.mode">

  572. <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>

  573. <dd><p>Return the single most common data point from discrete or nominal <em>data</em>.

  574. The mode (when it exists) is the most typical value and serves as a

  575. measure of central location.</p>

  576. <p>If there are multiple modes with the same frequency, returns the first one

  577. encountered in the <em>data</em>.  If the smallest or largest of those is

  578. desired 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>.

  579. If 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>

  580. <p><code class="docutils literal notranslate"><span class="pre">mode</span></code> assumes discrete data and returns a single value. This is the

  581. standard treatment of the mode as commonly taught in schools:</p>

  582. <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>

  583. <span class="go">3</span>

  584. </pre></div>

  585. </div>

  586. <p>The mode is unique in that it is the only statistic in this package that

  587. also applies to nominal (non-numeric) data:</p>

  588. <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>

  589. <span class="go">&#39;red&#39;</span>

  590. </pre></div>

  591. </div>

  592. <div class="versionchanged">

  593. <p><span class="versionmodified changed">Changed in version 3.8: </span>Now handles multimodal datasets by returning the first mode encountered.

  594. Formerly, 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 was

  595. found.</p>

  596. </div>

  597. </dd></dl>

  598. 

  599. <dl class="py function">

  600. <dt class="sig sig-object py" id="statistics.multimode">

  601. <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>

  602. <dd><p>Return a list of the most frequently occurring values in the order they

  603. were first encountered in the <em>data</em>.  Will return more than one result if

  604. there are multiple modes or an empty list if the <em>data</em> is empty:</p>

  605. <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>

  606. <span class="go">[&#39;b&#39;, &#39;d&#39;, &#39;f&#39;]</span>

  607. <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>

  608. <span class="go">[]</span>

  609. </pre></div>

  610. </div>

  611. <div class="versionadded">

  612. <p><span class="versionmodified added">New in version 3.8.</span></p>

  613. </div>

  614. </dd></dl>

  615. 

  616. <dl class="py function">

  617. <dt class="sig sig-object py" id="statistics.pstdev">

  618. <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>

  619. <dd><p>Return the population standard deviation (the square root of the population

  620. variance).  See <a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a> for arguments and other details.</p>

  621. <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>

  622. <span class="go">0.986893273527251</span>

  623. </pre></div>

  624. </div>

  625. </dd></dl>

  626. 

  627. <dl class="py function">

  628. <dt class="sig sig-object py" id="statistics.pvariance">

  629. <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>

  630. <dd><p>Return the population variance of <em>data</em>, a non-empty sequence or iterable

  631. of real-valued numbers.  Variance, or second moment about the mean, is a

  632. measure of the variability (spread or dispersion) of data.  A large

  633. variance indicates that the data is spread out; a small variance indicates

  634. it is clustered closely around the mean.</p>

  635. <p>If the optional second argument <em>mu</em> is given, it is typically the mean of

  636. the <em>data</em>.  It can also be used to compute the second moment around a

  637. point that is not the mean.  If it is missing or <code class="docutils literal notranslate"><span class="pre">None</span></code> (the default),

  638. the arithmetic mean is automatically calculated.</p>

  639. <p>Use this function to calculate the variance from the entire population.  To

  640. estimate the variance from a sample, the <a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate"><span class="pre">variance()</span></code></a> function is usually

  641. a better choice.</p>

  642. <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>

  643. <p>Examples:</p>

  644. <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>

  645. <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>

  646. <span class="go">1.25</span>

  647. </pre></div>

  648. </div>

  649. <p>If you have already calculated the mean of your data, you can pass it as the

  650. optional second argument <em>mu</em> to avoid recalculation:</p>

  651. <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>

  652. <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>

  653. <span class="go">1.25</span>

  654. </pre></div>

  655. </div>

  656. <p>Decimals and Fractions are supported:</p>

  657. <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="nn">decimal</span> <span class="kn">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>

  658. <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>

  659. <span class="go">Decimal(&#39;24.815&#39;)</span>

  660. 

  661. <span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="kn">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>

  662. <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>

  663. <span class="go">Fraction(13, 72)</span>

  664. </pre></div>

  665. </div>

  666. <div class="admonition note">

  667. <p class="admonition-title">Note</p>

  668. <p>When called with the entire population, this gives the population variance

  669. σ².  When called on a sample instead, this is the biased sample variance

  670. s², also known as variance with N degrees of freedom.</p>

  671. <p>If you somehow know the true population mean μ, you may use this

  672. function to calculate the variance of a sample, giving the known

  673. population mean as the second argument.  Provided the data points are a

  674. random sample of the population, the result will be an unbiased estimate

  675. of the population variance.</p>

  676. </div>

  677. </dd></dl>

  678. 

  679. <dl class="py function">

  680. <dt class="sig sig-object py" id="statistics.stdev">

  681. <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>

  682. <dd><p>Return the sample standard deviation (the square root of the sample

  683. variance).  See <a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate"><span class="pre">variance()</span></code></a> for arguments and other details.</p>

  684. <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>

  685. <span class="go">1.0810874155219827</span>

  686. </pre></div>

  687. </div>

  688. </dd></dl>

  689. 

  690. <dl class="py function">

  691. <dt class="sig sig-object py" id="statistics.variance">

  692. <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>

  693. <dd><p>Return the sample variance of <em>data</em>, an iterable of at least two real-valued

  694. numbers.  Variance, or second moment about the mean, is a measure of the

  695. variability (spread or dispersion) of data.  A large variance indicates that

  696. the data is spread out; a small variance indicates it is clustered closely

  697. around the mean.</p>

  698. <p>If the optional second argument <em>xbar</em> is given, it should be the mean of

  699. <em>data</em>.  If it is missing or <code class="docutils literal notranslate"><span class="pre">None</span></code> (the default), the mean is

  700. automatically calculated.</p>

  701. <p>Use this function when your data is a sample from a population. To calculate

  702. the variance from the entire population, see <a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a>.</p>

  703. <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>

  704. <p>Examples:</p>

  705. <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>

  706. <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>

  707. <span class="go">1.3720238095238095</span>

  708. </pre></div>

  709. </div>

  710. <p>If you have already calculated the mean of your data, you can pass it as the

  711. optional second argument <em>xbar</em> to avoid recalculation:</p>

  712. <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>

  713. <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>

  714. <span class="go">1.3720238095238095</span>

  715. </pre></div>

  716. </div>

  717. <p>This function does not attempt to verify that you have passed the actual mean

  718. as <em>xbar</em>.  Using arbitrary values for <em>xbar</em> can lead to invalid or

  719. impossible results.</p>

  720. <p>Decimal and Fraction values are supported:</p>

  721. <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="nn">decimal</span> <span class="kn">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>

  722. <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>

  723. <span class="go">Decimal(&#39;31.01875&#39;)</span>

  724. 

  725. <span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="kn">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>

  726. <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>

  727. <span class="go">Fraction(67, 108)</span>

  728. </pre></div>

  729. </div>

  730. <div class="admonition note">

  731. <p class="admonition-title">Note</p>

  732. <p>This is the sample variance s² with Bessel’s correction, also known as

  733. variance with N-1 degrees of freedom.  Provided that the data points are

  734. representative (e.g. independent and identically distributed), the result

  735. should be an unbiased estimate of the true population variance.</p>

  736. <p>If you somehow know the actual population mean μ you should pass it to the

  737. <a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a> function as the <em>mu</em> parameter to get the variance of a

  738. sample.</p>

  739. </div>

  740. </dd></dl>

  741. 

  742. <dl class="py function">

  743. <dt class="sig sig-object py" id="statistics.quantiles">

  744. <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="o"><span class="pre">*</span></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>

  745. <dd><p>Divide <em>data</em> into <em>n</em> continuous intervals with equal probability.

  746. Returns 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>

  747. <p>Set <em>n</em> to 4 for quartiles (the default).  Set <em>n</em> to 10 for deciles.  Set

  748. <em>n</em> to 100 for percentiles which gives the 99 cuts points that separate

  749. <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>

  750. is not least 1.</p>

  751. <p>The <em>data</em> can be any iterable containing sample data.  For meaningful

  752. results, the number of data points in <em>data</em> should be larger than <em>n</em>.

  753. 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 there are not at least two data points.</p>

  754. <p>The cut points are linearly interpolated from the

  755. two nearest data points.  For example, if a cut point falls one-third

  756. of 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>, the

  757. cut-point will evaluate to <code class="docutils literal notranslate"><span class="pre">104</span></code>.</p>

  758. <p>The <em>method</em> for computing quantiles can be varied depending on

  759. whether the <em>data</em> includes or excludes the lowest and

  760. highest possible values from the population.</p>

  761. <p>The default <em>method</em> is “exclusive” and is used for data sampled from

  762. a population that can have more extreme values than found in the

  763. samples.  The portion of the population falling below the <em>i-th</em> of

  764. <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 nine

  765. sample values, the method sorts them and assigns the following

  766. percentiles: 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%.</p>

  767. <p>Setting the <em>method</em> to “inclusive” is used for describing population

  768. data or for samples that are known to include the most extreme values

  769. from the population.  The minimum value in <em>data</em> is treated as the 0th

  770. percentile and the maximum value is treated as the 100th percentile.

  771. The portion of the population falling below the <em>i-th</em> of <em>m</em> sorted

  772. data 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 sample

  773. values, the method sorts them and assigns the following percentiles:

  774. 0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 100%.</p>

  775. <div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="go"># Decile cut points for empirically sampled data</span>

  776. <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>

  777. <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>

  778. <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>

  779. <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>

  780. <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>

  781. <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>

  782. <span class="go">[81.0, 86.2, 89.0, 99.4, 102.5, 103.6, 106.0, 109.8, 111.0]</span>

  783. </pre></div>

  784. </div>

  785. <div class="versionadded">

  786. <p><span class="versionmodified added">New in version 3.8.</span></p>

  787. </div>

  788. </dd></dl>

  789. 

  790. <dl class="py function">

  791. <dt class="sig sig-object py" id="statistics.covariance">

  792. <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="o"><span class="pre">/</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.covariance" title="Link to this definition">¶</a></dt>

  793. <dd><p>Return the sample covariance of two inputs <em>x</em> and <em>y</em>. Covariance

  794. is a measure of the joint variability of two inputs.</p>

  795. <p>Both inputs must be of the same length (no less than two), otherwise

  796. <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>

  797. <p>Examples:</p>

  798. <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>

  799. <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>

  800. <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>

  801. <span class="go">0.75</span>

  802. <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>

  803. <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>

  804. <span class="go">-7.5</span>

  805. <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>

  806. <span class="go">-7.5</span>

  807. </pre></div>

  808. </div>

  809. <div class="versionadded">

  810. <p><span class="versionmodified added">New in version 3.10.</span></p>

  811. </div>

  812. </dd></dl>

  813. 

  814. <dl class="py function">

  815. <dt class="sig sig-object py" id="statistics.correlation">

  816. <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="o"><span class="pre">/</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</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">'linear'</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.correlation" title="Link to this definition">¶</a></dt>

  817. <dd><p>Return the <a class="reference external" href="https://en.wikipedia.org/wiki/Pearson_correlation_coefficient">Pearson’s correlation coefficient</a>

  818. for two inputs. Pearson’s correlation coefficient <em>r</em> takes values

  819. between -1 and +1. It measures the strength and direction of a linear

  820. relationship.</p>

  821. <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>

  822. for two inputs. The data is replaced by ranks.  Ties are averaged so that

  823. equal values receive the same rank.  The resulting coefficient measures the

  824. strength of a monotonic relationship.</p>

  825. <p>Spearman’s correlation coefficient is appropriate for ordinal data or for

  826. continuous data that doesn’t meet the linear proportion requirement for

  827. Pearson’s correlation coefficient.</p>

  828. <p>Both inputs must be of the same length (no less than two), and need

  829. not 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>

  830. <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>

  831. <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>

  832. <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>

  833. <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>

  834. 

  835. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># Show that a perfect monotonic relationship exists</span>

  836. <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>

  837. <span class="go">1.0</span>

  838. 

  839. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># Observe that a linear relationship is imperfect</span>

  840. <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>

  841. <span class="go">0.9882</span>

  842. 

  843. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># Demonstrate Kepler&#39;s third law: There is a linear correlation</span>

  844. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># between the square of the orbital period and the cube of the</span>

  845. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># distance from the sun.</span>

  846. <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>

  847. <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>

  848. <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>

  849. <span class="go">1.0</span>

  850. </pre></div>

  851. </div>

  852. <div class="versionadded">

  853. <p><span class="versionmodified added">New in version 3.10.</span></p>

  854. </div>

  855. <div class="versionchanged">

  856. <p><span class="versionmodified changed">Changed in version 3.12: </span>Added support for Spearman’s rank correlation coefficient.</p>

  857. </div>

  858. </dd></dl>

  859. 

  860. <dl class="py function">

  861. <dt class="sig sig-object py" id="statistics.linear_regression">

  862. <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="o"><span class="pre">/</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></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>

  863. <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>

  864. parameters estimated using ordinary least squares. Simple linear

  865. regression describes the relationship between an independent variable <em>x</em> and

  866. a dependent variable <em>y</em> in terms of this linear function:</p>

  867. <blockquote>

  868. <div><p><em>y = slope * x + intercept + noise</em></p>

  869. </div></blockquote>

  870. <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 are

  871. estimated, and <code class="docutils literal notranslate"><span class="pre">noise</span></code> represents the

  872. variability of the data that was not explained by the linear regression

  873. (it is equal to the difference between predicted and actual values

  874. of the dependent variable).</p>

  875. <p>Both inputs must be of the same length (no less than two), and

  876. the independent variable <em>x</em> cannot be constant;

  877. otherwise 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>

  878. <p>For example, we can use the <a class="reference external" href="https://en.wikipedia.org/wiki/Monty_Python#Films">release dates of the Monty

  879. Python films</a>

  880. to predict the cumulative number of Monty Python films

  881. that would have been produced by 2019

  882. assuming that they had kept the pace.</p>

  883. <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>

  884. <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>

  885. <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>

  886. <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>

  887. <span class="go">16</span>

  888. </pre></div>

  889. </div>

  890. <p>If <em>proportional</em> is true, the independent variable <em>x</em> and the

  891. dependent variable <em>y</em> are assumed to be directly proportional.

  892. The data is fit to a line passing through the origin.

  893. Since the <em>intercept</em> will always be 0.0, the underlying linear

  894. function simplifies to:</p>

  895. <blockquote>

  896. <div><p><em>y = slope * x + noise</em></p>

  897. </div></blockquote>

  898. <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 see

  899. how well a model based on major planets can predict the orbital

  900. distances for dwarf planets:</p>

  901. <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>

  902. <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>

  903. 

  904. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># Dwarf planets:   Pluto,  Eris,    Makemake, Haumea, Ceres</span>

  905. <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>

  906. <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>

  907. <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>

  908. <span class="go">[5912, 10166, 6806, 6459, 414]</span>

  909. 

  910. <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>

  911. <span class="go">[5906, 10152, 6796, 6450, 414]</span>

  912. </pre></div>

  913. </div>

  914. <div class="versionadded">

  915. <p><span class="versionmodified added">New in version 3.10.</span></p>

  916. </div>

  917. <div class="versionchanged">

  918. <p><span class="versionmodified changed">Changed in version 3.11: </span>Added support for <em>proportional</em>.</p>

  919. </div>

  920. </dd></dl>

  921. 

  922. </section>

  923. <section id="exceptions">

  924. <h2>Exceptions<a class="headerlink" href="#exceptions" title="Link to this heading">¶</a></h2>

  925. <p>A single exception is defined:</p>

  926. <dl class="py exception">

  927. <dt class="sig sig-object py" id="statistics.StatisticsError">

  928. <em class="property"><span class="pre">exception</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>

  929. <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>

  930. </dd></dl>

  931. 

  932. </section>

  933. <section id="normaldist-objects">

  934. <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>

  935. <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 normal

  936. distributions of a <a class="reference external" href="http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm">random variable</a>.  It is a

  937. class that treats the mean and standard deviation of data

  938. measurements as a single entity.</p>

  939. <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 range

  940. of applications in statistics.</p>

  941. <dl class="py class">

  942. <dt class="sig sig-object py" id="statistics.NormalDist">

  943. <em class="property"><span class="pre">class</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>

  944. <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">arithmetic

  945. mean</a> and <em>sigma</em>

  946. represents the <a class="reference external" href="https://en.wikipedia.org/wiki/Standard_deviation">standard deviation</a>.</p>

  947. <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>

  948. <dl class="py attribute">

  949. <dt class="sig sig-object py" id="statistics.NormalDist.mean">

  950. <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>

  951. <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 normal

  952. distribution.</p>

  953. </dd></dl>

  954. 

  955. <dl class="py attribute">

  956. <dt class="sig sig-object py" id="statistics.NormalDist.median">

  957. <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>

  958. <dd><p>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Median">median</a> of a normal

  959. distribution.</p>

  960. </dd></dl>

  961. 

  962. <dl class="py attribute">

  963. <dt class="sig sig-object py" id="statistics.NormalDist.mode">

  964. <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>

  965. <dd><p>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Mode_(statistics)">mode</a> of a normal

  966. distribution.</p>

  967. </dd></dl>

  968. 

  969. <dl class="py attribute">

  970. <dt class="sig sig-object py" id="statistics.NormalDist.stdev">

  971. <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>

  972. <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 normal

  973. distribution.</p>

  974. </dd></dl>

  975. 

  976. <dl class="py attribute">

  977. <dt class="sig sig-object py" id="statistics.NormalDist.variance">

  978. <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>

  979. <dd><p>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Variance">variance</a> of a normal

  980. distribution. Equal to the square of the standard deviation.</p>

  981. </dd></dl>

  982. 

  983. <dl class="py method">

  984. <dt class="sig sig-object py" id="statistics.NormalDist.from_samples">

  985. <em class="property"><span class="pre">classmethod</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>

  986. <dd><p>Makes a normal distribution instance with <em>mu</em> and <em>sigma</em> parameters

  987. estimated 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>

  988. <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 values

  989. that 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 not

  990. contain 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 it

  991. takes at least one point to estimate a central value and at least two

  992. points to estimate dispersion.</p>

  993. </dd></dl>

  994. 

  995. <dl class="py method">

  996. <dt class="sig sig-object py" id="statistics.NormalDist.samples">

  997. <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="o"><span class="pre">*</span></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>

  998. <dd><p>Generates <em>n</em> random samples for a given mean and standard deviation.

  999. Returns 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>

  1000. <p>If <em>seed</em> is given, creates a new instance of the underlying random

  1001. number generator.  This is useful for creating reproducible results,

  1002. even in a multi-threading context.</p>

  1003. </dd></dl>

  1004. 

  1005. <dl class="py method">

  1006. <dt class="sig sig-object py" id="statistics.NormalDist.pdf">

  1007. <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>

  1008. <dd><p>Using a <a class="reference external" href="https://en.wikipedia.org/wiki/Probability_density_function">probability density function (pdf)</a>, compute

  1009. the relative likelihood that a random variable <em>X</em> will be near the

  1010. given 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>

  1011. <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>

  1012. <p>The relative likelihood is computed as the probability of a sample

  1013. occurring in a narrow range divided by the width of the range (hence

  1014. the word “density”).  Since the likelihood is relative to other points,

  1015. its value can be greater than <code class="docutils literal notranslate"><span class="pre">1.0</span></code>.</p>

  1016. </dd></dl>

  1017. 

  1018. <dl class="py method">

  1019. <dt class="sig sig-object py" id="statistics.NormalDist.cdf">

  1020. <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>

  1021. <dd><p>Using a <a class="reference external" href="https://en.wikipedia.org/wiki/Cumulative_distribution_function">cumulative distribution function (cdf)</a>,

  1022. compute the probability that a random variable <em>X</em> will be less than or

  1023. equal 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>

  1024. </dd></dl>

  1025. 

  1026. <dl class="py method">

  1027. <dt class="sig sig-object py" id="statistics.NormalDist.inv_cdf">

  1028. <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>

  1029. <dd><p>Compute the inverse cumulative distribution function, also known as the

  1030. <a class="reference external" href="https://en.wikipedia.org/wiki/Quantile_function">quantile function</a>

  1031. or the <a class="reference external" href="https://web.archive.org/web/20190203145224/https://www.statisticshowto.datasciencecentral.com/inverse-distribution-function/">percent-point</a>

  1032. function.  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>

  1033. <p>Finds the value <em>x</em> of the random variable <em>X</em> such that the

  1034. probability of the variable being less than or equal to that value

  1035. equals the given probability <em>p</em>.</p>

  1036. </dd></dl>

  1037. 

  1038. <dl class="py method">

  1039. <dt class="sig sig-object py" id="statistics.NormalDist.overlap">

  1040. <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>

  1041. <dd><p>Measures the agreement between two normal probability distributions.

  1042. Returns a value between 0.0 and 1.0 giving <a class="reference external" href="https://www.rasch.org/rmt/rmt101r.htm">the overlapping area for

  1043. the two probability density functions</a>.</p>

  1044. </dd></dl>

  1045. 

  1046. <dl class="py method">

  1047. <dt class="sig sig-object py" id="statistics.NormalDist.quantiles">

  1048. <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>

  1049. <dd><p>Divide the normal distribution into <em>n</em> continuous intervals with

  1050. equal probability.  Returns a list of (n - 1) cut points separating

  1051. the intervals.</p>

  1052. <p>Set <em>n</em> to 4 for quartiles (the default).  Set <em>n</em> to 10 for deciles.

  1053. Set <em>n</em> to 100 for percentiles which gives the 99 cuts points that

  1054. separate the normal distribution into 100 equal sized groups.</p>

  1055. </dd></dl>

  1056. 

  1057. <dl class="py method">

  1058. <dt class="sig sig-object py" id="statistics.NormalDist.zscore">

  1059. <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>

  1060. <dd><p>Compute the

  1061. <a class="reference external" href="https://www.statisticshowto.com/probability-and-statistics/z-score/">Standard Score</a>

  1062. describing <em>x</em> in terms of the number of standard deviations

  1063. above or below the mean of the normal distribution:

  1064. <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>

  1065. <div class="versionadded">

  1066. <p><span class="versionmodified added">New in version 3.9.</span></p>

  1067. </div>

  1068. </dd></dl>

  1069. 

  1070. <p>Instances of <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> support addition, subtraction,

  1071. multiplication and division by a constant.  These operations

  1072. are used for translation and scaling.  For example:</p>

  1073. <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>

  1074. <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>

  1075. <span class="go">NormalDist(mu=41.0, sigma=4.5)</span>

  1076. </pre></div>

  1077. </div>

  1078. <p>Dividing a constant by an instance of <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 not supported

  1079. because the result wouldn’t be normally distributed.</p>

  1080. <p>Since normal distributions arise from additive effects of independent

  1081. variables, 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 normally

  1082. distributed random variables</a>

  1083. represented as instances of <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>.  For example:</p>

  1084. <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>

  1085. <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>

  1086. <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>

  1087. <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>

  1088. <span class="go">3.1</span>

  1089. <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>

  1090. <span class="go">0.5</span>

  1091. </pre></div>

  1092. </div>

  1093. <div class="versionadded">

  1094. <p><span class="versionmodified added">New in version 3.8.</span></p>

  1095. </div>

  1096. </dd></dl>

  1097. 

  1098. </section>

  1099. <section id="examples-and-recipes">

  1100. <h2>Examples and Recipes<a class="headerlink" href="#examples-and-recipes" title="Link to this heading">¶</a></h2>

  1101. <section id="classic-probability-problems">

  1102. <h3>Classic probability problems<a class="headerlink" href="#classic-probability-problems" title="Link to this heading">¶</a></h3>

  1103. <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> readily solves classic probability problems.</p>

  1104. <p>For example, given <a class="reference external" href="https://nces.ed.gov/programs/digest/d17/tables/dt17_226.40.asp">historical data for SAT exams</a> showing

  1105. that scores are normally distributed with a mean of 1060 and a standard

  1106. deviation of 195, determine the percentage of students with test scores

  1107. between 1100 and 1200, after rounding to the nearest whole number:</p>

  1108. <div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">sat</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">1060</span><span class="p">,</span> <span class="mi">195</span><span class="p">)</span>

  1109. <span class="gp">&gt;&gt;&gt; </span><span class="n">fraction</span> <span class="o">=</span> <span class="n">sat</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="mi">1200</span> <span class="o">+</span> <span class="mf">0.5</span><span class="p">)</span> <span class="o">-</span> <span class="n">sat</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="mi">1100</span> <span class="o">-</span> <span class="mf">0.5</span><span class="p">)</span>

  1110. <span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">fraction</span> <span class="o">*</span> <span class="mf">100.0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>

  1111. <span class="go">18.4</span>

  1112. </pre></div>

  1113. </div>

  1114. <p>Find the <a class="reference external" href="https://en.wikipedia.org/wiki/Quartile">quartiles</a> and <a class="reference external" href="https://en.wikipedia.org/wiki/Decile">deciles</a> for the SAT scores:</p>

  1115. <div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><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">sat</span><span class="o">.</span><span class="n">quantiles</span><span class="p">()))</span>

  1116. <span class="go">[928, 1060, 1192]</span>

  1117. <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">sat</span><span class="o">.</span><span class="n">quantiles</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="mi">10</span><span class="p">)))</span>

  1118. <span class="go">[810, 896, 958, 1011, 1060, 1109, 1162, 1224, 1310]</span>

  1119. </pre></div>

  1120. </div>

  1121. </section>

  1122. <section id="monte-carlo-inputs-for-simulations">

  1123. <h3>Monte Carlo inputs for simulations<a class="headerlink" href="#monte-carlo-inputs-for-simulations" title="Link to this heading">¶</a></h3>

  1124. <p>To estimate the distribution for a model that isn’t easy to solve

  1125. analytically, <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> can generate input samples for a <a class="reference external" href="https://en.wikipedia.org/wiki/Monte_Carlo_method">Monte

  1126. Carlo simulation</a>:</p>

  1127. <div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">model</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>

  1128. <span class="gp">... </span>    <span class="k">return</span> <span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">7</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">11</span> <span class="o">*</span> <span class="n">z</span><span class="p">)</span>

  1129. <span class="gp">...</span>

  1130. <span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">100_000</span>

  1131. <span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span><span class="o">.</span><span class="n">samples</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">3652260728</span><span class="p">)</span>

  1132. <span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">)</span><span class="o">.</span><span class="n">samples</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">4582495471</span><span class="p">)</span>

  1133. <span class="gp">&gt;&gt;&gt; </span><span class="n">Z</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span> <span class="mf">1.25</span><span class="p">)</span><span class="o">.</span><span class="n">samples</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">6582483453</span><span class="p">)</span>

  1134. <span class="gp">&gt;&gt;&gt; </span><span class="n">quantiles</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span><span class="p">))</span>       

  1135. <span class="go">[1.4591308524824727, 1.8035946855390597, 2.175091447274739]</span>

  1136. </pre></div>

  1137. </div>

  1138. </section>

  1139. <section id="approximating-binomial-distributions">

  1140. <h3>Approximating binomial distributions<a class="headerlink" href="#approximating-binomial-distributions" title="Link to this heading">¶</a></h3>

  1141. <p>Normal distributions can be used to approximate <a class="reference external" href="https://mathworld.wolfram.com/BinomialDistribution.html">Binomial

  1142. distributions</a>

  1143. when the sample size is large and when the probability of a successful

  1144. trial is near 50%.</p>

  1145. <p>For example, an open source conference has 750 attendees and two rooms with a

  1146. 500 person capacity.  There is a talk about Python and another about Ruby.

  1147. In previous conferences, 65% of the attendees preferred to listen to Python

  1148. talks.  Assuming the population preferences haven’t changed, what is the

  1149. probability that the Python room will stay within its capacity limits?</p>

  1150. <div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">750</span>             <span class="c1"># Sample size</span>

  1151. <span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="mf">0.65</span>            <span class="c1"># Preference for Python</span>

  1152. <span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="o">-</span> <span class="n">p</span>         <span class="c1"># Preference for Ruby</span>

  1153. <span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="mi">500</span>             <span class="c1"># Room capacity</span>

  1154. 

  1155. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># Approximation using the cumulative normal distribution</span>

  1156. <span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">sqrt</span>

  1157. <span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">NormalDist</span><span class="p">(</span><span class="n">mu</span><span class="o">=</span><span class="n">n</span><span class="o">*</span><span class="n">p</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="p">))</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mf">0.5</span><span class="p">),</span> <span class="mi">4</span><span class="p">)</span>

  1158. <span class="go">0.8402</span>

  1159. 

  1160. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># Exact solution using the cumulative binomial distribution</span>

  1161. <span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">comb</span><span class="p">,</span> <span class="n">fsum</span>

  1162. <span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">fsum</span><span class="p">(</span><span class="n">comb</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span><span class="o">**</span><span class="n">r</span> <span class="o">*</span> <span class="n">q</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="n">r</span><span class="p">)</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)),</span> <span class="mi">4</span><span class="p">)</span>

  1163. <span class="go">0.8402</span>

  1164. 

  1165. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># Approximation using a simulation</span>

  1166. <span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">seed</span><span class="p">,</span> <span class="n">binomialvariate</span>

  1167. <span class="gp">&gt;&gt;&gt; </span><span class="n">seed</span><span class="p">(</span><span class="mi">8675309</span><span class="p">)</span>

  1168. <span class="gp">&gt;&gt;&gt; </span><span class="n">mean</span><span class="p">(</span><span class="n">binomialvariate</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">k</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_000</span><span class="p">))</span>

  1169. <span class="go">0.8406</span>

  1170. </pre></div>

  1171. </div>

  1172. </section>

  1173. <section id="naive-bayesian-classifier">

  1174. <h3>Naive bayesian classifier<a class="headerlink" href="#naive-bayesian-classifier" title="Link to this heading">¶</a></h3>

  1175. <p>Normal distributions commonly arise in machine learning problems.</p>

  1176. <p>Wikipedia has a <a class="reference external" href="https://en.wikipedia.org/wiki/Naive_Bayes_classifier#Person_classification">nice example of a Naive Bayesian Classifier</a>.

  1177. The challenge is to predict a person’s gender from measurements of normally

  1178. distributed features including height, weight, and foot size.</p>

  1179. <p>We’re given a training dataset with measurements for eight people.  The

  1180. measurements are assumed to be normally distributed, so we summarize the data

  1181. with <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>:</p>

  1182. <div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">height_male</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="mi">6</span><span class="p">,</span> <span class="mf">5.92</span><span class="p">,</span> <span class="mf">5.58</span><span class="p">,</span> <span class="mf">5.92</span><span class="p">])</span>

  1183. <span class="gp">&gt;&gt;&gt; </span><span class="n">height_female</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="mi">5</span><span class="p">,</span> <span class="mf">5.5</span><span class="p">,</span> <span class="mf">5.42</span><span class="p">,</span> <span class="mf">5.75</span><span class="p">])</span>

  1184. <span class="gp">&gt;&gt;&gt; </span><span class="n">weight_male</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="mi">180</span><span class="p">,</span> <span class="mi">190</span><span class="p">,</span> <span class="mi">170</span><span class="p">,</span> <span class="mi">165</span><span class="p">])</span>

  1185. <span class="gp">&gt;&gt;&gt; </span><span class="n">weight_female</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="mi">100</span><span class="p">,</span> <span class="mi">150</span><span class="p">,</span> <span class="mi">130</span><span class="p">,</span> <span class="mi">150</span><span class="p">])</span>

  1186. <span class="gp">&gt;&gt;&gt; </span><span class="n">foot_size_male</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="mi">12</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>

  1187. <span class="gp">&gt;&gt;&gt; </span><span class="n">foot_size_female</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="mi">6</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">9</span><span class="p">])</span>

  1188. </pre></div>

  1189. </div>

  1190. <p>Next, we encounter a new person whose feature measurements are known but whose

  1191. gender is unknown:</p>

  1192. <div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">ht</span> <span class="o">=</span> <span class="mf">6.0</span>        <span class="c1"># height</span>

  1193. <span class="gp">&gt;&gt;&gt; </span><span class="n">wt</span> <span class="o">=</span> <span class="mi">130</span>        <span class="c1"># weight</span>

  1194. <span class="gp">&gt;&gt;&gt; </span><span class="n">fs</span> <span class="o">=</span> <span class="mi">8</span>          <span class="c1"># foot size</span>

  1195. </pre></div>

  1196. </div>

  1197. <p>Starting with a 50% <a class="reference external" href="https://en.wikipedia.org/wiki/Prior_probability">prior probability</a> of being male or female,

  1198. we compute the posterior as the prior times the product of likelihoods for the

  1199. feature measurements given the gender:</p>

  1200. <div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">prior_male</span> <span class="o">=</span> <span class="mf">0.5</span>

  1201. <span class="gp">&gt;&gt;&gt; </span><span class="n">prior_female</span> <span class="o">=</span> <span class="mf">0.5</span>

  1202. <span class="gp">&gt;&gt;&gt; </span><span class="n">posterior_male</span> <span class="o">=</span> <span class="p">(</span><span class="n">prior_male</span> <span class="o">*</span> <span class="n">height_male</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">ht</span><span class="p">)</span> <span class="o">*</span>

  1203. <span class="gp">... </span>                  <span class="n">weight_male</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">wt</span><span class="p">)</span> <span class="o">*</span> <span class="n">foot_size_male</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">fs</span><span class="p">))</span>

  1204. 

  1205. <span class="gp">&gt;&gt;&gt; </span><span class="n">posterior_female</span> <span class="o">=</span> <span class="p">(</span><span class="n">prior_female</span> <span class="o">*</span> <span class="n">height_female</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">ht</span><span class="p">)</span> <span class="o">*</span>

  1206. <span class="gp">... </span>                    <span class="n">weight_female</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">wt</span><span class="p">)</span> <span class="o">*</span> <span class="n">foot_size_female</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">fs</span><span class="p">))</span>

  1207. </pre></div>

  1208. </div>

  1209. <p>The final prediction goes to the largest posterior. This is known as the

  1210. <a class="reference external" href="https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation">maximum a posteriori</a> or MAP:</p>

  1211. <div class="highlight-pycon notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="s1">&#39;male&#39;</span> <span class="k">if</span> <span class="n">posterior_male</span> <span class="o">&gt;</span> <span class="n">posterior_female</span> <span class="k">else</span> <span class="s1">&#39;female&#39;</span>

  1212. <span class="go">&#39;female&#39;</span>

  1213. </pre></div>

  1214. </div>

  1215. </section>

  1216. <section id="kernel-density-estimation">

  1217. <h3>Kernel density estimation<a class="headerlink" href="#kernel-density-estimation" title="Link to this heading">¶</a></h3>

  1218. <p>It is possible to estimate a continuous probability density function

  1219. from a fixed number of discrete samples.</p>

  1220. <p>The basic idea is to smooth the data using <a class="reference external" href="https://en.wikipedia.org/wiki/Kernel_(statistics)#Kernel_functions_in_common_use">a kernel function such as a

  1221. normal distribution, triangular distribution, or uniform distribution</a>.

  1222. The degree of smoothing is controlled by a scaling parameter, <code class="docutils literal notranslate"><span class="pre">h</span></code>,

  1223. which is called the <em>bandwidth</em>.</p>

  1224. <div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">kde_normal</span><span class="p">(</span><span class="n">sample</span><span class="p">,</span> <span class="n">h</span><span class="p">):</span>

  1225.     <span class="s2">&quot;Create a continuous probability density function from a sample.&quot;</span>

  1226.     <span class="c1"># Smooth the sample with a normal distribution kernel scaled by h.</span>

  1227.     <span class="n">kernel_h</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">h</span><span class="p">)</span><span class="o">.</span><span class="n">pdf</span>

  1228.     <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">sample</span><span class="p">)</span>

  1229.     <span class="k">def</span> <span class="nf">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>

  1230.         <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="n">kernel_h</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">x_i</span><span class="p">)</span> <span class="k">for</span> <span class="n">x_i</span> <span class="ow">in</span> <span class="n">sample</span><span class="p">)</span> <span class="o">/</span> <span class="n">n</span>

  1231.     <span class="k">return</span> <span class="n">pdf</span>

  1232. </pre></div>

  1233. </div>

  1234. <p><a class="reference external" href="https://en.wikipedia.org/wiki/Kernel_density_estimation#Example">Wikipedia has an example</a>

  1235. where we can use the <code class="docutils literal notranslate"><span class="pre">kde_normal()</span></code> recipe to generate and plot

  1236. a probability density function estimated from a small sample:</p>

  1237. <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>

  1238. <span class="gp">&gt;&gt;&gt; </span><span class="n">f_hat</span> <span class="o">=</span> <span class="n">kde_normal</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>

  1239. <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>

  1240. <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>

  1241. </pre></div>

  1242. </div>

  1243. <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>

  1244. <img alt="Scatter plot of the estimated probability density function." src="../_images/kde_example.png" />

  1245. </section>

  1246. </section>

  1247. </section>

  1248. 

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  1259. <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>

  1260. <li><a class="reference internal" href="#averages-and-measures-of-central-location">Averages and measures of central location</a></li>

  1261. <li><a class="reference internal" href="#measures-of-spread">Measures of spread</a></li>

  1262. <li><a class="reference internal" href="#statistics-for-relations-between-two-inputs">Statistics for relations between two inputs</a></li>

  1263. <li><a class="reference internal" href="#function-details">Function details</a></li>

  1264. <li><a class="reference internal" href="#exceptions">Exceptions</a></li>

  1265. <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>

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  1271. <li><a class="reference internal" href="#kernel-density-estimation">Kernel density estimation</a></li>

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