595 lines
17 KiB
Python
595 lines
17 KiB
Python
## Module statistics.py
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##
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## Copyright (c) 2013 Steven D'Aprano <steve+python@pearwood.info>.
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##
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## Licensed under the Apache License, Version 2.0 (the "License");
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## you may not use this file except in compliance with the License.
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## You may obtain a copy of the License at
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##
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## http://www.apache.org/licenses/LICENSE-2.0
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##
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## Unless required by applicable law or agreed to in writing, software
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## distributed under the License is distributed on an "AS IS" BASIS,
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## WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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## See the License for the specific language governing permissions and
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## limitations under the License.
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"""
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Basic statistics module.
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This module provides functions for calculating statistics of data, including
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averages, variance, and standard deviation.
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Calculating averages
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--------------------
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================== =============================================
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Function Description
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================== =============================================
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mean Arithmetic mean (average) of data.
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median Median (middle value) of data.
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median_low Low median of data.
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median_high High median of data.
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median_grouped Median, or 50th percentile, of grouped data.
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mode Mode (most common value) of data.
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================== =============================================
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Calculate the arithmetic mean ("the average") of data:
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>>> mean([-1.0, 2.5, 3.25, 5.75])
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2.625
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Calculate the standard median of discrete data:
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>>> median([2, 3, 4, 5])
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3.5
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Calculate the median, or 50th percentile, of data grouped into class intervals
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centred on the data values provided. E.g. if your data points are rounded to
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the nearest whole number:
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>>> median_grouped([2, 2, 3, 3, 3, 4]) #doctest: +ELLIPSIS
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2.8333333333...
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This should be interpreted in this way: you have two data points in the class
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interval 1.5-2.5, three data points in the class interval 2.5-3.5, and one in
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the class interval 3.5-4.5. The median of these data points is 2.8333...
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Calculating variability or spread
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---------------------------------
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================== =============================================
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Function Description
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================== =============================================
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pvariance Population variance of data.
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variance Sample variance of data.
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pstdev Population standard deviation of data.
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stdev Sample standard deviation of data.
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================== =============================================
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Calculate the standard deviation of sample data:
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>>> stdev([2.5, 3.25, 5.5, 11.25, 11.75]) #doctest: +ELLIPSIS
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4.38961843444...
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If you have previously calculated the mean, you can pass it as the optional
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second argument to the four "spread" functions to avoid recalculating it:
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>>> data = [1, 2, 2, 4, 4, 4, 5, 6]
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>>> mu = mean(data)
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>>> pvariance(data, mu)
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2.5
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Exceptions
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----------
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A single exception is defined: StatisticsError is a subclass of ValueError.
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"""
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__all__ = [ 'StatisticsError',
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'pstdev', 'pvariance', 'stdev', 'variance',
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'median', 'median_low', 'median_high', 'median_grouped',
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'mean', 'mode',
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]
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import collections
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import math
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from fractions import Fraction
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from decimal import Decimal
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# === Exceptions ===
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class StatisticsError(ValueError):
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pass
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# === Private utilities ===
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def _sum(data, start=0):
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"""_sum(data [, start]) -> value
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Return a high-precision sum of the given numeric data. If optional
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argument ``start`` is given, it is added to the total. If ``data`` is
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empty, ``start`` (defaulting to 0) is returned.
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Examples
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--------
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>>> _sum([3, 2.25, 4.5, -0.5, 1.0], 0.75)
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11.0
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Some sources of round-off error will be avoided:
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>>> _sum([1e50, 1, -1e50] * 1000) # Built-in sum returns zero.
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1000.0
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Fractions and Decimals are also supported:
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>>> from fractions import Fraction as F
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>>> _sum([F(2, 3), F(7, 5), F(1, 4), F(5, 6)])
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Fraction(63, 20)
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>>> from decimal import Decimal as D
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>>> data = [D("0.1375"), D("0.2108"), D("0.3061"), D("0.0419")]
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>>> _sum(data)
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Decimal('0.6963')
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Mixed types are currently treated as an error, except that int is
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allowed.
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"""
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# We fail as soon as we reach a value that is not an int or the type of
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# the first value which is not an int. E.g. _sum([int, int, float, int])
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# is okay, but sum([int, int, float, Fraction]) is not.
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allowed_types = {int, type(start)}
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n, d = _exact_ratio(start)
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partials = {d: n} # map {denominator: sum of numerators}
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# Micro-optimizations.
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exact_ratio = _exact_ratio
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partials_get = partials.get
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# Add numerators for each denominator.
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for x in data:
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_check_type(type(x), allowed_types)
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n, d = exact_ratio(x)
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partials[d] = partials_get(d, 0) + n
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# Find the expected result type. If allowed_types has only one item, it
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# will be int; if it has two, use the one which isn't int.
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assert len(allowed_types) in (1, 2)
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if len(allowed_types) == 1:
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assert allowed_types.pop() is int
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T = int
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else:
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T = (allowed_types - {int}).pop()
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if None in partials:
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assert issubclass(T, (float, Decimal))
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assert not math.isfinite(partials[None])
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return T(partials[None])
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total = Fraction()
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for d, n in sorted(partials.items()):
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total += Fraction(n, d)
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if issubclass(T, int):
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assert total.denominator == 1
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return T(total.numerator)
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if issubclass(T, Decimal):
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return T(total.numerator)/total.denominator
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return T(total)
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def _check_type(T, allowed):
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if T not in allowed:
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if len(allowed) == 1:
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allowed.add(T)
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else:
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types = ', '.join([t.__name__ for t in allowed] + [T.__name__])
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raise TypeError("unsupported mixed types: %s" % types)
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def _exact_ratio(x):
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"""Convert Real number x exactly to (numerator, denominator) pair.
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>>> _exact_ratio(0.25)
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(1, 4)
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x is expected to be an int, Fraction, Decimal or float.
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"""
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try:
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try:
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# int, Fraction
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return (x.numerator, x.denominator)
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except AttributeError:
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# float
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try:
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return x.as_integer_ratio()
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except AttributeError:
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# Decimal
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try:
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return _decimal_to_ratio(x)
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except AttributeError:
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msg = "can't convert type '{}' to numerator/denominator"
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raise TypeError(msg.format(type(x).__name__)) from None
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except (OverflowError, ValueError):
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# INF or NAN
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if __debug__:
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# Decimal signalling NANs cannot be converted to float :-(
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if isinstance(x, Decimal):
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assert not x.is_finite()
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else:
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assert not math.isfinite(x)
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return (x, None)
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# FIXME This is faster than Fraction.from_decimal, but still too slow.
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def _decimal_to_ratio(d):
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"""Convert Decimal d to exact integer ratio (numerator, denominator).
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>>> from decimal import Decimal
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>>> _decimal_to_ratio(Decimal("2.6"))
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(26, 10)
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"""
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sign, digits, exp = d.as_tuple()
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if exp in ('F', 'n', 'N'): # INF, NAN, sNAN
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assert not d.is_finite()
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raise ValueError
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num = 0
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for digit in digits:
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num = num*10 + digit
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if exp < 0:
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den = 10**-exp
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else:
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num *= 10**exp
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den = 1
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if sign:
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num = -num
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return (num, den)
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def _counts(data):
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# Generate a table of sorted (value, frequency) pairs.
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table = collections.Counter(iter(data)).most_common()
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if not table:
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return table
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# Extract the values with the highest frequency.
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maxfreq = table[0][1]
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for i in range(1, len(table)):
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if table[i][1] != maxfreq:
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table = table[:i]
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break
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return table
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# === Measures of central tendency (averages) ===
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def mean(data):
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"""Return the sample arithmetic mean of data.
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>>> mean([1, 2, 3, 4, 4])
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2.8
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>>> from fractions import Fraction as F
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>>> mean([F(3, 7), F(1, 21), F(5, 3), F(1, 3)])
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Fraction(13, 21)
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>>> from decimal import Decimal as D
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>>> mean([D("0.5"), D("0.75"), D("0.625"), D("0.375")])
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Decimal('0.5625')
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If ``data`` is empty, StatisticsError will be raised.
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"""
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if iter(data) is data:
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data = list(data)
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n = len(data)
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if n < 1:
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raise StatisticsError('mean requires at least one data point')
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return _sum(data)/n
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# FIXME: investigate ways to calculate medians without sorting? Quickselect?
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def median(data):
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"""Return the median (middle value) of numeric data.
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When the number of data points is odd, return the middle data point.
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When the number of data points is even, the median is interpolated by
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taking the average of the two middle values:
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>>> median([1, 3, 5])
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3
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>>> median([1, 3, 5, 7])
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4.0
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"""
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data = sorted(data)
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n = len(data)
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if n == 0:
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raise StatisticsError("no median for empty data")
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if n%2 == 1:
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return data[n//2]
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else:
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i = n//2
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return (data[i - 1] + data[i])/2
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def median_low(data):
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"""Return the low median of numeric data.
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When the number of data points is odd, the middle value is returned.
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When it is even, the smaller of the two middle values is returned.
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>>> median_low([1, 3, 5])
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3
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>>> median_low([1, 3, 5, 7])
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3
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"""
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data = sorted(data)
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n = len(data)
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if n == 0:
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raise StatisticsError("no median for empty data")
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if n%2 == 1:
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return data[n//2]
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else:
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return data[n//2 - 1]
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def median_high(data):
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"""Return the high median of data.
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When the number of data points is odd, the middle value is returned.
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When it is even, the larger of the two middle values is returned.
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>>> median_high([1, 3, 5])
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3
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>>> median_high([1, 3, 5, 7])
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5
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"""
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data = sorted(data)
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n = len(data)
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if n == 0:
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raise StatisticsError("no median for empty data")
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return data[n//2]
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def median_grouped(data, interval=1):
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""""Return the 50th percentile (median) of grouped continuous data.
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>>> median_grouped([1, 2, 2, 3, 4, 4, 4, 4, 4, 5])
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3.7
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>>> median_grouped([52, 52, 53, 54])
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52.5
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This calculates the median as the 50th percentile, and should be
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used when your data is continuous and grouped. In the above example,
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the values 1, 2, 3, etc. actually represent the midpoint of classes
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0.5-1.5, 1.5-2.5, 2.5-3.5, etc. The middle value falls somewhere in
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class 3.5-4.5, and interpolation is used to estimate it.
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Optional argument ``interval`` represents the class interval, and
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defaults to 1. Changing the class interval naturally will change the
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interpolated 50th percentile value:
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>>> median_grouped([1, 3, 3, 5, 7], interval=1)
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3.25
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>>> median_grouped([1, 3, 3, 5, 7], interval=2)
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3.5
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This function does not check whether the data points are at least
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``interval`` apart.
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"""
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data = sorted(data)
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n = len(data)
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if n == 0:
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raise StatisticsError("no median for empty data")
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elif n == 1:
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return data[0]
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# Find the value at the midpoint. Remember this corresponds to the
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# centre of the class interval.
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x = data[n//2]
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for obj in (x, interval):
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if isinstance(obj, (str, bytes)):
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raise TypeError('expected number but got %r' % obj)
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try:
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L = x - interval/2 # The lower limit of the median interval.
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except TypeError:
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# Mixed type. For now we just coerce to float.
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L = float(x) - float(interval)/2
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cf = data.index(x) # Number of values below the median interval.
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# FIXME The following line could be more efficient for big lists.
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f = data.count(x) # Number of data points in the median interval.
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return L + interval*(n/2 - cf)/f
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def mode(data):
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"""Return the most common data point from discrete or nominal data.
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``mode`` assumes discrete data, and returns a single value. This is the
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standard treatment of the mode as commonly taught in schools:
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>>> mode([1, 1, 2, 3, 3, 3, 3, 4])
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3
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This also works with nominal (non-numeric) data:
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>>> mode(["red", "blue", "blue", "red", "green", "red", "red"])
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'red'
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If there is not exactly one most common value, ``mode`` will raise
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StatisticsError.
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"""
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# Generate a table of sorted (value, frequency) pairs.
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table = _counts(data)
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if len(table) == 1:
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return table[0][0]
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elif table:
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raise StatisticsError(
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'no unique mode; found %d equally common values' % len(table)
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)
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else:
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raise StatisticsError('no mode for empty data')
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# === Measures of spread ===
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# See http://mathworld.wolfram.com/Variance.html
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# http://mathworld.wolfram.com/SampleVariance.html
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# http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance
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#
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# Under no circumstances use the so-called "computational formula for
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# variance", as that is only suitable for hand calculations with a small
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# amount of low-precision data. It has terrible numeric properties.
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#
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# See a comparison of three computational methods here:
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# http://www.johndcook.com/blog/2008/09/26/comparing-three-methods-of-computing-standard-deviation/
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def _ss(data, c=None):
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"""Return sum of square deviations of sequence data.
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If ``c`` is None, the mean is calculated in one pass, and the deviations
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from the mean are calculated in a second pass. Otherwise, deviations are
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calculated from ``c`` as given. Use the second case with care, as it can
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lead to garbage results.
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"""
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if c is None:
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c = mean(data)
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ss = _sum((x-c)**2 for x in data)
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# The following sum should mathematically equal zero, but due to rounding
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# error may not.
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ss -= _sum((x-c) for x in data)**2/len(data)
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assert not ss < 0, 'negative sum of square deviations: %f' % ss
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return ss
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def variance(data, xbar=None):
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"""Return the sample variance of data.
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data should be an iterable of Real-valued numbers, with at least two
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values. The optional argument xbar, if given, should be the mean of
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the data. If it is missing or None, the mean is automatically calculated.
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Use this function when your data is a sample from a population. To
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calculate the variance from the entire population, see ``pvariance``.
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Examples:
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>>> data = [2.75, 1.75, 1.25, 0.25, 0.5, 1.25, 3.5]
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>>> variance(data)
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1.3720238095238095
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If you have already calculated the mean of your data, you can pass it as
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the optional second argument ``xbar`` to avoid recalculating it:
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>>> m = mean(data)
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>>> variance(data, m)
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1.3720238095238095
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This function does not check that ``xbar`` is actually the mean of
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``data``. Giving arbitrary values for ``xbar`` may lead to invalid or
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impossible results.
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Decimals and Fractions are supported:
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>>> from decimal import Decimal as D
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>>> variance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")])
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Decimal('31.01875')
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>>> from fractions import Fraction as F
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>>> variance([F(1, 6), F(1, 2), F(5, 3)])
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Fraction(67, 108)
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"""
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if iter(data) is data:
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data = list(data)
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n = len(data)
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if n < 2:
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raise StatisticsError('variance requires at least two data points')
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ss = _ss(data, xbar)
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return ss/(n-1)
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def pvariance(data, mu=None):
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"""Return the population variance of ``data``.
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data should be an iterable of Real-valued numbers, with at least one
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value. The optional argument mu, if given, should be the mean of
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the data. If it is missing or None, the mean is automatically calculated.
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Use this function to calculate the variance from the entire population.
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To estimate the variance from a sample, the ``variance`` function is
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usually a better choice.
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Examples:
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>>> data = [0.0, 0.25, 0.25, 1.25, 1.5, 1.75, 2.75, 3.25]
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>>> pvariance(data)
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1.25
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If you have already calculated the mean of the data, you can pass it as
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the optional second argument to avoid recalculating it:
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>>> mu = mean(data)
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>>> pvariance(data, mu)
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1.25
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This function does not check that ``mu`` is actually the mean of ``data``.
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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)
|