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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

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

.. testsetup:: *

   from statistics import *
   import math
   __name__ = '<doctest>'

--------------

This module provides functions for calculating mathematical statistics of
numeric (:class:`~numbers.Real`-valued) data.

The module is not intended to be a competitor to third-party libraries such
as `NumPy <https://numpy.org>`_, `SciPy <https://scipy.org/>`_, 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 :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.  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 :func:`map` to ensure a consistent result, for
example: ``map(float, input_data)``.

Some datasets use ``NaN`` (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 ``median()``, ``median_low()``,
``median_high()``, ``median_grouped()``, ``mode()``, ``multimode()``, and
``quantiles()``.  The ``NaN`` values should be stripped before calling these
functions::

    >>> from statistics import median
    >>> from math import isnan
    >>> from itertools import filterfalse

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

    >>> sum(map(isnan, data))    # Number of missing values
    2
    >>> clean = list(filterfalse(isnan, data))  # Strip NaN values
    >>> clean
    [20.7, 19.2, 18.3, 14.4]
    >>> sorted(clean)  # Sorting now works as expected
    [14.4, 18.3, 19.2, 20.7]
    >>> median(clean)       # This result is now well defined
    18.75


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:`fmean`            Fast, floating-point arithmetic mean, with optional weighting.
:func:`geometric_mean`   Geometric mean of data.
:func:`harmonic_mean`    Harmonic mean 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 (50th percentile) of grouped data.
:func:`mode`             Single mode (most common value) of discrete or nominal data.
:func:`multimode`        List of modes (most common values) of discrete or nominal data.
:func:`quantiles`        Divide data into intervals with equal probability.
=======================  ===============================================================

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

Statistics for relations between two inputs
-------------------------------------------

These functions calculate statistics regarding relations between two inputs.

=========================  =====================================================
:func:`covariance`         Sample covariance for two variables.
:func:`correlation`        Pearson and Spearman's correlation coefficients.
:func:`linear_regression`  Slope and intercept for simple linear regression.
=========================  =====================================================


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 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, :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
      <https://en.wikipedia.org/wiki/Outlier>`_ and is not necessarily a
      typical example of the data points. For a more robust, although less
      efficient, measure of `central tendency
      <https://en.wikipedia.org/wiki/Central_tendency>`_, see :func:`median`.

      The sample mean gives an unbiased estimate of the true population mean,
      so that when 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:: fmean(data, weights=None)

   Convert *data* to floats and compute the arithmetic mean.

   This runs faster than the :func:`mean` function and it always returns a
   :class:`float`.  The *data* may be a sequence or iterable.  If the input
   dataset is empty, raises a :exc:`StatisticsError`.

   .. doctest::

      >>> fmean([3.5, 4.0, 5.25])
      4.25

   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%:

   .. doctest::

      >>> grades = [85, 92, 83, 91]
      >>> weights = [0.20, 0.20, 0.30, 0.30]
      >>> fmean(grades, weights)
      87.6

   If *weights* is supplied, it must be the same length as the *data* or
   a :exc:`ValueError` will be raised.

   .. versionadded:: 3.8

   .. versionchanged:: 3.11
      Added support for *weights*.


.. function:: geometric_mean(data)

   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 :exc:`StatisticsError` 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.)

   .. doctest::

      >>> round(geometric_mean([54, 24, 36]), 1)
      36.0

   .. versionadded:: 3.8


.. function:: harmonic_mean(data, weights=None)

   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 :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)``.  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?

   .. doctest::

      >>> harmonic_mean([40, 60])
      48.0

   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?

   .. doctest::

      >>> harmonic_mean([40, 60], weights=[5, 30])
      56.0

   :exc:`StatisticsError` 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.)

   .. versionadded:: 3.6

   .. versionchanged:: 3.10
      Added support for *weights*.

.. 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 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:

   .. 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 the data is ordinal (supports order operations) but not numeric (doesn't
   support addition), consider using :func:`median_low` or :func:`median_high`
   instead.

.. 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 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.

   .. 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 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.

   .. 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=1.0)

   Estimates the median for numeric data that has been `grouped or binned
   <https://en.wikipedia.org/wiki/Data_binning>`_ 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:

   .. doctest::

      >>> from collections import Counter
      >>> 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
      ... })
      ...

   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 :func:`median` 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
   :func:`median_grouped`:

   .. doctest::

       >>> data = list(demographics.elements())
       >>> median(data)
       35
       >>> round(median_grouped(data, interval=10), 1)
       37.5

   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.


.. function:: mode(data)

   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 ``min(multimode(data))`` or ``max(multimode(data))``.
   If the input *data* is empty, :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 in this package that
   also applies to nominal (non-numeric) data:

   .. doctest::

      >>> mode(["red", "blue", "blue", "red", "green", "red", "red"])
      'red'

   Only hashable inputs are supported.  To handle type :class:`set`,
   consider casting to :class:`frozenset`.  To handle type :class:`list`,
   consider casting to :class:`tuple`.  For mixed or nested inputs, consider
   using this slower quadratic algorithm that only depends on equality tests:
   ``max(data, key=data.count)``.

   .. versionchanged:: 3.8
      Now handles multimodal datasets by returning the first mode encountered.
      Formerly, it raised :exc:`StatisticsError` when more than one mode was
      found.


.. function:: multimode(data)

   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:

   .. doctest::

        >>> multimode('aabbbbccddddeeffffgg')
        ['b', 'd', 'f']
        >>> multimode('')
        []

   .. versionadded:: 3.8


.. 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 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 should be the *population*
   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 ``None`` (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 :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

   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 a
      random sample of the population, 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 *sample*
   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 sample 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.

.. function:: quantiles(data, *, n=4, method='exclusive')

   Divide *data* 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
   *data* into 100 equal sized groups.  Raises :exc:`StatisticsError` 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 :exc:`StatisticsError` 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, ``100`` and ``112``, the
   cut-point will evaluate to ``104``.

   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 ``i / (m + 1)``.  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 ``(i - 1) / (m - 1)``.  Given 11 sample
   values, the method sorts them and assigns the following percentiles:
   0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 100%.

   .. doctest::

        # Decile cut points for empirically sampled data
        >>> 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]
        >>> [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]

   .. versionadded:: 3.8

.. function:: covariance(x, y, /)

   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
   :exc:`StatisticsError` is raised.

   Examples:

   .. doctest::

      >>> x = [1, 2, 3, 4, 5, 6, 7, 8, 9]
      >>> y = [1, 2, 3, 1, 2, 3, 1, 2, 3]
      >>> covariance(x, y)
      0.75
      >>> z = [9, 8, 7, 6, 5, 4, 3, 2, 1]
      >>> covariance(x, z)
      -7.5
      >>> covariance(z, x)
      -7.5

   .. versionadded:: 3.10

.. function:: correlation(x, y, /, *, method='linear')

   Return the `Pearson's correlation coefficient
   <https://en.wikipedia.org/wiki/Pearson_correlation_coefficient>`_
   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 `Spearman's rank correlation coefficient
   <https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient>`_
   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 :exc:`StatisticsError` is raised.

   Example with `Kepler's laws of planetary motion
   <https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion>`_:

   .. doctest::

      >>> # Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and  Neptune
      >>> orbital_period = [88, 225, 365, 687, 4331, 10_756, 30_687, 60_190]    # days
      >>> dist_from_sun = [58, 108, 150, 228, 778, 1_400, 2_900, 4_500] # million km

      >>> # Show that a perfect monotonic relationship exists
      >>> correlation(orbital_period, dist_from_sun, method='ranked')
      1.0

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

      >>> # Demonstrate Kepler's third law: There is a linear correlation
      >>> # between the square of the orbital period and the cube of the
      >>> # distance from the sun.
      >>> period_squared = [p * p for p in orbital_period]
      >>> dist_cubed = [d * d * d for d in dist_from_sun]
      >>> round(correlation(period_squared, dist_cubed), 4)
      1.0

   .. versionadded:: 3.10

   .. versionchanged:: 3.12
      Added support for Spearman's rank correlation coefficient.

.. function:: linear_regression(x, y, /, *, proportional=False)

   Return the slope and intercept of `simple linear regression
   <https://en.wikipedia.org/wiki/Simple_linear_regression>`_
   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:

      *y = slope \* x + intercept + noise*

   where ``slope`` and ``intercept`` are the regression parameters that are
   estimated, and ``noise`` 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 :exc:`StatisticsError` is raised.

   For example, we can use the `release dates of the Monty
   Python films <https://en.wikipedia.org/wiki/Monty_Python#Films>`_
   to predict the cumulative number of Monty Python films
   that would have been produced by 2019
   assuming that they had kept the pace.

   .. doctest::

      >>> year = [1971, 1975, 1979, 1982, 1983]
      >>> films_total = [1, 2, 3, 4, 5]
      >>> slope, intercept = linear_regression(year, films_total)
      >>> round(slope * 2019 + intercept)
      16

   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:

      *y = slope \* x + noise*

   Continuing the example from :func:`correlation`, we look to see
   how well a model based on major planets can predict the orbital
   distances for dwarf planets:

   .. doctest::

      >>> model = linear_regression(period_squared, dist_cubed, proportional=True)
      >>> slope = model.slope

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

      >>> [5_906, 10_152, 6_796, 6_450, 414]  # actual distance in million km
      [5906, 10152, 6796, 6450, 414]

   .. versionadded:: 3.10

   .. versionchanged:: 3.11
      Added support for *proportional*.

Exceptions
----------

A single exception is defined:

.. exception:: StatisticsError

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


:class:`NormalDist` objects
---------------------------

:class:`NormalDist` is a tool for creating and manipulating normal
distributions of a `random variable
<http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm>`_.  It is a
class that treats the mean and standard deviation of data
measurements as a single entity.

Normal distributions arise from the `Central Limit Theorem
<https://en.wikipedia.org/wiki/Central_limit_theorem>`_ and have a wide range
of applications in statistics.

.. class:: NormalDist(mu=0.0, sigma=1.0)

    Returns a new *NormalDist* object where *mu* represents the `arithmetic
    mean <https://en.wikipedia.org/wiki/Arithmetic_mean>`_ and *sigma*
    represents the `standard deviation
    <https://en.wikipedia.org/wiki/Standard_deviation>`_.

    If *sigma* is negative, raises :exc:`StatisticsError`.

    .. attribute:: mean

       A read-only property for the `arithmetic mean
       <https://en.wikipedia.org/wiki/Arithmetic_mean>`_ of a normal
       distribution.

    .. attribute:: median

       A read-only property for the `median
       <https://en.wikipedia.org/wiki/Median>`_ of a normal
       distribution.

    .. attribute:: mode

       A read-only property for the `mode
       <https://en.wikipedia.org/wiki/Mode_(statistics)>`_ of a normal
       distribution.

    .. attribute:: stdev

       A read-only property for the `standard deviation
       <https://en.wikipedia.org/wiki/Standard_deviation>`_ of a normal
       distribution.

    .. attribute:: variance

       A read-only property for the `variance
       <https://en.wikipedia.org/wiki/Variance>`_ of a normal
       distribution. Equal to the square of the standard deviation.

    .. classmethod:: NormalDist.from_samples(data)

       Makes a normal distribution instance with *mu* and *sigma* parameters
       estimated from the *data* using :func:`fmean` and :func:`stdev`.

       The *data* can be any :term:`iterable` and should consist of values
       that can be converted to type :class:`float`.  If *data* does not
       contain at least two elements, raises :exc:`StatisticsError` because it
       takes at least one point to estimate a central value and at least two
       points to estimate dispersion.

    .. method:: NormalDist.samples(n, *, seed=None)

       Generates *n* random samples for a given mean and standard deviation.
       Returns a :class:`list` of :class:`float` 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.

    .. method:: NormalDist.pdf(x)

       Using a `probability density function (pdf)
       <https://en.wikipedia.org/wiki/Probability_density_function>`_, compute
       the relative likelihood that a random variable *X* will be near the
       given value *x*.  Mathematically, it is the limit of the ratio ``P(x <=
       X < x+dx) / dx`` 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 ``1.0``.

    .. method:: NormalDist.cdf(x)

       Using a `cumulative distribution function (cdf)
       <https://en.wikipedia.org/wiki/Cumulative_distribution_function>`_,
       compute the probability that a random variable *X* will be less than or
       equal to *x*.  Mathematically, it is written ``P(X <= x)``.

    .. method:: NormalDist.inv_cdf(p)

       Compute the inverse cumulative distribution function, also known as the
       `quantile function <https://en.wikipedia.org/wiki/Quantile_function>`_
       or the `percent-point
       <https://web.archive.org/web/20190203145224/https://www.statisticshowto.datasciencecentral.com/inverse-distribution-function/>`_
       function.  Mathematically, it is written ``x : P(X <= x) = p``.

       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*.

    .. method:: NormalDist.overlap(other)

       Measures the agreement between two normal probability distributions.
       Returns a value between 0.0 and 1.0 giving `the overlapping area for
       the two probability density functions
       <https://www.rasch.org/rmt/rmt101r.htm>`_.

    .. method:: NormalDist.quantiles(n=4)

        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.

    .. method:: NormalDist.zscore(x)

        Compute the
        `Standard Score <https://www.statisticshowto.com/probability-and-statistics/z-score/>`_
        describing *x* in terms of the number of standard deviations
        above or below the mean of the normal distribution:
        ``(x - mean) / stdev``.

        .. versionadded:: 3.9

    Instances of :class:`NormalDist` support addition, subtraction,
    multiplication and division by a constant.  These operations
    are used for translation and scaling.  For example:

    .. doctest::

        >>> temperature_february = NormalDist(5, 2.5)             # Celsius
        >>> temperature_february * (9/5) + 32                     # Fahrenheit
        NormalDist(mu=41.0, sigma=4.5)

    Dividing a constant by an instance of :class:`NormalDist` 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 `add and subtract two independent normally
    distributed random variables
    <https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables>`_
    represented as instances of :class:`NormalDist`.  For example:

    .. doctest::

        >>> birth_weights = NormalDist.from_samples([2.5, 3.1, 2.1, 2.4, 2.7, 3.5])
        >>> drug_effects = NormalDist(0.4, 0.15)
        >>> combined = birth_weights + drug_effects
        >>> round(combined.mean, 1)
        3.1
        >>> round(combined.stdev, 1)
        0.5

    .. versionadded:: 3.8


Examples and Recipes
--------------------


Classic probability problems
****************************

:class:`NormalDist` readily solves classic probability problems.

For example, given `historical data for SAT exams
<https://nces.ed.gov/programs/digest/d17/tables/dt17_226.40.asp>`_ 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:

.. doctest::

    >>> sat = NormalDist(1060, 195)
    >>> fraction = sat.cdf(1200 + 0.5) - sat.cdf(1100 - 0.5)
    >>> round(fraction * 100.0, 1)
    18.4

Find the `quartiles <https://en.wikipedia.org/wiki/Quartile>`_ and `deciles
<https://en.wikipedia.org/wiki/Decile>`_ for the SAT scores:

.. doctest::

    >>> list(map(round, sat.quantiles()))
    [928, 1060, 1192]
    >>> list(map(round, sat.quantiles(n=10)))
    [810, 896, 958, 1011, 1060, 1109, 1162, 1224, 1310]


Monte Carlo inputs for simulations
**********************************

To estimate the distribution for a model that isn't easy to solve
analytically, :class:`NormalDist` can generate input samples for a `Monte
Carlo simulation <https://en.wikipedia.org/wiki/Monte_Carlo_method>`_:

.. doctest::

    >>> def model(x, y, z):
    ...     return (3*x + 7*x*y - 5*y) / (11 * z)
    ...
    >>> n = 100_000
    >>> X = NormalDist(10, 2.5).samples(n, seed=3652260728)
    >>> Y = NormalDist(15, 1.75).samples(n, seed=4582495471)
    >>> Z = NormalDist(50, 1.25).samples(n, seed=6582483453)
    >>> quantiles(map(model, X, Y, Z))       # doctest: +SKIP
    [1.4591308524824727, 1.8035946855390597, 2.175091447274739]

Approximating binomial distributions
************************************

Normal distributions can be used to approximate `Binomial
distributions <https://mathworld.wolfram.com/BinomialDistribution.html>`_
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?

.. doctest::

    >>> n = 750             # Sample size
    >>> p = 0.65            # Preference for Python
    >>> q = 1.0 - p         # Preference for Ruby
    >>> k = 500             # Room capacity

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

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

    >>> # Approximation using a simulation
    >>> from random import seed, binomialvariate
    >>> seed(8675309)
    >>> mean(binomialvariate(n, p) <= k for i in range(10_000))
    0.8406


Naive bayesian classifier
*************************

Normal distributions commonly arise in machine learning problems.

Wikipedia has a `nice example of a Naive Bayesian Classifier
<https://en.wikipedia.org/wiki/Naive_Bayes_classifier#Person_classification>`_.
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 :class:`NormalDist`:

.. doctest::

    >>> height_male = NormalDist.from_samples([6, 5.92, 5.58, 5.92])
    >>> height_female = NormalDist.from_samples([5, 5.5, 5.42, 5.75])
    >>> weight_male = NormalDist.from_samples([180, 190, 170, 165])
    >>> weight_female = NormalDist.from_samples([100, 150, 130, 150])
    >>> foot_size_male = NormalDist.from_samples([12, 11, 12, 10])
    >>> foot_size_female = NormalDist.from_samples([6, 8, 7, 9])

Next, we encounter a new person whose feature measurements are known but whose
gender is unknown:

.. doctest::

    >>> ht = 6.0        # height
    >>> wt = 130        # weight
    >>> fs = 8          # foot size

Starting with a 50% `prior probability
<https://en.wikipedia.org/wiki/Prior_probability>`_ of being male or female,
we compute the posterior as the prior times the product of likelihoods for the
feature measurements given the gender:

.. doctest::

   >>> prior_male = 0.5
   >>> prior_female = 0.5
   >>> posterior_male = (prior_male * height_male.pdf(ht) *
   ...                   weight_male.pdf(wt) * foot_size_male.pdf(fs))

   >>> posterior_female = (prior_female * height_female.pdf(ht) *
   ...                     weight_female.pdf(wt) * foot_size_female.pdf(fs))

The final prediction goes to the largest posterior. This is known as the
`maximum a posteriori
<https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation>`_ or MAP:

.. doctest::

  >>> 'male' if posterior_male > posterior_female else 'female'
  'female'


Kernel density estimation
*************************

It is possible to estimate a continuous probability distribution
from a fixed number of discrete samples.

The basic idea is to smooth the data using `a kernel function such as a
normal distribution, triangular distribution, or uniform distribution
<https://en.wikipedia.org/wiki/Kernel_(statistics)#Kernel_functions_in_common_use>`_.
The degree of smoothing is controlled by a scaling parameter, ``h``,
which is called the *bandwidth*.

.. testcode::

   from random import choice, random

   def kde_normal(data, h):
       "Create a continuous probability distribution from discrete samples."

       # Smooth the data with a normal distribution kernel scaled by h.
       K_h = NormalDist(0.0, h)

       def pdf(x):
           'Probability density function.  P(x <= X < x+dx) / dx'
           return sum(K_h.pdf(x - x_i) for x_i in data) / len(data)

       def cdf(x):
           'Cumulative distribution function.  P(X <= x)'
           return sum(K_h.cdf(x - x_i) for x_i in data) / len(data)

       def rand():
           'Random selection from the probability distribution.'
           return choice(data) + K_h.inv_cdf(random())

       return pdf, cdf, rand

`Wikipedia has an example
<https://en.wikipedia.org/wiki/Kernel_density_estimation#Example>`_
where we can use the ``kde_normal()`` recipe to generate and plot
a probability density function estimated from a small sample:

.. doctest::

   >>> sample = [-2.1, -1.3, -0.4, 1.9, 5.1, 6.2]
   >>> pdf, cdf, rand = kde_normal(sample, h=1.5)
   >>> xarr = [i/100 for i in range(-750, 1100)]
   >>> yarr = [pdf(x) for x in xarr]

The points in ``xarr`` and ``yarr`` can be used to make a PDF plot:

.. image:: kde_example.png
   :alt: Scatter plot of the estimated probability density function.

`Resample <https://en.wikipedia.org/wiki/Resampling_(statistics)>`_
the data to produce 100 new selections:

.. doctest::

    >>> new_selections = [rand() for i in range(100)]

Determine the probability of a new selection being below ``2.0``:

.. doctest::

   >>> round(cdf(2.0), 4)
   0.5794

Add a new sample data point and find the new CDF at ``2.0``:

.. doctest::

   >>> sample.append(4.9)
   >>> round(cdf(2.0), 4)
   0.5005


..
   # 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;