: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.

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.
=======================  =============================================

:func:`mean`
~~~~~~~~~~~~

The :func:`mean` function calculates the arithmetic mean, commonly known
as the average, of its iterable argument:

.. 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.

   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 effected 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.

   If ``data`` is empty, :exc:`StatisticsError` will be raised.

:func:`median`
~~~~~~~~~~~~~~

The :func:`median` function calculates the median, or middle, data point,
using the common "mean of middle two" method.

   .. seealso::

      :func:`median_low`

      :func:`median_high`

      :func:`median_grouped`

.. function:: median(data)

   Return the median (middle value) of numeric data.

   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.

   If data is empty, :exc:`StatisticsError` is raised.

:func:`median_low`
~~~~~~~~~~~~~~~~~~

The :func:`median_low` function calculates the low median without
interpolation.

.. function:: median_low(data)

   Return the low median of numeric data.

   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.

   If data is empty, :exc:`StatisticsError` is raised.

:func:`median_high`
~~~~~~~~~~~~~~~~~~~

The :func:`median_high` function calculates the high median without
interpolation.

.. function:: median_high(data)

   Return the high median of data.

   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.

   If data is empty, :exc:`StatisticsError` is raised.

:func:`median_grouped`
~~~~~~~~~~~~~~~~~~~~~~

The :func:`median_grouped` function calculates the median of grouped data
as the 50th percentile, using interpolation.

.. function:: median_grouped(data [, interval])

   Return the median of grouped continuous data, calculated as the
   50th percentile.

   .. 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>`_.

   If data is empty, :exc:`StatisticsError` is raised.

:func:`mode`
~~~~~~~~~~~~

The :func:`mode` function calculates the mode, or most common element, of
discrete or nominal data. The mode (when it exists) is the most typical
value, and is a robust measure of central location.

.. function:: mode(data)

   Return the most common data point from discrete or nominal data.

   ``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'

   If data is empty, or if there is not exactly one most common value,
   :exc:`StatisticsError` is raised.

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.
=======================  =============================================

:func:`pstdev`
~~~~~~~~~~~~~~

The :func:`pstdev` function calculates the standard deviation of a
population. The standard deviation is equivalent to the square root of
the variance.

.. function:: pstdev(data [, mu])

   Return 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

:func:`pvariance`
~~~~~~~~~~~~~~~~~

The :func:`pvariance` function calculates the variance of a population.
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.

.. function:: pvariance(data [, mu])

   Return the population variance of *data*, a non-empty iterable of
   real-valued numbers.

   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.

   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.

   Raises :exc:`StatisticsError` if *data* is empty.

:func:`stdev`
~~~~~~~~~~~~~~

The :func:`stdev` function calculates the standard deviation of a sample.
The standard deviation is equivalent to the square root of the variance.

.. function:: stdev(data [, xbar])

   Return 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

:func:`variance`
~~~~~~~~~~~~~~~~~

The :func:`variance` function calculates the variance of a sample. 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.

.. function:: variance(data [, xbar])

   Return the sample variance of *data*, an iterable of at least two
   real-valued numbers.

   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`.

   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.

   Raises :exc:`StatisticsError` if *data* has fewer than two values.

Exceptions
----------

A single exception is defined:

:exc:`StatisticsError`

Subclass of :exc:`ValueError` for statistics-related exceptions.

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