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+:mod:`statistics` --- Mathematical statistics functions
+=======================================================
+
+.. module:: statistics
+   :synopsis: mathematical statistics functions
+.. moduleauthor:: Steven D'Aprano <steve+python@pearwood.info>
+.. sectionauthor:: Steven D'Aprano <steve+python@pearwood.info>
+
+.. versionadded:: 3.4
+
+.. testsetup:: *
+
+   from statistics import *
+   __name__ = '<doctest>'
+
+**Source code:** :source:`Lib/statistics.py`
+
+--------------
+
+This module provides functions for calculating mathematical statistics of
+numeric (:class:`Real`-valued) data.
+
+.. note::
+
+   Unless explicitly noted otherwise, these functions support :class:`int`,
+   :class:`float`, :class:`decimal.Decimal` and :class:`fractions.Fraction`.
+   Behaviour with other types (whether in the numeric tower or not) is
+   currently unsupported.  Mixed types are also undefined and
+   implementation-dependent.  If your input data consists of mixed types,
+   you may be able to use :func:`map` to ensure a consistent result, e.g.
+   ``map(float, input_data)``.
+
+Averages and measures of central location
+-----------------------------------------
+
+These functions calculate an average or typical value from a population
+or sample.
+
+=======================  =============================================
+:func:`mean`             Arithmetic mean ("average") of data.
+:func:`median`           Median (middle value) of data.
+:func:`median_low`       Low median of data.
+:func:`median_high`      High median of data.
+:func:`median_grouped`   Median, or 50th percentile, of grouped data.
+:func:`mode`             Mode (most common value) of discrete data.
+=======================  =============================================
+
+Measures of spread
+------------------
+
+These functions calculate a measure of how much the population or sample
+tends to deviate from the typical or average values.
+
+=======================  =============================================
+:func:`pstdev`           Population standard deviation of data.
+:func:`pvariance`        Population variance of data.
+:func:`stdev`            Sample standard deviation of data.
+:func:`variance`         Sample variance of data.
+=======================  =============================================
+
+
+Function details
+----------------
+
+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.
+
+.. function:: mean(data)
+
+   Return the sample arithmetic mean of *data*, a sequence or iterator of
+   real-valued numbers.
+
+   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, :exc:`StatisticsError` will be raised.
+
+   Some examples of use:
+
+   .. doctest::
+
+      >>> mean([1, 2, 3, 4, 4])
+      2.8
+      >>> mean([-1.0, 2.5, 3.25, 5.75])
+      2.625
+
+      >>> from fractions import Fraction as F
+      >>> mean([F(3, 7), F(1, 21), F(5, 3), F(1, 3)])
+      Fraction(13, 21)
+
+      >>> from decimal import Decimal as D
+      >>> mean([D("0.5"), D("0.75"), D("0.625"), D("0.375")])
+      Decimal('0.5625')
+
+   .. note::
+
+      The mean is strongly affected by outliers and is not a robust estimator
+      for central location: the mean is not necessarily a typical example of the
+      data points.  For more robust, although less efficient, measures of
+      central location, see :func:`median` and :func:`mode`.  (In this case,
+      "efficient" refers to statistical efficiency rather than computational
+      efficiency.)
+
+      The sample mean gives an unbiased estimate of the true population mean,
+      which means that, taken on average over all the possible samples,
+      ``mean(sample)`` converges on the true mean of the entire population.  If
+      *data* represents the entire population rather than a sample, then
+      ``mean(data)`` is equivalent to calculating the true population mean μ.
+
+
+.. function:: median(data)
+
+   Return the median (middle value) of numeric data, using the common "mean of
+   middle two" method.  If *data* is empty, :exc:`StatisticsError` is raised.
+
+   The median is a robust measure of central location, and is less affected by
+   the presence of outliers in your data.  When the number of data points is
+   odd, the middle data point is returned:
+
+   .. doctest::
+
+      >>> median([1, 3, 5])
+      3
+
+   When the number of data points is even, the median is interpolated by taking
+   the average of the two middle values:
+
+   .. doctest::
+
+      >>> median([1, 3, 5, 7])
+      4.0
+
+   This is suited for when your data is discrete, and you don't mind that the
+   median may not be an actual data point.
+
+   .. seealso:: :func:`median_low`, :func:`median_high`, :func:`median_grouped`
+
+
+.. function:: median_low(data)
+
+   Return the low median of numeric data.  If *data* is empty,
+   :exc:`StatisticsError` is raised.
+
+   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.
+
+   .. doctest::
+
+      >>> median_low([1, 3, 5])
+      3
+      >>> median_low([1, 3, 5, 7])
+      3
+
+   Use the low median when your data are discrete and you prefer the median to
+   be an actual data point rather than interpolated.
+
+
+.. function:: median_high(data)
+
+   Return the high median of data.  If *data* is empty, :exc:`StatisticsError`
+   is raised.
+
+   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.
+
+   .. doctest::
+
+      >>> median_high([1, 3, 5])
+      3
+      >>> median_high([1, 3, 5, 7])
+      5
+
+   Use the high median when your data are discrete and you prefer the median to
+   be an actual data point rather than interpolated.
+
+
+.. function:: median_grouped(data, interval=1)
+
+   Return the median of grouped continuous data, calculated as the 50th
+   percentile, using interpolation.  If *data* is empty, :exc:`StatisticsError`
+   is raised.
+
+   .. doctest::
+
+      >>> median_grouped([52, 52, 53, 54])
+      52.5
+
+   In the following example, the data are rounded, so that each value represents
+   the midpoint of data classes, e.g. 1 is the midpoint of the class 0.5-1.5, 2
+   is the midpoint of 1.5-2.5, 3 is the midpoint of 2.5-3.5, etc.  With the data
+   given, the middle value falls somewhere in the class 3.5-4.5, and
+   interpolation is used to estimate it:
+
+   .. doctest::
+
+      >>> median_grouped([1, 2, 2, 3, 4, 4, 4, 4, 4, 5])
+      3.7
+
+   Optional argument *interval* represents the class interval, and defaults
+   to 1.  Changing the class interval naturally will change the interpolation:
+
+   .. doctest::
+
+      >>> median_grouped([1, 3, 3, 5, 7], interval=1)
+      3.25
+      >>> median_grouped([1, 3, 3, 5, 7], interval=2)
+      3.5
+
+   This function does not check whether the data points are at least
+   *interval* apart.
+
+   .. impl-detail::
+
+      Under some circumstances, :func:`median_grouped` may coerce data points to
+      floats.  This behaviour is likely to change in the future.
+
+   .. seealso::
+
+      * "Statistics for the Behavioral Sciences", Frederick J Gravetter and
+        Larry B Wallnau (8th Edition).
+
+      * Calculating the `median <http://www.ualberta.ca/~opscan/median.html>`_.
+
+      * The `SSMEDIAN
+        <https://projects.gnome.org/gnumeric/doc/gnumeric-function-SSMEDIAN.shtml>`_
+        function in the Gnome Gnumeric spreadsheet, including `this discussion
+        <https://mail.gnome.org/archives/gnumeric-list/2011-April/msg00018.html>`_.
+
+
+.. function:: mode(data)
+
+   Return the most common data point from discrete or nominal *data*.  The mode
+   (when it exists) is the most typical value, and is a robust measure of
+   central location.
+
+   If *data* is empty, or if there is not exactly one most common value,
+   :exc:`StatisticsError` is raised.
+
+   ``mode`` assumes discrete data, and returns a single value. This is the
+   standard treatment of the mode as commonly taught in schools:
+
+   .. doctest::
+
+      >>> mode([1, 1, 2, 3, 3, 3, 3, 4])
+      3
+
+   The mode is unique in that it is the only statistic which also applies
+   to nominal (non-numeric) data:
+
+   .. doctest::
+
+      >>> mode(["red", "blue", "blue", "red", "green", "red", "red"])
+      'red'
+
+
+.. function:: pstdev(data, mu=None)
+
+   Return the population standard deviation (the square root of the population
+   variance).  See :func:`pvariance` for arguments and other details.
+
+   .. doctest::
+
+      >>> pstdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
+      0.986893273527251
+
+
+.. function:: pvariance(data, mu=None)
+
+   Return the population variance of *data*, a non-empty iterable of real-valued
+   numbers.  Variance, or second moment about the mean, is a measure of the
+   variability (spread or dispersion) of data.  A large variance indicates that
+   the data is spread out; a small variance indicates it is clustered closely
+   around the mean.
+
+   If the optional second argument *mu* is given, it should be the mean of
+   *data*.  If it is missing or ``None`` (the default), the mean is
+   automatically calculated.
+
+   Use this function to calculate the variance from the entire population.  To
+   estimate the variance from a sample, the :func:`variance` function is usually
+   a better choice.
+
+   Raises :exc:`StatisticsError` if *data* is empty.
+
+   Examples:
+
+   .. doctest::
+
+      >>> data = [0.0, 0.25, 0.25, 1.25, 1.5, 1.75, 2.75, 3.25]
+      >>> pvariance(data)
+      1.25
+
+   If you have already calculated the mean of your data, you can pass it as the
+   optional second argument *mu* to avoid recalculation:
+
+   .. doctest::
+
+      >>> mu = mean(data)
+      >>> pvariance(data, mu)
+      1.25
+
+   This function does not attempt to verify that you have passed the actual mean
+   as *mu*.  Using arbitrary values for *mu* may lead to invalid or impossible
+   results.
+
+   Decimals and Fractions are supported:
+
+   .. doctest::
+
+      >>> from decimal import Decimal as D
+      >>> pvariance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")])
+      Decimal('24.815')
+
+      >>> from fractions import Fraction as F
+      >>> pvariance([F(1, 4), F(5, 4), F(1, 2)])
+      Fraction(13, 72)
+
+   .. 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 representative
+      (e.g. independent and identically distributed), the result will be an
+      unbiased estimate of the population variance.
+
+
+.. function:: stdev(data, xbar=None)
+
+   Return the sample standard deviation (the square root of the sample
+   variance).  See :func:`variance` for arguments and other details.
+
+   .. doctest::
+
+      >>> stdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
+      1.0810874155219827
+
+
+.. function:: variance(data, xbar=None)
+
+   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 ``None`` (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 :func:`pvariance`.
+
+   Raises :exc:`StatisticsError` if *data* has fewer than two values.
+
+   Examples:
+
+   .. doctest::
+
+      >>> data = [2.75, 1.75, 1.25, 0.25, 0.5, 1.25, 3.5]
+      >>> variance(data)
+      1.3720238095238095
+
+   If you have already calculated the mean of your data, you can pass it as the
+   optional second argument *xbar* to avoid recalculation:
+
+   .. doctest::
+
+      >>> m = mean(data)
+      >>> variance(data, m)
+      1.3720238095238095
+
+   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:
+
+   .. doctest::
+
+      >>> from decimal import Decimal as D
+      >>> variance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")])
+      Decimal('31.01875')
+
+      >>> from fractions import Fraction as F
+      >>> variance([F(1, 6), F(1, 2), F(5, 3)])
+      Fraction(67, 108)
+
+   .. 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
+      :func:`pvariance` function as the *mu* parameter to get the variance of a
+      sample.
+
+Exceptions
+----------
+
+A single exception is defined:
+
+.. exception:: StatisticsError
+
+   Subclass of :exc:`ValueError` for statistics-related exceptions.
+
+..
+   # This modelines must appear within the last ten lines of the file.
+   kate: indent-width 3; remove-trailing-space on; replace-tabs on; encoding utf-8;