: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

Source code: :source:`Lib/statistics.py`

.. testsetup:: *

 from statistics import *
 __name__ = '<doctest>'

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.

Measures of spread

These functions calculate a measure of how much the population or sample tends to deviate from the typical or average values.

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* which can be a sequence or iterator.

 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:: harmonic_mean(data)

 Return the harmonic mean of *data*, a sequence or iterator of
 real-valued numbers.

 The harmonic mean, sometimes called the subcontrary mean, is the
 reciprocal of the arithmetic :func:`mean` of the reciprocals of the
 data. For example, the harmonic mean of three values *a*, *b* and *c*
 will be equivalent to ``3/(1/a + 1/b + 1/c)``.

 The harmonic mean is a type of average, a measure of the central
 location of the data. It is often appropriate when averaging quantities
 which are rates or ratios, for example speeds. For example:

 Suppose an investor purchases an equal value of shares in each of
 three companies, with P/E (price/earning) ratios of 2.5, 3 and 10.
 What is the average P/E ratio for the investor's portfolio?

 .. doctest::

 >>> harmonic_mean([2.5, 3, 10]) # For an equal investment portfolio.
 3.6

 Using the arithmetic mean would give an average of about 5.167, which
 is too high.

 :exc:`StatisticsError` is raised if *data* is empty, or any element
 is less than zero.

 .. versionadded:: 3.6

.. 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.
 *data* can be a sequence or iterator.

 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 your data is ordinal (supports order operations) but not numeric (doesn't
 support addition), you should use :func:`median_low` or :func:`median_high`
 instead.

 .. 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. *data* can be a sequence or iterator.

 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. *data* can be a sequence or iterator.

 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. *data* can be a sequence or iterator.

 .. 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).

 * The `SSMEDIAN
 <https://help.gnome.org/users/gnumeric/stable/gnumeric.html#gnumeric-function-SSMEDIAN>`_
 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.

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