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python3.8.1 Doc library statistics.rst

: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:`~numbers.Real`-valued) data.

The module is not intended to be a competitor to third-party libraries such as SciPy, 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).

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 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 and is not a robust estimator
 for central location: the mean is not necessarily a typical example of
 the data points. For more robust measures of central location, see
 :func:`median` and :func:`mode`.

 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)

 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

 .. versionadded:: 3.8

.. 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)

 Return the harmonic mean of *data*, a sequence or iterable 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)``. 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
 rates or ratios, 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 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

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

 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

.. 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)

 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 iterable.

 .. 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 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'

 .. 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 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 ``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 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.

.. 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

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 Central Limit Theorem and have a wide range of applications in statistics.

Returns a new NormalDist object where mu represents the 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://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.

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

>>> 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 :class:`NormalDist`. For example:

>>> 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

:class:`NormalDist` Examples and Recipes

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

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

Find the deciles for the SAT scores:

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

To estimate the distribution for a model than isn't easy to solve analytically, :class:`NormalDist` can generate input samples for a >>> 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]

Normal distributions commonly arise in machine learning problems.

Wikipedia has a :class:`NormalDist`:

>>> 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:

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

Starting with a 50% >>> 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 >>> 'male' if posterior_male > posterior_female else 'female' 'female'

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