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 <h3><a href="../contents.html">Table of Contents</a></h3>
 <ul>
<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate">statistics</code> — Mathematical statistics functions</a><ul>
<li><a class="reference internal" href="#averages-and-measures-of-central-location">Averages and measures of central location</a></li>
<li><a class="reference internal" href="#measures-of-spread">Measures of spread</a></li>
<li><a class="reference internal" href="#statistics-for-relations-between-two-inputs">Statistics for relations between two inputs</a></li>
<li><a class="reference internal" href="#function-details">Function details</a></li>
<li><a class="reference internal" href="#exceptions">Exceptions</a></li>
<li><a class="reference internal" href="#normaldist-objects"><code class="xref py py-class docutils literal notranslate">NormalDist</code> objects</a></li>
<li><a class="reference internal" href="#examples-and-recipes">Examples and Recipes</a><ul>
<li><a class="reference internal" href="#classic-probability-problems">Classic probability problems</a></li>
<li><a class="reference internal" href="#monte-carlo-inputs-for-simulations">Monte Carlo inputs for simulations</a></li>
<li><a class="reference internal" href="#approximating-binomial-distributions">Approximating binomial distributions</a></li>
<li><a class="reference internal" href="#naive-bayesian-classifier">Naive bayesian classifier</a></li>
<li><a class="reference internal" href="#kernel-density-estimation">Kernel density estimation</a></li>
</ul>
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</ul>
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 <section id="module-statistics">
<h1><a class="reference internal" href="#module-statistics" title="statistics: Mathematical statistics functions"><code class="xref py py-mod docutils literal notranslate">statistics</code></a> — Mathematical statistics functions<a class="headerlink" href="#module-statistics" title="Link to this heading">¶</a></h1>
<div class="versionadded">
New in version 3.4.
</div>
Source code: <a class="reference external" href="https://github.com/python/cpython/tree/3.12/Lib/statistics.py">Lib/statistics.py</a>
<hr class="docutils" />
This module provides functions for calculating mathematical statistics of
numeric (<a class="reference internal" href="numbers.html#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate">Real</code></a>-valued) data.
The module is not intended to be a competitor to third-party libraries such
as <a class="reference external" href="https://numpy.org">NumPy</a>, <a class="reference external" href="https://scipy.org/">SciPy</a>, or
proprietary full-featured statistics packages aimed at professional
statisticians such as Minitab, SAS and Matlab. It is aimed at the level of
graphing and scientific calculators.
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">int</code></a>,
<a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate">float</code></a>, <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate">Decimal</code></a> and <a class="reference internal" href="fractions.html#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate">Fraction</code></a>.
Behaviour with other types (whether in the numeric tower or not) is
currently unsupported. Collections with a mix of types are also undefined
and implementation-dependent. If your input data consists of mixed types,
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">map()</code></a> to ensure a consistent result, for
example: <code class="docutils literal notranslate">map(float, input_data)</code>.
Some datasets use <code class="docutils literal notranslate">NaN</code> (not a number) values to represent missing data.
Since NaNs have unusual comparison semantics, they cause surprising or
undefined behaviors in the statistics functions that sort data or that count
occurrences. The functions affected are <code class="docutils literal notranslate">median()</code>, <code class="docutils literal notranslate">median_low()</code>,
<code class="docutils literal notranslate">median_high()</code>, <code class="docutils literal notranslate">median_grouped()</code>, <code class="docutils literal notranslate">mode()</code>, <code class="docutils literal notranslate">multimode()</code>, and
<code class="docutils literal notranslate">quantiles()</code>. The <code class="docutils literal notranslate">NaN</code> values should be stripped before calling these
functions:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; from statistics import median
&gt;&gt;&gt; from math import isnan
&gt;&gt;&gt; from itertools import filterfalse

&gt;&gt;&gt; data = [20.7, float(&#39;NaN&#39;),19.2, 18.3, float(&#39;NaN&#39;), 14.4]
&gt;&gt;&gt; sorted(data) # This has surprising behavior
[20.7, nan, 14.4, 18.3, 19.2, nan]
&gt;&gt;&gt; median(data) # This result is unexpected
16.35

&gt;&gt;&gt; sum(map(isnan, data)) # Number of missing values
2
&gt;&gt;&gt; clean = list(filterfalse(isnan, data)) # Strip NaN values
&gt;&gt;&gt; clean
[20.7, 19.2, 18.3, 14.4]
&gt;&gt;&gt; sorted(clean) # Sorting now works as expected
[14.4, 18.3, 19.2, 20.7]
&gt;&gt;&gt; median(clean) # This result is now well defined
18.75
</pre></div>
</div>
<section id="averages-and-measures-of-central-location">
<h2>Averages and measures of central location<a class="headerlink" href="#averages-and-measures-of-central-location" title="Link to this heading">¶</a></h2>
These functions calculate an average or typical value from a population
or sample.
<table class="docutils align-default">
<tbody>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.mean" title="statistics.mean"><code class="xref py py-func docutils literal notranslate">mean()</code></a></td>
<td>Arithmetic mean (“average”) of data.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.fmean" title="statistics.fmean"><code class="xref py py-func docutils literal notranslate">fmean()</code></a></td>
<td>Fast, floating point arithmetic mean, with optional weighting.</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.geometric_mean" title="statistics.geometric_mean"><code class="xref py py-func docutils literal notranslate">geometric_mean()</code></a></td>
<td>Geometric mean of data.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.harmonic_mean" title="statistics.harmonic_mean"><code class="xref py py-func docutils literal notranslate">harmonic_mean()</code></a></td>
<td>Harmonic mean of data.</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.median" title="statistics.median"><code class="xref py py-func docutils literal notranslate">median()</code></a></td>
<td>Median (middle value) of data.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.median_low" title="statistics.median_low"><code class="xref py py-func docutils literal notranslate">median_low()</code></a></td>
<td>Low median of data.</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.median_high" title="statistics.median_high"><code class="xref py py-func docutils literal notranslate">median_high()</code></a></td>
<td>High median of data.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.median_grouped" title="statistics.median_grouped"><code class="xref py py-func docutils literal notranslate">median_grouped()</code></a></td>
<td>Median (50th percentile) of grouped data.</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.mode" title="statistics.mode"><code class="xref py py-func docutils literal notranslate">mode()</code></a></td>
<td>Single mode (most common value) of discrete or nominal data.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.multimode" title="statistics.multimode"><code class="xref py py-func docutils literal notranslate">multimode()</code></a></td>
<td>List of modes (most common values) of discrete or nominal data.</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.quantiles" title="statistics.quantiles"><code class="xref py py-func docutils literal notranslate">quantiles()</code></a></td>
<td>Divide data into intervals with equal probability.</td>
</tr>
</tbody>
</table>
</section>
<section id="measures-of-spread">
<h2>Measures of spread<a class="headerlink" href="#measures-of-spread" title="Link to this heading">¶</a></h2>
These functions calculate a measure of how much the population or sample
tends to deviate from the typical or average values.
<table class="docutils align-default">
<tbody>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.pstdev" title="statistics.pstdev"><code class="xref py py-func docutils literal notranslate">pstdev()</code></a></td>
<td>Population standard deviation of data.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate">pvariance()</code></a></td>
<td>Population variance of data.</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.stdev" title="statistics.stdev"><code class="xref py py-func docutils literal notranslate">stdev()</code></a></td>
<td>Sample standard deviation of data.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate">variance()</code></a></td>
<td>Sample variance of data.</td>
</tr>
</tbody>
</table>
</section>
<section id="statistics-for-relations-between-two-inputs">
<h2>Statistics for relations between two inputs<a class="headerlink" href="#statistics-for-relations-between-two-inputs" title="Link to this heading">¶</a></h2>
These functions calculate statistics regarding relations between two inputs.
<table class="docutils align-default">
<tbody>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.covariance" title="statistics.covariance"><code class="xref py py-func docutils literal notranslate">covariance()</code></a></td>
<td>Sample covariance for two variables.</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#statistics.correlation" title="statistics.correlation"><code class="xref py py-func docutils literal notranslate">correlation()</code></a></td>
<td>Pearson and Spearman’s correlation coefficients.</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#statistics.linear_regression" title="statistics.linear_regression"><code class="xref py py-func docutils literal notranslate">linear_regression()</code></a></td>
<td>Slope and intercept for simple linear regression.</td>
</tr>
</tbody>
</table>
</section>
<section id="function-details">
<h2>Function details<a class="headerlink" href="#function-details" title="Link to this heading">¶</a></h2>
Note: The functions do not require the data given to them to be sorted.
However, for reading convenience, most of the examples show sorted sequences.
<dl class="py function">
<dt class="sig sig-object py" id="statistics.mean">
statistics.mean(data)<a class="headerlink" href="#statistics.mean" title="Link to this definition">¶</a></dt>
<dd>Return the sample arithmetic mean of data which can be a sequence or iterable.
The arithmetic mean is the sum of the data divided by the number of data
points. It is commonly called “the average”, although it is only one of many
different mathematical averages. It is a measure of the central location of
the data.
If data is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> will be raised.
Some examples of use:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; mean([1, 2, 3, 4, 4])
2.8
&gt;&gt;&gt; mean([-1.0, 2.5, 3.25, 5.75])
2.625

&gt;&gt;&gt; from decimal import Decimal as D
&gt;&gt;&gt; mean([D(&quot;0.5&quot;), D(&quot;0.75&quot;), D(&quot;0.625&quot;), D(&quot;0.375&quot;)])
Decimal(&#39;0.5625&#39;)
</pre></div>
</div>
<div class="admonition note">
Note
The mean is strongly affected by <a class="reference external" href="https://en.wikipedia.org/wiki/Outlier">outliers</a> and is not necessarily a
typical example of the data points. For a more robust, although less
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">median()</code></a>.
The sample mean gives an unbiased estimate of the true population mean,
so that when taken on average over all the possible samples,
<code class="docutils literal notranslate">mean(sample)</code> converges on the true mean of the entire population. If
data represents the entire population rather than a sample, then
<code class="docutils literal notranslate">mean(data)</code> is equivalent to calculating the true population mean μ.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.fmean">
statistics.fmean(data, weights=None)<a class="headerlink" href="#statistics.fmean" title="Link to this definition">¶</a></dt>
<dd>Convert data to floats and compute the arithmetic mean.
This runs faster than the <a class="reference internal" href="#statistics.mean" title="statistics.mean"><code class="xref py py-func docutils literal notranslate">mean()</code></a> function and it always returns a
<a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate">float</code></a>. The data may be a sequence or iterable. If the input
dataset is empty, raises a <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a>.
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; fmean([3.5, 4.0, 5.25])
4.25
</pre></div>
</div>
Optional weighting is supported. For example, a professor assigns a
grade for a course by weighting quizzes at 20%, homework at 20%, a
midterm exam at 30%, and a final exam at 30%:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; grades = [85, 92, 83, 91]
&gt;&gt;&gt; weights = [0.20, 0.20, 0.30, 0.30]
&gt;&gt;&gt; fmean(grades, weights)
87.6
</pre></div>
</div>
If weights is supplied, it must be the same length as the data or
a <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate">ValueError</code></a> will be raised.
<div class="versionadded">
New in version 3.8.
</div>
<div class="versionchanged">
Changed in version 3.11: Added support for weights.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.geometric_mean">
statistics.geometric_mean(data)<a class="headerlink" href="#statistics.geometric_mean" title="Link to this definition">¶</a></dt>
<dd>Convert data to floats and compute the geometric mean.
The geometric mean indicates the central tendency or typical value of the
data using the product of the values (as opposed to the arithmetic mean
which uses their sum).
Raises a <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> if the input dataset is empty,
if it contains a zero, or if it contains a negative value.
The data may be a sequence or iterable.
No special efforts are made to achieve exact results.
(However, this may change in the future.)
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; round(geometric_mean([54, 24, 36]), 1)
36.0
</pre></div>
</div>
<div class="versionadded">
New in version 3.8.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.harmonic_mean">
statistics.harmonic_mean(data, weights=None)<a class="headerlink" href="#statistics.harmonic_mean" title="Link to this definition">¶</a></dt>
<dd>Return the harmonic mean of data, a sequence or iterable of
real-valued numbers. If weights is omitted or None, then
equal weighting is assumed.
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">mean()</code></a> of the
reciprocals of the data. For example, the harmonic mean of three values a,
b and c will be equivalent to <code class="docutils literal notranslate">3/(1/a + 1/b + 1/c)</code>. If one of the
values is zero, the result will be zero.
The harmonic mean is a type of average, a measure of the central
location of the data. It is often appropriate when averaging
ratios or rates, for example speeds.
Suppose a car travels 10 km at 40 km/hr, then another 10 km at 60 km/hr.
What is the average speed?
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; harmonic_mean([40, 60])
48.0
</pre></div>
</div>
Suppose a car travels 40 km/hr for 5 km, and when traffic clears,
speeds-up to 60 km/hr for the remaining 30 km of the journey. What
is the average speed?
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; harmonic_mean([40, 60], weights=[5, 30])
56.0
</pre></div>
</div>
<a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> is raised if data is empty, any element
is less than zero, or if the weighted sum isn’t positive.
The current algorithm has an early-out when it encounters a zero
in the input. This means that the subsequent inputs are not tested
for validity. (This behavior may change in the future.)
<div class="versionadded">
New in version 3.6.
</div>
<div class="versionchanged">
Changed in version 3.10: Added support for weights.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.median">
statistics.median(data)<a class="headerlink" href="#statistics.median" title="Link to this definition">¶</a></dt>
<dd>Return the median (middle value) of numeric data, using the common “mean of
middle two” method. If data is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> is raised.
data can be a sequence or iterable.
The median is a robust measure of central location and is less affected by
the presence of outliers. When the number of data points is odd, the
middle data point is returned:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; median([1, 3, 5])
3
</pre></div>
</div>
When the number of data points is even, the median is interpolated by taking
the average of the two middle values:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; median([1, 3, 5, 7])
4.0
</pre></div>
</div>
This is suited for when your data is discrete, and you don’t mind that the
median may not be an actual data point.
If the data is ordinal (supports order operations) but not numeric (doesn’t
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">median_low()</code></a> or <a class="reference internal" href="#statistics.median_high" title="statistics.median_high"><code class="xref py py-func docutils literal notranslate">median_high()</code></a>
instead.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.median_low">
statistics.median_low(data)<a class="headerlink" href="#statistics.median_low" title="Link to this definition">¶</a></dt>
<dd>Return the low median of numeric data. If data is empty,
<a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> is raised. data can be a sequence or iterable.
The low median is always a member of the data set. When the number of data
points is odd, the middle value is returned. When it is even, the smaller of
the two middle values is returned.
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; median_low([1, 3, 5])
3
&gt;&gt;&gt; median_low([1, 3, 5, 7])
3
</pre></div>
</div>
Use the low median when your data are discrete and you prefer the median to
be an actual data point rather than interpolated.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.median_high">
statistics.median_high(data)<a class="headerlink" href="#statistics.median_high" title="Link to this definition">¶</a></dt>
<dd>Return the high median of data. If data is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a>
is raised. data can be a sequence or iterable.
The high median is always a member of the data set. When the number of data
points is odd, the middle value is returned. When it is even, the larger of
the two middle values is returned.
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; median_high([1, 3, 5])
3
&gt;&gt;&gt; median_high([1, 3, 5, 7])
5
</pre></div>
</div>
Use the high median when your data are discrete and you prefer the median to
be an actual data point rather than interpolated.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.median_grouped">
statistics.median_grouped(data, interval=1.0)<a class="headerlink" href="#statistics.median_grouped" title="Link to this definition">¶</a></dt>
<dd>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
of consecutive, fixed-width intervals.
The data can be any iterable of numeric data with each value being
exactly the midpoint of a bin. At least one value must be present.
The interval is the width of each bin.
For example, demographic information may have been summarized into
consecutive ten-year age groups with each group being represented
by the 5-year midpoints of the intervals:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; from collections import Counter
&gt;&gt;&gt; demographics = Counter({
... 25: 172, # 20 to 30 years old
... 35: 484, # 30 to 40 years old
... 45: 387, # 40 to 50 years old
... 55: 22, # 50 to 60 years old
... 65: 6, # 60 to 70 years old
... })
...
</pre></div>
</div>
The 50th percentile (median) is the 536th person out of the 1071
member cohort. That person is in the 30 to 40 year old age group.
The regular <a class="reference internal" href="#statistics.median" title="statistics.median"><code class="xref py py-func docutils literal notranslate">median()</code></a> function would assume that everyone in the
tricenarian age group was exactly 35 years old. A more tenable
assumption is that the 484 members of that age group are evenly
distributed between 30 and 40. For that, we use
<a class="reference internal" href="#statistics.median_grouped" title="statistics.median_grouped"><code class="xref py py-func docutils literal notranslate">median_grouped()</code></a>:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; data = list(demographics.elements())
&gt;&gt;&gt; median(data)
35
&gt;&gt;&gt; round(median_grouped(data, interval=10), 1)
37.5
</pre></div>
</div>
The caller is responsible for making sure the data points are separated
by exact multiples of interval. This is essential for getting a
correct result. The function does not check this precondition.
Inputs may be any numeric type that can be coerced to a float during
the interpolation step.
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.mode">
statistics.mode(data)<a class="headerlink" href="#statistics.mode" title="Link to this definition">¶</a></dt>
<dd>Return the single most common data point from discrete or nominal data.
The mode (when it exists) is the most typical value and serves as a
measure of central location.
If there are multiple modes with the same frequency, returns the first one
encountered in the data. If the smallest or largest of those is
desired instead, use <code class="docutils literal notranslate">min(multimode(data))</code> or <code class="docutils literal notranslate">max(multimode(data))</code>.
If the input data is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> is raised.
<code class="docutils literal notranslate">mode</code> assumes discrete data and returns a single value. This is the
standard treatment of the mode as commonly taught in schools:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; mode([1, 1, 2, 3, 3, 3, 3, 4])
3
</pre></div>
</div>
The mode is unique in that it is the only statistic in this package that
also applies to nominal (non-numeric) data:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; mode([&quot;red&quot;, &quot;blue&quot;, &quot;blue&quot;, &quot;red&quot;, &quot;green&quot;, &quot;red&quot;, &quot;red&quot;])
&#39;red&#39;
</pre></div>
</div>
<div class="versionchanged">
Changed in version 3.8: Now handles multimodal datasets by returning the first mode encountered.
Formerly, it raised <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> when more than one mode was
found.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.multimode">
statistics.multimode(data)<a class="headerlink" href="#statistics.multimode" title="Link to this definition">¶</a></dt>
<dd>Return a list of the most frequently occurring values in the order they
were first encountered in the data. Will return more than one result if
there are multiple modes or an empty list if the data is empty:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; multimode(&#39;aabbbbccddddeeffffgg&#39;)
[&#39;b&#39;, &#39;d&#39;, &#39;f&#39;]
&gt;&gt;&gt; multimode(&#39;&#39;)
[]
</pre></div>
</div>
<div class="versionadded">
New in version 3.8.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.pstdev">
statistics.pstdev(data, mu=None)<a class="headerlink" href="#statistics.pstdev" title="Link to this definition">¶</a></dt>
<dd>Return the population standard deviation (the square root of the population
variance). See <a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate">pvariance()</code></a> for arguments and other details.
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; pstdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
0.986893273527251
</pre></div>
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.pvariance">
statistics.pvariance(data, mu=None)<a class="headerlink" href="#statistics.pvariance" title="Link to this definition">¶</a></dt>
<dd>Return the population variance of data, a non-empty sequence or iterable
of real-valued numbers. Variance, or second moment about the mean, is a
measure of the variability (spread or dispersion) of data. A large
variance indicates that the data is spread out; a small variance indicates
it is clustered closely around the mean.
If the optional second argument mu is given, it is typically the mean of
the data. It can also be used to compute the second moment around a
point that is not the mean. If it is missing or <code class="docutils literal notranslate">None</code> (the default),
the arithmetic mean is automatically calculated.
Use this function to calculate the variance from the entire population. To
estimate the variance from a sample, the <a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate">variance()</code></a> function is usually
a better choice.
Raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> if data is empty.
Examples:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; data = [0.0, 0.25, 0.25, 1.25, 1.5, 1.75, 2.75, 3.25]
&gt;&gt;&gt; pvariance(data)
1.25
</pre></div>
</div>
If you have already calculated the mean of your data, you can pass it as the
optional second argument mu to avoid recalculation:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; mu = mean(data)
&gt;&gt;&gt; pvariance(data, mu)
1.25
</pre></div>
</div>
Decimals and Fractions are supported:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; from decimal import Decimal as D
&gt;&gt;&gt; pvariance([D(&quot;27.5&quot;), D(&quot;30.25&quot;), D(&quot;30.25&quot;), D(&quot;34.5&quot;), D(&quot;41.75&quot;)])
Decimal(&#39;24.815&#39;)

&gt;&gt;&gt; from fractions import Fraction as F
&gt;&gt;&gt; pvariance([F(1, 4), F(5, 4), F(1, 2)])
Fraction(13, 72)
</pre></div>
</div>
<div class="admonition note">
Note
When called with the entire population, this gives the population variance
σ². When called on a sample instead, this is the biased sample variance
s², also known as variance with N degrees of freedom.
If you somehow know the true population mean μ, you may use this
function to calculate the variance of a sample, giving the known
population mean as the second argument. Provided the data points are a
random sample of the population, the result will be an unbiased estimate
of the population variance.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.stdev">
statistics.stdev(data, xbar=None)<a class="headerlink" href="#statistics.stdev" title="Link to this definition">¶</a></dt>
<dd>Return the sample standard deviation (the square root of the sample
variance). See <a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate">variance()</code></a> for arguments and other details.
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; stdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
1.0810874155219827
</pre></div>
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.variance">
statistics.variance(data, xbar=None)<a class="headerlink" href="#statistics.variance" title="Link to this definition">¶</a></dt>
<dd>Return the sample variance of data, an iterable of at least two real-valued
numbers. Variance, or second moment about the mean, is a measure of the
variability (spread or dispersion) of data. A large variance indicates that
the data is spread out; a small variance indicates it is clustered closely
around the mean.
If the optional second argument xbar is given, it should be the mean of
data. If it is missing or <code class="docutils literal notranslate">None</code> (the default), the mean is
automatically calculated.
Use this function when your data is a sample from a population. To calculate
the variance from the entire population, see <a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate">pvariance()</code></a>.
Raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> if data has fewer than two values.
Examples:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; data = [2.75, 1.75, 1.25, 0.25, 0.5, 1.25, 3.5]
&gt;&gt;&gt; variance(data)
1.3720238095238095
</pre></div>
</div>
If you have already calculated the mean of your data, you can pass it as the
optional second argument xbar to avoid recalculation:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; m = mean(data)
&gt;&gt;&gt; variance(data, m)
1.3720238095238095
</pre></div>
</div>
This function does not attempt to verify that you have passed the actual mean
as xbar. Using arbitrary values for xbar can lead to invalid or
impossible results.
Decimal and Fraction values are supported:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; from decimal import Decimal as D
&gt;&gt;&gt; variance([D(&quot;27.5&quot;), D(&quot;30.25&quot;), D(&quot;30.25&quot;), D(&quot;34.5&quot;), D(&quot;41.75&quot;)])
Decimal(&#39;31.01875&#39;)

&gt;&gt;&gt; from fractions import Fraction as F
&gt;&gt;&gt; variance([F(1, 6), F(1, 2), F(5, 3)])
Fraction(67, 108)
</pre></div>
</div>
<div class="admonition note">
Note
This is the sample variance s² with Bessel’s correction, also known as
variance with N-1 degrees of freedom. Provided that the data points are
representative (e.g. independent and identically distributed), the result
should be an unbiased estimate of the true population variance.
If you somehow know the actual population mean μ you should pass it to the
<a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate">pvariance()</code></a> function as the mu parameter to get the variance of a
sample.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.quantiles">
statistics.quantiles(data, *, n=4, method='exclusive')<a class="headerlink" href="#statistics.quantiles" title="Link to this definition">¶</a></dt>
<dd>Divide data into n continuous intervals with equal probability.
Returns a list of <code class="docutils literal notranslate">n - 1</code> cut points separating the intervals.
Set n to 4 for quartiles (the default). Set n to 10 for deciles. Set
n to 100 for percentiles which gives the 99 cuts points that separate
data 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">StatisticsError</code></a> if n
is not least 1.
The data can be any iterable containing sample data. For meaningful
results, the number of data points in data should be larger than n.
Raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> if there are not at least two data points.
The cut points are linearly interpolated from the
two nearest data points. For example, if a cut point falls one-third
of the distance between two sample values, <code class="docutils literal notranslate">100</code> and <code class="docutils literal notranslate">112</code>, the
cut-point will evaluate to <code class="docutils literal notranslate">104</code>.
The method for computing quantiles can be varied depending on
whether the data includes or excludes the lowest and
highest possible values from the population.
The default method is “exclusive” and is used for data sampled from
a population that can have more extreme values than found in the
samples. The portion of the population falling below the i-th of
m sorted data points is computed as <code class="docutils literal notranslate">i / (m + 1)</code>. Given nine
sample values, the method sorts them and assigns the following
percentiles: 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%.
Setting the method to “inclusive” is used for describing population
data or for samples that are known to include the most extreme values
from the population. The minimum value in data is treated as the 0th
percentile and the maximum value is treated as the 100th percentile.
The portion of the population falling below the i-th of m sorted
data points is computed as <code class="docutils literal notranslate">(i - 1) / (m - 1)</code>. Given 11 sample
values, the method sorts them and assigns the following percentiles:
0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 100%.
<div class="highlight-pycon notranslate"><div class="highlight"><pre># Decile cut points for empirically sampled data
&gt;&gt;&gt; data = [105, 129, 87, 86, 111, 111, 89, 81, 108, 92, 110,
... 100, 75, 105, 103, 109, 76, 119, 99, 91, 103, 129,
... 106, 101, 84, 111, 74, 87, 86, 103, 103, 106, 86,
... 111, 75, 87, 102, 121, 111, 88, 89, 101, 106, 95,
... 103, 107, 101, 81, 109, 104]
&gt;&gt;&gt; [round(q, 1) for q in quantiles(data, n=10)]
[81.0, 86.2, 89.0, 99.4, 102.5, 103.6, 106.0, 109.8, 111.0]
</pre></div>
</div>
<div class="versionadded">
New in version 3.8.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.covariance">
statistics.covariance(x, y, /)<a class="headerlink" href="#statistics.covariance" title="Link to this definition">¶</a></dt>
<dd>Return the sample covariance of two inputs x and y. Covariance
is a measure of the joint variability of two inputs.
Both inputs must be of the same length (no less than two), otherwise
<a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> is raised.
Examples:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; x = [1, 2, 3, 4, 5, 6, 7, 8, 9]
&gt;&gt;&gt; y = [1, 2, 3, 1, 2, 3, 1, 2, 3]
&gt;&gt;&gt; covariance(x, y)
0.75
&gt;&gt;&gt; z = [9, 8, 7, 6, 5, 4, 3, 2, 1]
&gt;&gt;&gt; covariance(x, z)
-7.5
&gt;&gt;&gt; covariance(z, x)
-7.5
</pre></div>
</div>
<div class="versionadded">
New in version 3.10.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.correlation">
statistics.correlation(x, y, /, *, method='linear')<a class="headerlink" href="#statistics.correlation" title="Link to this definition">¶</a></dt>
<dd>Return the <a class="reference external" href="https://en.wikipedia.org/wiki/Pearson_correlation_coefficient">Pearson’s correlation coefficient</a>
for two inputs. Pearson’s correlation coefficient r takes values
between -1 and +1. It measures the strength and direction of a linear
relationship.
If method is “ranked”, computes <a class="reference external" href="https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient">Spearman’s rank correlation coefficient</a>
for two inputs. The data is replaced by ranks. Ties are averaged so that
equal values receive the same rank. The resulting coefficient measures the
strength of a monotonic relationship.
Spearman’s correlation coefficient is appropriate for ordinal data or for
continuous data that doesn’t meet the linear proportion requirement for
Pearson’s correlation coefficient.
Both inputs must be of the same length (no less than two), and need
not to be constant, otherwise <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> is raised.
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>:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; # Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune
&gt;&gt;&gt; orbital_period = [88, 225, 365, 687, 4331, 10_756, 30_687, 60_190] # days
&gt;&gt;&gt; dist_from_sun = [58, 108, 150, 228, 778, 1_400, 2_900, 4_500] # million km

&gt;&gt;&gt; # Show that a perfect monotonic relationship exists
&gt;&gt;&gt; correlation(orbital_period, dist_from_sun, method=&#39;ranked&#39;)
1.0

&gt;&gt;&gt; # Observe that a linear relationship is imperfect
&gt;&gt;&gt; round(correlation(orbital_period, dist_from_sun), 4)
0.9882

&gt;&gt;&gt; # Demonstrate Kepler&#39;s third law: There is a linear correlation
&gt;&gt;&gt; # between the square of the orbital period and the cube of the
&gt;&gt;&gt; # distance from the sun.
&gt;&gt;&gt; period_squared = [p * p for p in orbital_period]
&gt;&gt;&gt; dist_cubed = [d * d * d for d in dist_from_sun]
&gt;&gt;&gt; round(correlation(period_squared, dist_cubed), 4)
1.0
</pre></div>
</div>
<div class="versionadded">
New in version 3.10.
</div>
<div class="versionchanged">
Changed in version 3.12: Added support for Spearman’s rank correlation coefficient.
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="statistics.linear_regression">
statistics.linear_regression(x, y, /, *, proportional=False)<a class="headerlink" href="#statistics.linear_regression" title="Link to this definition">¶</a></dt>
<dd>Return the slope and intercept of <a class="reference external" href="https://en.wikipedia.org/wiki/Simple_linear_regression">simple linear regression</a>
parameters estimated using ordinary least squares. Simple linear
regression describes the relationship between an independent variable x and
a dependent variable y in terms of this linear function:
<blockquote>
<div>y = slope * x + intercept + noise
</div></blockquote>
where <code class="docutils literal notranslate">slope</code> and <code class="docutils literal notranslate">intercept</code> are the regression parameters that are
estimated, and <code class="docutils literal notranslate">noise</code> represents the
variability of the data that was not explained by the linear regression
(it is equal to the difference between predicted and actual values
of the dependent variable).
Both inputs must be of the same length (no less than two), and
the independent variable x cannot be constant;
otherwise a <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a> is raised.
For example, we can use the <a class="reference external" href="https://en.wikipedia.org/wiki/Monty_Python#Films">release dates of the Monty
Python films</a>
to predict the cumulative number of Monty Python films
that would have been produced by 2019
assuming that they had kept the pace.
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; year = [1971, 1975, 1979, 1982, 1983]
&gt;&gt;&gt; films_total = [1, 2, 3, 4, 5]
&gt;&gt;&gt; slope, intercept = linear_regression(year, films_total)
&gt;&gt;&gt; round(slope * 2019 + intercept)
16
</pre></div>
</div>
If proportional is true, the independent variable x and the
dependent variable y are assumed to be directly proportional.
The data is fit to a line passing through the origin.
Since the intercept will always be 0.0, the underlying linear
function simplifies to:
<blockquote>
<div>y = slope * x + noise
</div></blockquote>
Continuing the example from <a class="reference internal" href="#statistics.correlation" title="statistics.correlation"><code class="xref py py-func docutils literal notranslate">correlation()</code></a>, we look to see
how well a model based on major planets can predict the orbital
distances for dwarf planets:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; model = linear_regression(period_squared, dist_cubed, proportional=True)
&gt;&gt;&gt; slope = model.slope

&gt;&gt;&gt; # Dwarf planets: Pluto, Eris, Makemake, Haumea, Ceres
&gt;&gt;&gt; orbital_periods = [90_560, 204_199, 111_845, 103_410, 1_680] # days
&gt;&gt;&gt; predicted_dist = [math.cbrt(slope * (p * p)) for p in orbital_periods]
&gt;&gt;&gt; list(map(round, predicted_dist))
[5912, 10166, 6806, 6459, 414]

&gt;&gt;&gt; [5_906, 10_152, 6_796, 6_450, 414] # actual distance in million km
[5906, 10152, 6796, 6450, 414]
</pre></div>
</div>
<div class="versionadded">
New in version 3.10.
</div>
<div class="versionchanged">
Changed in version 3.11: Added support for proportional.
</div>
</dd></dl>

</section>
<section id="exceptions">
<h2>Exceptions<a class="headerlink" href="#exceptions" title="Link to this heading">¶</a></h2>
A single exception is defined:
<dl class="py exception">
<dt class="sig sig-object py" id="statistics.StatisticsError">
exception statistics.StatisticsError<a class="headerlink" href="#statistics.StatisticsError" title="Link to this definition">¶</a></dt>
<dd>Subclass of <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate">ValueError</code></a> for statistics-related exceptions.
</dd></dl>

</section>
<section id="normaldist-objects">
<h2><a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a> objects<a class="headerlink" href="#normaldist-objects" title="Link to this heading">¶</a></h2>
<a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a> is a tool for creating and manipulating normal
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
class that treats the mean and standard deviation of data
measurements as a single entity.
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
of applications in statistics.
<dl class="py class">
<dt class="sig sig-object py" id="statistics.NormalDist">
class statistics.NormalDist(mu=0.0, sigma=1.0)<a class="headerlink" href="#statistics.NormalDist" title="Link to this definition">¶</a></dt>
<dd>Returns a new NormalDist object where mu represents the <a class="reference external" href="https://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic
mean</a> and sigma
represents the <a class="reference external" href="https://en.wikipedia.org/wiki/Standard_deviation">standard deviation</a>.
If sigma is negative, raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate">StatisticsError</code></a>.
<dl class="py attribute">
<dt class="sig sig-object py" id="statistics.NormalDist.mean">
mean<a class="headerlink" href="#statistics.NormalDist.mean" title="Link to this definition">¶</a></dt>
<dd>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of a normal
distribution.
</dd></dl>

<dl class="py attribute">
<dt class="sig sig-object py" id="statistics.NormalDist.median">
median<a class="headerlink" href="#statistics.NormalDist.median" title="Link to this definition">¶</a></dt>
<dd>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Median">median</a> of a normal
distribution.
</dd></dl>

<dl class="py attribute">
<dt class="sig sig-object py" id="statistics.NormalDist.mode">
mode<a class="headerlink" href="#statistics.NormalDist.mode" title="Link to this definition">¶</a></dt>
<dd>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Mode_(statistics)">mode</a> of a normal
distribution.
</dd></dl>

<dl class="py attribute">
<dt class="sig sig-object py" id="statistics.NormalDist.stdev">
stdev<a class="headerlink" href="#statistics.NormalDist.stdev" title="Link to this definition">¶</a></dt>
<dd>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Standard_deviation">standard deviation</a> of a normal
distribution.
</dd></dl>

<dl class="py attribute">
<dt class="sig sig-object py" id="statistics.NormalDist.variance">
variance<a class="headerlink" href="#statistics.NormalDist.variance" title="Link to this definition">¶</a></dt>
<dd>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Variance">variance</a> of a normal
distribution. Equal to the square of the standard deviation.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.from_samples">
classmethod from_samples(data)<a class="headerlink" href="#statistics.NormalDist.from_samples" title="Link to this definition">¶</a></dt>
<dd>Makes a normal distribution instance with mu and sigma parameters
estimated from the data using <a class="reference internal" href="#statistics.fmean" title="statistics.fmean"><code class="xref py py-func docutils literal notranslate">fmean()</code></a> and <a class="reference internal" href="#statistics.stdev" title="statistics.stdev"><code class="xref py py-func docutils literal notranslate">stdev()</code></a>.
The data can be any <a class="reference internal" href="../glossary.html#term-iterable">iterable</a> and should consist of values
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">float</code></a>. If data does not
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">StatisticsError</code></a> because it
takes at least one point to estimate a central value and at least two
points to estimate dispersion.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.samples">
samples(n, *, seed=None)<a class="headerlink" href="#statistics.NormalDist.samples" title="Link to this definition">¶</a></dt>
<dd>Generates n random samples for a given mean and standard deviation.
Returns a <a class="reference internal" href="stdtypes.html#list" title="list"><code class="xref py py-class docutils literal notranslate">list</code></a> of <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate">float</code></a> values.
If seed is given, creates a new instance of the underlying random
number generator. This is useful for creating reproducible results,
even in a multi-threading context.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.pdf">
pdf(x)<a class="headerlink" href="#statistics.NormalDist.pdf" title="Link to this definition">¶</a></dt>
<dd>Using a <a class="reference external" href="https://en.wikipedia.org/wiki/Probability_density_function">probability density function (pdf)</a>, compute
the relative likelihood that a random variable X will be near the
given value x. Mathematically, it is the limit of the ratio <code class="docutils literal notranslate">P(x &lt;=
X &lt; x+dx) / dx</code> as dx approaches zero.
The relative likelihood is computed as the probability of a sample
occurring in a narrow range divided by the width of the range (hence
the word “density”). Since the likelihood is relative to other points,
its value can be greater than <code class="docutils literal notranslate">1.0</code>.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.cdf">
cdf(x)<a class="headerlink" href="#statistics.NormalDist.cdf" title="Link to this definition">¶</a></dt>
<dd>Using a <a class="reference external" href="https://en.wikipedia.org/wiki/Cumulative_distribution_function">cumulative distribution function (cdf)</a>,
compute the probability that a random variable X will be less than or
equal to x. Mathematically, it is written <code class="docutils literal notranslate">P(X &lt;= x)</code>.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.inv_cdf">
inv_cdf(p)<a class="headerlink" href="#statistics.NormalDist.inv_cdf" title="Link to this definition">¶</a></dt>
<dd>Compute the inverse cumulative distribution function, also known as the
<a class="reference external" href="https://en.wikipedia.org/wiki/Quantile_function">quantile function</a>
or the <a class="reference external" href="https://web.archive.org/web/20190203145224/https://www.statisticshowto.datasciencecentral.com/inverse-distribution-function/">percent-point</a>
function. Mathematically, it is written <code class="docutils literal notranslate">x : P(X &lt;= x) = p</code>.
Finds the value x of the random variable X such that the
probability of the variable being less than or equal to that value
equals the given probability p.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.overlap">
overlap(other)<a class="headerlink" href="#statistics.NormalDist.overlap" title="Link to this definition">¶</a></dt>
<dd>Measures the agreement between two normal probability distributions.
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
the two probability density functions</a>.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.quantiles">
quantiles(n=4)<a class="headerlink" href="#statistics.NormalDist.quantiles" title="Link to this definition">¶</a></dt>
<dd>Divide the normal distribution into n continuous intervals with
equal probability. Returns a list of (n - 1) cut points separating
the intervals.
Set n to 4 for quartiles (the default). Set n to 10 for deciles.
Set n to 100 for percentiles which gives the 99 cuts points that
separate the normal distribution into 100 equal sized groups.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="statistics.NormalDist.zscore">
zscore(x)<a class="headerlink" href="#statistics.NormalDist.zscore" title="Link to this definition">¶</a></dt>
<dd>Compute the
<a class="reference external" href="https://www.statisticshowto.com/probability-and-statistics/z-score/">Standard Score</a>
describing x in terms of the number of standard deviations
above or below the mean of the normal distribution:
<code class="docutils literal notranslate">(x - mean) / stdev</code>.
<div class="versionadded">
New in version 3.9.
</div>
</dd></dl>

Instances of <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a> support addition, subtraction,
multiplication and division by a constant. These operations
are used for translation and scaling. For example:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; temperature_february = NormalDist(5, 2.5) # Celsius
&gt;&gt;&gt; temperature_february * (9/5) + 32 # Fahrenheit
NormalDist(mu=41.0, sigma=4.5)
</pre></div>
</div>
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">NormalDist</code></a> is not supported
because the result wouldn’t be normally distributed.
Since normal distributions arise from additive effects of independent
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
distributed random variables</a>
represented as instances of <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a>. For example:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; birth_weights = NormalDist.from_samples([2.5, 3.1, 2.1, 2.4, 2.7, 3.5])
&gt;&gt;&gt; drug_effects = NormalDist(0.4, 0.15)
&gt;&gt;&gt; combined = birth_weights + drug_effects
&gt;&gt;&gt; round(combined.mean, 1)
3.1
&gt;&gt;&gt; round(combined.stdev, 1)
0.5
</pre></div>
</div>
<div class="versionadded">
New in version 3.8.
</div>
</dd></dl>

</section>
<section id="examples-and-recipes">
<h2>Examples and Recipes<a class="headerlink" href="#examples-and-recipes" title="Link to this heading">¶</a></h2>
<section id="classic-probability-problems">
<h3>Classic probability problems<a class="headerlink" href="#classic-probability-problems" title="Link to this heading">¶</a></h3>
<a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a> readily solves classic probability problems.
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
that scores are normally distributed with a mean of 1060 and a standard
deviation of 195, determine the percentage of students with test scores
between 1100 and 1200, after rounding to the nearest whole number:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; sat = NormalDist(1060, 195)
&gt;&gt;&gt; fraction = sat.cdf(1200 + 0.5) - sat.cdf(1100 - 0.5)
&gt;&gt;&gt; round(fraction * 100.0, 1)
18.4
</pre></div>
</div>
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:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; list(map(round, sat.quantiles()))
[928, 1060, 1192]
&gt;&gt;&gt; list(map(round, sat.quantiles(n=10)))
[810, 896, 958, 1011, 1060, 1109, 1162, 1224, 1310]
</pre></div>
</div>
</section>
<section id="monte-carlo-inputs-for-simulations">
<h3>Monte Carlo inputs for simulations<a class="headerlink" href="#monte-carlo-inputs-for-simulations" title="Link to this heading">¶</a></h3>
To estimate the distribution for a model that isn’t easy to solve
analytically, <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a> can generate input samples for a <a class="reference external" href="https://en.wikipedia.org/wiki/Monte_Carlo_method">Monte
Carlo simulation</a>:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; def model(x, y, z):
... return (3*x + 7*x*y - 5*y) / (11 * z)
...
&gt;&gt;&gt; n = 100_000
&gt;&gt;&gt; X = NormalDist(10, 2.5).samples(n, seed=3652260728)
&gt;&gt;&gt; Y = NormalDist(15, 1.75).samples(n, seed=4582495471)
&gt;&gt;&gt; Z = NormalDist(50, 1.25).samples(n, seed=6582483453)
&gt;&gt;&gt; quantiles(map(model, X, Y, Z)) 
[1.4591308524824727, 1.8035946855390597, 2.175091447274739]
</pre></div>
</div>
</section>
<section id="approximating-binomial-distributions">
<h3>Approximating binomial distributions<a class="headerlink" href="#approximating-binomial-distributions" title="Link to this heading">¶</a></h3>
Normal distributions can be used to approximate <a class="reference external" href="https://mathworld.wolfram.com/BinomialDistribution.html">Binomial
distributions</a>
when the sample size is large and when the probability of a successful
trial is near 50%.
For example, an open source conference has 750 attendees and two rooms with a
500 person capacity. There is a talk about Python and another about Ruby.
In previous conferences, 65% of the attendees preferred to listen to Python
talks. Assuming the population preferences haven’t changed, what is the
probability that the Python room will stay within its capacity limits?
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; n = 750 # Sample size
&gt;&gt;&gt; p = 0.65 # Preference for Python
&gt;&gt;&gt; q = 1.0 - p # Preference for Ruby
&gt;&gt;&gt; k = 500 # Room capacity

&gt;&gt;&gt; # Approximation using the cumulative normal distribution
&gt;&gt;&gt; from math import sqrt
&gt;&gt;&gt; round(NormalDist(mu=n*p, sigma=sqrt(n*p*q)).cdf(k + 0.5), 4)
0.8402

&gt;&gt;&gt; # Exact solution using the cumulative binomial distribution
&gt;&gt;&gt; from math import comb, fsum
&gt;&gt;&gt; round(fsum(comb(n, r) * p**r * q**(n-r) for r in range(k+1)), 4)
0.8402

&gt;&gt;&gt; # Approximation using a simulation
&gt;&gt;&gt; from random import seed, binomialvariate
&gt;&gt;&gt; seed(8675309)
&gt;&gt;&gt; mean(binomialvariate(n, p) &lt;= k for i in range(10_000))
0.8406
</pre></div>
</div>
</section>
<section id="naive-bayesian-classifier">
<h3>Naive bayesian classifier<a class="headerlink" href="#naive-bayesian-classifier" title="Link to this heading">¶</a></h3>
Normal distributions commonly arise in machine learning problems.
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>.
The challenge is to predict a person’s gender from measurements of normally
distributed features including height, weight, and foot size.
We’re given a training dataset with measurements for eight people. The
measurements are assumed to be normally distributed, so we summarize the data
with <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate">NormalDist</code></a>:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; height_male = NormalDist.from_samples([6, 5.92, 5.58, 5.92])
&gt;&gt;&gt; height_female = NormalDist.from_samples([5, 5.5, 5.42, 5.75])
&gt;&gt;&gt; weight_male = NormalDist.from_samples([180, 190, 170, 165])
&gt;&gt;&gt; weight_female = NormalDist.from_samples([100, 150, 130, 150])
&gt;&gt;&gt; foot_size_male = NormalDist.from_samples([12, 11, 12, 10])
&gt;&gt;&gt; foot_size_female = NormalDist.from_samples([6, 8, 7, 9])
</pre></div>
</div>
Next, we encounter a new person whose feature measurements are known but whose
gender is unknown:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; ht = 6.0 # height
&gt;&gt;&gt; wt = 130 # weight
&gt;&gt;&gt; fs = 8 # foot size
</pre></div>
</div>
Starting with a 50% <a class="reference external" href="https://en.wikipedia.org/wiki/Prior_probability">prior probability</a> of being male or female,
we compute the posterior as the prior times the product of likelihoods for the
feature measurements given the gender:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; prior_male = 0.5
&gt;&gt;&gt; prior_female = 0.5
&gt;&gt;&gt; posterior_male = (prior_male * height_male.pdf(ht) *
... weight_male.pdf(wt) * foot_size_male.pdf(fs))

&gt;&gt;&gt; posterior_female = (prior_female * height_female.pdf(ht) *
... weight_female.pdf(wt) * foot_size_female.pdf(fs))
</pre></div>
</div>
The final prediction goes to the largest posterior. This is known as the
<a class="reference external" href="https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation">maximum a posteriori</a> or MAP:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; &#39;male&#39; if posterior_male &gt; posterior_female else &#39;female&#39;
&#39;female&#39;
</pre></div>
</div>
</section>
<section id="kernel-density-estimation">
<h3>Kernel density estimation<a class="headerlink" href="#kernel-density-estimation" title="Link to this heading">¶</a></h3>
It is possible to estimate a continuous probability density function
from a fixed number of discrete samples.
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
normal distribution, triangular distribution, or uniform distribution</a>.
The degree of smoothing is controlled by a scaling parameter, <code class="docutils literal notranslate">h</code>,
which is called the bandwidth.
<div class="highlight-python notranslate"><div class="highlight"><pre>def kde_normal(sample, h):
 &quot;Create a continuous probability density function from a sample.&quot;
 # Smooth the sample with a normal distribution kernel scaled by h.
 kernel_h = NormalDist(0.0, h).pdf
 n = len(sample)
 def pdf(x):
 return sum(kernel_h(x - x_i) for x_i in sample) / n
 return pdf
</pre></div>
</div>
<a class="reference external" href="https://en.wikipedia.org/wiki/Kernel_density_estimation#Example">Wikipedia has an example</a>
where we can use the <code class="docutils literal notranslate">kde_normal()</code> recipe to generate and plot
a probability density function estimated from a small sample:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; sample = [-2.1, -1.3, -0.4, 1.9, 5.1, 6.2]
&gt;&gt;&gt; f_hat = kde_normal(sample, h=1.5)
&gt;&gt;&gt; xarr = [i/100 for i in range(-750, 1100)]
&gt;&gt;&gt; yarr = [f_hat(x) for x in xarr]
</pre></div>
</div>
The points in <code class="docutils literal notranslate">xarr</code> and <code class="docutils literal notranslate">yarr</code> can be used to make a PDF plot:
<img alt="Scatter plot of the estimated probability density function." src="../_images/kde_example.png" />
</section>
</section>
</section>

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 <div>
 <h3><a href="../contents.html">Table of Contents</a></h3>
 <ul>
<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate">statistics</code> — Mathematical statistics functions</a><ul>
<li><a class="reference internal" href="#averages-and-measures-of-central-location">Averages and measures of central location</a></li>
<li><a class="reference internal" href="#measures-of-spread">Measures of spread</a></li>
<li><a class="reference internal" href="#statistics-for-relations-between-two-inputs">Statistics for relations between two inputs</a></li>
<li><a class="reference internal" href="#function-details">Function details</a></li>
<li><a class="reference internal" href="#exceptions">Exceptions</a></li>
<li><a class="reference internal" href="#normaldist-objects"><code class="xref py py-class docutils literal notranslate">NormalDist</code> objects</a></li>
<li><a class="reference internal" href="#examples-and-recipes">Examples and Recipes</a><ul>
<li><a class="reference internal" href="#classic-probability-problems">Classic probability problems</a></li>
<li><a class="reference internal" href="#monte-carlo-inputs-for-simulations">Monte Carlo inputs for simulations</a></li>
<li><a class="reference internal" href="#approximating-binomial-distributions">Approximating binomial distributions</a></li>
<li><a class="reference internal" href="#naive-bayesian-classifier">Naive bayesian classifier</a></li>
<li><a class="reference internal" href="#kernel-density-estimation">Kernel density estimation</a></li>
</ul>
</li>
</ul>
</li>
</ul>

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