statistics — Mathematical statistics functions

New in version 3.4.

Source code: Lib/statistics.py


This module provides functions for calculating mathematical statistics of numeric (Real-valued) data.

Note

Unless explicitly noted otherwise, these functions support int, float, decimal.Decimal and fractions.Fraction. Behaviour with other types (whether in the numeric tower or not) is currently unsupported. Mixed types are also undefined and implementation-dependent. If your input data consists of mixed types, you may be able to use map() to ensure a consistent result, e.g. map(float, input_data).

Averages and measures of central location

These functions calculate an average or typical value from a population or sample.

mean() Arithmetic mean (“average”) of data.
fmean() Fast, floating point arithmetic mean.
geometric_mean() Geometric mean of data.
harmonic_mean() Harmonic mean of data.
median() Median (middle value) of data.
median_low() Low median of data.
median_high() High median of data.
median_grouped() Median, or 50th percentile, of grouped data.
mode() Single mode (most common value) of discrete or nominal data.
multimode() List of modes (most common values) of discrete or nomimal data.

Measures of spread

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

pstdev() Population standard deviation of data.
pvariance() Population variance of data.
stdev() Sample standard deviation of data.
variance() Sample variance of data.

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.

statistics.mean(data)

Return the sample arithmetic mean of data which can be a sequence or iterator.

The arithmetic mean is the sum of the data divided by the number of data points. It is commonly called “the average”, although it is only one of many different mathematical averages. It is a measure of the central location of the data.

If data is empty, StatisticsError will be raised.

Some examples of use:

>>> mean([1, 2, 3, 4, 4])
2.8
>>> mean([-1.0, 2.5, 3.25, 5.75])
2.625

>>> from fractions import Fraction as F
>>> mean([F(3, 7), F(1, 21), F(5, 3), F(1, 3)])
Fraction(13, 21)

>>> from decimal import Decimal as D
>>> mean([D("0.5"), D("0.75"), D("0.625"), D("0.375")])
Decimal('0.5625')

Note

The mean is strongly affected by outliers and is not a robust estimator for central location: the mean is not necessarily a typical example of the data points. For more robust, although less efficient, measures of central location, see median() and mode(). (In this case, “efficient” refers to statistical efficiency rather than computational efficiency.)

The sample mean gives an unbiased estimate of the true population mean, which means that, taken on average over all the possible samples, mean(sample) converges on the true mean of the entire population. If data represents the entire population rather than a sample, then mean(data) is equivalent to calculating the true population mean μ.

statistics.fmean(data)

Convert data to floats and compute the arithmetic mean.

This runs faster than the mean() function and it always returns a float. The result is highly accurate but not as perfect as mean(). If the input dataset is empty, raises a StatisticsError.

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

New in version 3.8.

statistics.geometric_mean(data)

Convert data to floats and compute the geometric mean.

Raises a StatisticsError if the input dataset is empty, if it contains a zero, or if it contains a negative value.

No special efforts are made to achieve exact results. (However, this may change in the future.)

>>> round(geometric_mean([54, 24, 36]), 9)
36.0

New in version 3.8.

statistics.harmonic_mean(data)

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

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

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

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

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

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

StatisticsError is raised if data is empty, or any element is less than zero.

New in version 3.6.

statistics.median(data)

Return the median (middle value) of numeric data, using the common “mean of middle two” method. If data is empty, StatisticsError is raised. data can be a sequence or iterator.

The median is a robust measure of central location, and is less affected by the presence of outliers in your data. When the number of data points is odd, the middle data point is returned:

>>> 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:

>>> median([1, 3, 5, 7])
4.0

This is suited for when your data is discrete, and you don’t mind that the median may not be an actual data point.

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

statistics.median_low(data)

Return the low median of numeric data. If data is empty, StatisticsError is raised. data can be a sequence or iterator.

The low median is always a member of the data set. When the number of data points is odd, the middle value is returned. When it is even, the smaller of the two middle values is returned.

>>> 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.

statistics.median_high(data)

Return the high median of data. If data is empty, StatisticsError is raised. data can be a sequence or iterator.

The high median is always a member of the data set. When the number of data points is odd, the middle value is returned. When it is even, the larger of the two middle values is returned.

>>> 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.

statistics.median_grouped(data, interval=1)

Return the median of grouped continuous data, calculated as the 50th percentile, using interpolation. If data is empty, StatisticsError is raised. data can be a sequence or iterator.

>>> 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:

>>> 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:

>>> 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.

CPython implementation detail: Under some circumstances, median_grouped() may coerce data points to floats. This behaviour is likely to change in the future.

See also

  • “Statistics for the Behavioral Sciences”, Frederick J Gravetter and Larry B Wallnau (8th Edition).
  • The SSMEDIAN function in the Gnome Gnumeric spreadsheet, including this discussion.
statistics.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, returns the first one encountered in the data. If data is empty, 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:

>>> mode([1, 1, 2, 3, 3, 3, 3, 4])
3

The mode is unique in that it is the only statistic which also applies to nominal (non-numeric) data:

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

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

statistics.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:

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

New in version 3.8.

statistics.pstdev(data, mu=None)

Return the population standard deviation (the square root of the population variance). See pvariance() for arguments and other details.

>>> pstdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
0.986893273527251
statistics.pvariance(data, mu=None)

Return the population variance of data, a non-empty iterable of real-valued numbers. Variance, or second moment about the mean, is a measure of the variability (spread or dispersion) of data. A large variance indicates that the data is spread out; a small variance indicates it is clustered closely around the mean.

If the optional second argument mu is given, it should be the mean of data. If it is missing or None (the default), the mean is automatically calculated.

Use this function to calculate the variance from the entire population. To estimate the variance from a sample, the variance() function is usually a better choice.

Raises StatisticsError if data is empty.

Examples:

>>> 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:

>>> mu = mean(data)
>>> pvariance(data, mu)
1.25

This function does not attempt to verify that you have passed the actual mean as mu. Using arbitrary values for mu may lead to invalid or impossible results.

Decimals and Fractions are supported:

>>> from decimal import Decimal as D
>>> pvariance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")])
Decimal('24.815')

>>> from fractions import Fraction as F
>>> pvariance([F(1, 4), F(5, 4), F(1, 2)])
Fraction(13, 72)

Note

When called with the entire population, this gives the population variance σ². When called on a sample instead, this is the biased sample variance s², also known as variance with N degrees of freedom.

If you somehow know the true population mean μ, you may use this function to calculate the variance of a sample, giving the known population mean as the second argument. Provided the data points are representative (e.g. independent and identically distributed), the result will be an unbiased estimate of the population variance.

statistics.stdev(data, xbar=None)

Return the sample standard deviation (the square root of the sample variance). See variance() for arguments and other details.

>>> stdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
1.0810874155219827
statistics.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 pvariance().

Raises StatisticsError if data has fewer than two values.

Examples:

>>> 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:

>>> 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:

>>> 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 pvariance() function as the mu parameter to get the variance of a sample.

Exceptions

A single exception is defined:

exception statistics.StatisticsError

Subclass of ValueError for statistics-related exceptions.

NormalDist objects

NormalDist is a tool for creating and manipulating normal distributions of a random variable. It is a composite class that treats the mean and standard deviation of data measurements as a single entity.

Normal distributions arise from the Central Limit Theorem and have a wide range of applications in statistics.

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

Returns a new NormalDist object where mu represents the arithmetic mean and sigma represents the standard deviation.

If sigma is negative, raises StatisticsError.

mean

A read-only property for the arithmetic mean of a normal distribution.

stdev

A read-only property for the standard deviation of a normal distribution.

variance

A read-only property for the variance of a normal distribution. Equal to the square of the standard deviation.

classmethod from_samples(data)

Makes a normal distribution instance computed from sample data. The data can be any iterable and should consist of values that can be converted to type float.

If data does not contain at least two elements, raises StatisticsError because it takes at least one point to estimate a central value and at least two points to estimate dispersion.

samples(n, seed=None)

Generates n random samples for a given mean and standard deviation. Returns a list of 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.

pdf(x)

Using a probability density function (pdf), compute the relative likelihood that a random variable X will be near the given value x. Mathematically, it is the ratio P(x <= X < x+dx) / dx.

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.

cdf(x)

Using a cumulative distribution function (cdf), compute the probability that a random variable X will be less than or equal to x. Mathematically, it is written P(X <= x).

inv_cdf(p)

Compute the inverse cumulative distribution function, also known as the quantile function or the percent-point 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.

overlap(other)

Compute the overlapping coefficient (OVL) between two normal distributions, giving a measure of agreement. Returns a value between 0.0 and 1.0 giving the overlapping area for two probability density functions.

Instances of 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 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 represented as instances of 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

New in version 3.8.

NormalDist Examples and Recipes

NormalDist readily solves classic probability problems.

For example, given historical data for SAT exams showing that scores are normally distributed with a mean of 1060 and a standard deviation of 192, determine the percentage of students with scores between 1100 and 1200, after rounding to the nearest whole number:

>>> 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 and deciles for the SAT scores:

>>> [round(sat.inv_cdf(p)) for p in (0.25, 0.50, 0.75)]
[928, 1060, 1192]
>>> [round(sat.inv_cdf(p / 10)) for p in range(1, 10)]
[810, 896, 958, 1011, 1060, 1109, 1162, 1224, 1310]

What percentage of men and women will have the same height in two normally distributed populations with known means and standard deviations?

>>> men = NormalDist(70, 4)
>>> women = NormalDist(65, 3.5)
>>> ovl = men.overlap(women)
>>> round(ovl * 100.0, 1)
50.3

To estimate the distribution for a model than isn’t easy to solve analytically, NormalDist can generate input samples for a Monte Carlo simulation:

>>> 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)
>>> Y = NormalDist(15, 1.75).samples(n)
>>> Z = NormalDist(5, 1.25).samples(n)
>>> NormalDist.from_samples(map(model, X, Y, Z))     
NormalDist(mu=19.640137307085507, sigma=47.03273142191088)

Normal distributions commonly arise in machine learning problems.

Wikipedia has a nice example of a Naive Bayesian Classifier. 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 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 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:

>>> 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 or MAP:

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