Issue #18606: Add the new "statistics" module (PEP 450). Contributed
by Steven D'Aprano.
diff --git a/Lib/statistics.py b/Lib/statistics.py
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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)