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Issue #18606: Add the new "statistics" module (PEP 450). Contributed 

by Steven D'Aprano.
This commit is contained in:
Larry Hastings 2013-10-19 11:50:09 -07:00
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@ -23,3 +23,4 @@ The following modules are documented in this chapter:
decimal.rst
fractions.rst
random.rst
statistics.rst

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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
.. testsetup:: *
from statistics import *
__name__ = '<doctest>'
**Source code:** :source:`Lib/statistics.py`
--------------
This module provides functions for calculating mathematical statistics of
numeric (:class:`Real`-valued) data.
Averages and measures of central location
-----------------------------------------
These functions calculate an average or typical value from a population
or sample.
======================= =============================================
:func:`mean` Arithmetic mean ("average") 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, or 50th percentile, of grouped data.
:func:`mode` Mode (most common value) of discrete data.
======================= =============================================
:func:`mean`
~~~~~~~~~~~~
The :func:`mean` function calculates the arithmetic mean, commonly known
as the average, of its iterable argument:
.. function:: mean(data)
Return the sample arithmetic mean of *data*, a sequence or iterator
of real-valued numbers.
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.
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 effected by outliers and is not a robust
estimator for central location: the mean is not necessarily a
typical example of the data points. For more robust, although less
efficient, measures of central location, see :func:`median` and
:func:`mode`. (In this case, "efficient" refers to statistical
efficiency rather than computational efficiency.)
The sample mean gives an unbiased estimate of the true population
mean, which means that, taken on average over all the possible
samples, ``mean(sample)`` converges on the true mean of the entire
population. If *data* represents the entire population rather than
a sample, then ``mean(data)`` is equivalent to calculating the true
population mean μ.
If ``data`` is empty, :exc:`StatisticsError` will be raised.
:func:`median`
~~~~~~~~~~~~~~
The :func:`median` function calculates the median, or middle, data point,
using the common "mean of middle two" method.
.. seealso::
:func:`median_low`
:func:`median_high`
:func:`median_grouped`
.. function:: median(data)
Return the median (middle value) of numeric data.
The median is a robust measure of central location, and is less affected
by the presence of outliers in your data. When the number of data points
is odd, the middle data point is returned:
.. doctest::
>>> median([1, 3, 5])
3
When the number of data points is even, the median is interpolated by
taking the average of the two middle values:
.. doctest::
>>> median([1, 3, 5, 7])
4.0
This is suited for when your data is discrete, and you don't mind that
the median may not be an actual data point.
If data is empty, :exc:`StatisticsError` is raised.
:func:`median_low`
~~~~~~~~~~~~~~~~~~
The :func:`median_low` function calculates the low median without
interpolation.
.. function:: median_low(data)
Return the low median of numeric data.
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.
If data is empty, :exc:`StatisticsError` is raised.
:func:`median_high`
~~~~~~~~~~~~~~~~~~~
The :func:`median_high` function calculates the high median without
interpolation.
.. function:: median_high(data)
Return the high median of data.
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.
If data is empty, :exc:`StatisticsError` is raised.
:func:`median_grouped`
~~~~~~~~~~~~~~~~~~~~~~
The :func:`median_grouped` function calculates the median of grouped data
as the 50th percentile, using interpolation.
.. function:: median_grouped(data [, interval])
Return the median of grouped continuous data, calculated as the
50th percentile.
.. 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).
* Calculating the `median <http://www.ualberta.ca/~opscan/median.html>`_.
* The `SSMEDIAN <https://projects.gnome.org/gnumeric/doc/gnumeric-function-SSMEDIAN.shtml>`_
function in the Gnome Gnumeric spreadsheet, including
`this discussion <https://mail.gnome.org/archives/gnumeric-list/2011-April/msg00018.html>`_.
If data is empty, :exc:`StatisticsError` is raised.
:func:`mode`
~~~~~~~~~~~~
The :func:`mode` function calculates the mode, or most common element, of
discrete or nominal data. The mode (when it exists) is the most typical
value, and is a robust measure of central location.
.. function:: mode(data)
Return the most common data point from discrete or nominal data.
``mode`` assumes discrete data, and returns a single value. This is the
standard treatment of the mode as commonly taught in schools:
.. doctest::
>>> mode([1, 1, 2, 3, 3, 3, 3, 4])
3
The mode is unique in that it is the only statistic which also applies
to nominal (non-numeric) data:
.. doctest::
>>> mode(["red", "blue", "blue", "red", "green", "red", "red"])
'red'
If data is empty, or if there is not exactly one most common value,
:exc:`StatisticsError` is raised.
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.
======================= =============================================
:func:`pstdev`
~~~~~~~~~~~~~~
The :func:`pstdev` function calculates the standard deviation of a
population. The standard deviation is equivalent to the square root of
the variance.
.. function:: pstdev(data [, mu])
Return 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
:func:`pvariance`
~~~~~~~~~~~~~~~~~
The :func:`pvariance` function calculates the variance of a population.
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.
.. function:: pvariance(data [, mu])
Return the population variance of *data*, a non-empty iterable of
real-valued numbers.
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 caclulated.
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.
Examples:
.. doctest::
>>> data = [0.0, 0.25, 0.25, 1.25, 1.5, 1.75, 2.75, 3.25]
>>> pvariance(data)
1.25
If you have already calculated the mean of your data, you can pass
it as the optional second argument *mu* to avoid recalculation:
.. doctest::
>>> mu = mean(data)
>>> pvariance(data, mu)
1.25
This function does not attempt to verify that you have passed the actual
mean as *mu*. Using arbitrary values for *mu* may lead to invalid or
impossible results.
Decimals and Fractions are supported:
.. doctest::
>>> from decimal import Decimal as D
>>> pvariance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")])
Decimal('24.815')
>>> from fractions import Fraction as F
>>> pvariance([F(1, 4), F(5, 4), F(1, 2)])
Fraction(13, 72)
.. note::
When called with the entire population, this gives the population
variance σ². When called on a sample instead, this is the biased
sample variance s², also known as variance with N degrees of freedom.
If you somehow know the true population mean μ, you may use this
function to calculate the variance of a sample, giving the known
population mean as the second argument. Provided the data points are
representative (e.g. independent and identically distributed), the
result will be an unbiased estimate of the population variance.
Raises :exc:`StatisticsError` if *data* is empty.
:func:`stdev`
~~~~~~~~~~~~~~
The :func:`stdev` function calculates the standard deviation of a sample.
The standard deviation is equivalent to the square root of the variance.
.. function:: stdev(data [, xbar])
Return 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
:func:`variance`
~~~~~~~~~~~~~~~~~
The :func:`variance` function calculates the variance of a sample. 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.
.. function:: variance(data [, xbar])
Return the sample variance of *data*, an iterable of at least two
real-valued numbers.
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 caclulated.
Use this function when your data is a sample from a population. To
calculate the variance from the entire population, see :func:`pvariance`.
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.
Raises :exc:`StatisticsError` if *data* has fewer than two values.
Exceptions
----------
A single exception is defined:
:exc:`StatisticsError`
Subclass of :exc:`ValueError` for statistics-related exceptions.
..
# 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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## Module statistics.py
##
## Copyright (c) 2013 Steven D'Aprano <steve+python@pearwood.info>.
##
## Licensed under the Apache License, Version 2.0 (the "License");
## you may not use this file except in compliance with the License.
## You may obtain a copy of the License at
##
## http://www.apache.org/licenses/LICENSE-2.0
##
## Unless required by applicable law or agreed to in writing, software
## distributed under the License is distributed on an "AS IS" BASIS,
## WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
## See the License for the specific language governing permissions and
## limitations under the License.
"""
Basic statistics module.
This module provides functions for calculating statistics of data, including
averages, variance, and standard deviation.
Calculating averages
--------------------
================== =============================================
Function Description
================== =============================================
mean Arithmetic mean (average) 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 Mode (most common value) of data.
================== =============================================
Calculate the arithmetic mean ("the average") of data:
>>> mean([-1.0, 2.5, 3.25, 5.75])
2.625
Calculate the standard median of discrete data:
>>> median([2, 3, 4, 5])
3.5
Calculate the median, or 50th percentile, of data grouped into class intervals
centred on the data values provided. E.g. if your data points are rounded to
the nearest whole number:
>>> median_grouped([2, 2, 3, 3, 3, 4]) #doctest: +ELLIPSIS
2.8333333333...
This should be interpreted in this way: you have two data points in the class
interval 1.5-2.5, three data points in the class interval 2.5-3.5, and one in
the class interval 3.5-4.5. The median of these data points is 2.8333...
Calculating variability or spread
---------------------------------
================== =============================================
Function Description
================== =============================================
pvariance Population variance of data.
variance Sample variance of data.
pstdev Population standard deviation of data.
stdev Sample standard deviation of data.
================== =============================================
Calculate the standard deviation of sample data:
>>> stdev([2.5, 3.25, 5.5, 11.25, 11.75]) #doctest: +ELLIPSIS
4.38961843444...
If you have previously calculated the mean, you can pass it as the optional
second argument to the four "spread" functions to avoid recalculating it:
>>> data = [1, 2, 2, 4, 4, 4, 5, 6]
>>> mu = mean(data)
>>> pvariance(data, mu)
2.5
Exceptions
----------
A single exception is defined: StatisticsError is a subclass of ValueError.
"""
__all__ = [ 'StatisticsError',
'pstdev', 'pvariance', 'stdev', 'variance',
'median', 'median_low', 'median_high', 'median_grouped',
'mean', 'mode',
]
import collections
import math
import numbers
import operator
from fractions import Fraction
from decimal import Decimal
# === Exceptions ===
class StatisticsError(ValueError):
pass
# === Private utilities ===
def _sum(data, start=0):
"""_sum(data [, start]) -> value
Return a high-precision sum of the given numeric data. If optional
argument ``start`` is given, it is added to the total. If ``data`` is
empty, ``start`` (defaulting to 0) is returned.
Examples
--------
>>> _sum([3, 2.25, 4.5, -0.5, 1.0], 0.75)
11.0
Some sources of round-off error will be avoided:
>>> _sum([1e50, 1, -1e50] * 1000) # Built-in sum returns zero.
1000.0
Fractions and Decimals are also supported:
>>> from fractions import Fraction as F
>>> _sum([F(2, 3), F(7, 5), F(1, 4), F(5, 6)])
Fraction(63, 20)
>>> from decimal import Decimal as D
>>> data = [D("0.1375"), D("0.2108"), D("0.3061"), D("0.0419")]
>>> _sum(data)
Decimal('0.6963')
"""
n, d = _exact_ratio(start)
T = type(start)
partials = {d: n} # map {denominator: sum of numerators}
# Micro-optimizations.
coerce_types = _coerce_types
exact_ratio = _exact_ratio
partials_get = partials.get
# Add numerators for each denominator, and track the "current" type.
for x in data:
T = _coerce_types(T, type(x))
n, d = exact_ratio(x)
partials[d] = partials_get(d, 0) + n
if None in partials:
assert issubclass(T, (float, Decimal))
assert not math.isfinite(partials[None])
return T(partials[None])
total = Fraction()
for d, n in sorted(partials.items()):
total += Fraction(n, d)
if issubclass(T, int):
assert total.denominator == 1
return T(total.numerator)
if issubclass(T, Decimal):
return T(total.numerator)/total.denominator
return T(total)
def _exact_ratio(x):
"""Convert Real number x exactly to (numerator, denominator) pair.
>>> _exact_ratio(0.25)
(1, 4)
x is expected to be an int, Fraction, Decimal or float.
"""
try:
try:
# int, Fraction
return (x.numerator, x.denominator)
except AttributeError:
# float
try:
return x.as_integer_ratio()
except AttributeError:
# Decimal
try:
return _decimal_to_ratio(x)
except AttributeError:
msg = "can't convert type '{}' to numerator/denominator"
raise TypeError(msg.format(type(x).__name__)) from None
except (OverflowError, ValueError):
# INF or NAN
if __debug__:
# Decimal signalling NANs cannot be converted to float :-(
if isinstance(x, Decimal):
assert not x.is_finite()
else:
assert not math.isfinite(x)
return (x, None)
# FIXME This is faster than Fraction.from_decimal, but still too slow.
def _decimal_to_ratio(d):
"""Convert Decimal d to exact integer ratio (numerator, denominator).
>>> from decimal import Decimal
>>> _decimal_to_ratio(Decimal("2.6"))
(26, 10)
"""
sign, digits, exp = d.as_tuple()
if exp in ('F', 'n', 'N'): # INF, NAN, sNAN
assert not d.is_finite()
raise ValueError
num = 0
for digit in digits:
num = num*10 + digit
if sign:
num = -num
den = 10**-exp
return (num, den)
def _coerce_types(T1, T2):
"""Coerce types T1 and T2 to a common type.
>>> _coerce_types(int, float)
<class 'float'>
Coercion is performed according to this table, where "N/A" means
that a TypeError exception is raised.
+----------+-----------+-----------+-----------+----------+
| | int | Fraction | Decimal | float |
+----------+-----------+-----------+-----------+----------+
| int | int | Fraction | Decimal | float |
| Fraction | Fraction | Fraction | N/A | float |
| Decimal | Decimal | N/A | Decimal | float |
| float | float | float | float | float |
+----------+-----------+-----------+-----------+----------+
Subclasses trump their parent class; two subclasses of the same
base class will be coerced to the second of the two.
"""
# Get the common/fast cases out of the way first.
if T1 is T2: return T1
if T1 is int: return T2
if T2 is int: return T1
# Subclasses trump their parent class.
if issubclass(T2, T1): return T2
if issubclass(T1, T2): return T1
# Floats trump everything else.
if issubclass(T2, float): return T2
if issubclass(T1, float): return T1
# Subclasses of the same base class give priority to the second.
if T1.__base__ is T2.__base__: return T2
# Otherwise, just give up.
raise TypeError('cannot coerce types %r and %r' % (T1, T2))
def _counts(data):
# Generate a table of sorted (value, frequency) pairs.
if data is None:
raise TypeError('None is not iterable')
table = collections.Counter(data).most_common()
if not table:
return table
# Extract the values with the highest frequency.
maxfreq = table[0][1]
for i in range(1, len(table)):
if table[i][1] != maxfreq:
table = table[:i]
break
return table
# === Measures of central tendency (averages) ===
def mean(data):
"""Return the sample arithmetic mean of data.
>>> mean([1, 2, 3, 4, 4])
2.8
>>> 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')
If ``data`` is empty, StatisticsError will be raised.
"""
if iter(data) is data:
data = list(data)
n = len(data)
if n < 1:
raise StatisticsError('mean requires at least one data point')
return _sum(data)/n
# FIXME: investigate ways to calculate medians without sorting? Quickselect?
def median(data):
"""Return the median (middle value) of numeric data.
When the number of data points is odd, return the middle data point.
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])
3
>>> median([1, 3, 5, 7])
4.0
"""
data = sorted(data)
n = len(data)
if n == 0:
raise StatisticsError("no median for empty data")
if n%2 == 1:
return data[n//2]
else:
i = n//2
return (data[i - 1] + data[i])/2
def median_low(data):
"""Return the low median of numeric data.
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
"""
data = sorted(data)
n = len(data)
if n == 0:
raise StatisticsError("no median for empty data")
if n%2 == 1:
return data[n//2]
else:
return data[n//2 - 1]
def median_high(data):
"""Return the high median of data.
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
"""
data = sorted(data)
n = len(data)
if n == 0:
raise StatisticsError("no median for empty data")
return data[n//2]
def median_grouped(data, interval=1):
""""Return the 50th percentile (median) of grouped continuous data.
>>> median_grouped([1, 2, 2, 3, 4, 4, 4, 4, 4, 5])
3.7
>>> median_grouped([52, 52, 53, 54])
52.5
This calculates the median as the 50th percentile, and should be
used when your data is continuous and grouped. In the above example,
the values 1, 2, 3, etc. actually represent the midpoint of classes
0.5-1.5, 1.5-2.5, 2.5-3.5, etc. The middle value falls somewhere in
class 3.5-4.5, and interpolation is used to estimate it.
Optional argument ``interval`` represents the class interval, and
defaults to 1. Changing the class interval naturally will change the
interpolated 50th percentile value:
>>> 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.
"""
data = sorted(data)
n = len(data)
if n == 0:
raise StatisticsError("no median for empty data")
elif n == 1:
return data[0]
# Find the value at the midpoint. Remember this corresponds to the
# centre of the class interval.
x = data[n//2]
for obj in (x, interval):
if isinstance(obj, (str, bytes)):
raise TypeError('expected number but got %r' % obj)
try:
L = x - interval/2 # The lower limit of the median interval.
except TypeError:
# Mixed type. For now we just coerce to float.
L = float(x) - float(interval)/2
cf = data.index(x) # Number of values below the median interval.
# FIXME The following line could be more efficient for big lists.
f = data.count(x) # Number of data points in the median interval.
return L + interval*(n/2 - cf)/f
def mode(data):
"""Return the most common data point from discrete or nominal data.
``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
This also works with nominal (non-numeric) data:
>>> mode(["red", "blue", "blue", "red", "green", "red", "red"])
'red'
If there is not exactly one most common value, ``mode`` will raise
StatisticsError.
"""
# Generate a table of sorted (value, frequency) pairs.
table = _counts(data)
if len(table) == 1:
return table[0][0]
elif table:
raise StatisticsError(
'no unique mode; found %d equally common values' % len(table)
)
else:
raise StatisticsError('no mode for empty data')
# === Measures of spread ===
# See http://mathworld.wolfram.com/Variance.html
# http://mathworld.wolfram.com/SampleVariance.html
# http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance
#
# Under no circumstances use the so-called "computational formula for
# variance", as that is only suitable for hand calculations with a small
# amount of low-precision data. It has terrible numeric properties.
#
# See a comparison of three computational methods here:
# http://www.johndcook.com/blog/2008/09/26/comparing-three-methods-of-computing-standard-deviation/
def _ss(data, c=None):
"""Return sum of square deviations of sequence data.
If ``c`` is None, the mean is calculated in one pass, and the deviations
from the mean are calculated in a second pass. Otherwise, deviations are
calculated from ``c`` as given. Use the second case with care, as it can
lead to garbage results.
"""
if c is None:
c = mean(data)
ss = _sum((x-c)**2 for x in data)
# The following sum should mathematically equal zero, but due to rounding
# error may not.
ss -= _sum((x-c) for x in data)**2/len(data)
assert not ss < 0, 'negative sum of square deviations: %f' % ss
return ss
def variance(data, xbar=None):
"""Return the sample variance of data.
data should be an iterable of Real-valued numbers, with at least two
values. The optional argument xbar, if given, should be the mean of
the data. If it is missing or None, 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``.
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 recalculating it:
>>> m = mean(data)
>>> variance(data, m)
1.3720238095238095
This function does not check that ``xbar`` is actually the mean of
``data``. Giving arbitrary values for ``xbar`` may lead to invalid or
impossible results.
Decimals and Fractions 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)
"""
if iter(data) is data:
data = list(data)
n = len(data)
if n < 2:
raise StatisticsError('variance requires at least two data points')
ss = _ss(data, xbar)
return ss/(n-1)
def pvariance(data, mu=None):
"""Return the population variance of ``data``.
data should be an iterable of Real-valued numbers, with at least one
value. The optional argument mu, if given, should be the mean of
the data. If it is missing or None, 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.
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 the data, you can pass it as
the optional second argument to avoid recalculating it:
>>> mu = mean(data)
>>> pvariance(data, mu)
1.25
This function does not check that ``mu`` is actually the mean of ``data``.
Giving 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)
"""
if iter(data) is data:
data = list(data)
n = len(data)
if n < 1:
raise StatisticsError('pvariance requires at least one data point')
ss = _ss(data, mu)
return ss/n
def stdev(data, xbar=None):
"""Return 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
"""
var = variance(data, xbar)
try:
return var.sqrt()
except AttributeError:
return math.sqrt(var)
def pstdev(data, mu=None):
"""Return 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
"""
var = pvariance(data, mu)
try:
return var.sqrt()
except AttributeError:
return math.sqrt(var)

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Misc/NEWS
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@ -59,6 +59,9 @@ Core and Builtins
Library
-------
- Issue #18606: Add the new "statistics" module (PEP 450). Contributed
by Steven D'Aprano.
- Issue #12866: The audioop module now supports 24-bit samples.
- Issue #19254: Provide an optimized Python implementation of pbkdf2_hmac.