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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:`kde` Estimate the probability density distribution of the data.
:func:`kde_random` Random sampling from the PDF generated by kde().
: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:: kde(data, h, kernel='normal', *, cumulative=False)
`Kernel Density Estimation (KDE)
<https://www.itm-conferences.org/articles/itmconf/pdf/2018/08/itmconf_sam2018_00037.pdf>`_:
Create a continuous probability density function or cumulative
distribution function from discrete samples.
The basic idea is to smooth the data using `a kernel function
<https://en.wikipedia.org/wiki/Kernel_(statistics)>`_.
to help draw inferences about a population from a sample.
The degree of smoothing is controlled by the scaling parameter *h*
which is called the bandwidth. Smaller values emphasize local
features while larger values give smoother results.
The *kernel* determines the relative weights of the sample data
points. Generally, the choice of kernel shape does not matter
as much as the more influential bandwidth smoothing parameter.
Kernels that give some weight to every sample point include
*normal* (*gauss*), *logistic*, and *sigmoid*.
Kernels that only give weight to sample points within the bandwidth
include *rectangular* (*uniform*), *triangular*, *parabolic*
(*epanechnikov*), *quartic* (*biweight*), *triweight*, and *cosine*.
If *cumulative* is true, will return a cumulative distribution function.
A :exc:`StatisticsError` will be raised if the *data* sequence is empty.
`Wikipedia has an example
<https://en.wikipedia.org/wiki/Kernel_density_estimation#Example>`_
where we can use :func:`kde` 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]
>>> f_hat = kde(sample, h=1.5)
>>> xarr = [i/100 for i in range(-750, 1100)]
>>> yarr = [f_hat(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.
.. versionadded:: 3.13
.. function:: kde_random(data, h, kernel='normal', *, seed=None)
Return a function that makes a random selection from the estimated
probability density function produced by ``kde(data, h, kernel)``.
Providing a *seed* allows reproducible selections. In the future, the
values may change slightly as more accurate kernel inverse CDF estimates
are implemented. The seed may be an integer, float, str, or bytes.
A :exc:`StatisticsError` will be raised if the *data* sequence is empty.
Continuing the example for :func:`kde`, we can use
:func:`kde_random` to generate new random selections from an
estimated probability density function:
>>> data = [-2.1, -1.3, -0.4, 1.9, 5.1, 6.2]
>>> rand = kde_random(data, h=1.5, seed=8675309)
>>> new_selections = [rand() for i in range(10)]
>>> [round(x, 1) for x in new_selections]
[0.7, 6.2, 1.2, 6.9, 7.0, 1.8, 2.5, -0.5, -1.8, 5.6]
.. versionadded:: 3.13
.. 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'
.. 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 is not at least one data point.
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
.. versionchanged:: 3.13
No longer raises an exception for an input with only a single data point.
This allows quantile estimates to be built up one sample point
at a time becoming gradually more refined with each new data point.
.. 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.
.. versionchanged:: 3.13
Switched to a faster algorithm. To reproduce samples from previous
versions, use :func:`random.seed` and :func:`random.gauss`.
.. 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'
..
# 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;
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