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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
from fractions import Fraction
from decimal import Decimal
from itertools import groupby
from bisect import bisect_left, bisect_right
# === Exceptions ===
class StatisticsError(ValueError):
pass
# === Private utilities ===
def _sum(data, start=0):
"""_sum(data [, start]) -> (type, sum, count)
Return a high-precision sum of the given numeric data as a fraction,
together with the type to be converted to and the count of items.
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)
(<class 'float' ...>, Fraction(11, 1), 5)
Some sources of round-off error will be avoided:
>>> _sum([1e50, 1, -1e50] * 1000) # Built-in sum returns zero.
(<class 'float' ...>, Fraction(1000, 1), 3000)
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)])
(<class 'fractions.Fraction' ...>, Fraction(63, 20), 4)
>>> from decimal import Decimal as D
>>> data = [D("0.1375"), D("0.2108"), D("0.3061"), D("0.0419")]
>>> _sum(data)
(<class 'decimal.Decimal' ...>, Fraction(6963, 10000), 4)
Mixed types are currently treated as an error, except that int is
allowed.
"""
count = 0
n, d = _exact_ratio(start)
partials = {d: n}
partials_get = partials.get
T = _coerce(int, type(start))
for typ, values in groupby(data, type):
T = _coerce(T, typ) # or raise TypeError
for n,d in map(_exact_ratio, values):
count += 1
partials[d] = partials_get(d, 0) + n
if None in partials:
# The sum will be a NAN or INF. We can ignore all the finite
# partials, and just look at this special one.
total = partials[None]
assert not _isfinite(total)
else:
# Sum all the partial sums using builtin sum.
# FIXME is this faster if we sum them in order of the denominator?
total = sum(Fraction(n, d) for d, n in sorted(partials.items()))
return (T, total, count)
def _isfinite(x):
try:
return x.is_finite() # Likely a Decimal.
except AttributeError:
return math.isfinite(x) # Coerces to float first.
def _coerce(T, S):
"""Coerce types T and S to a common type, or raise TypeError.
Coercion rules are currently an implementation detail. See the CoerceTest
test class in test_statistics for details.
"""
# See http://bugs.python.org/issue24068.
assert T is not bool, "initial type T is bool"
# If the types are the same, no need to coerce anything. Put this
# first, so that the usual case (no coercion needed) happens as soon
# as possible.
if T is S: return T
# Mixed int & other coerce to the other type.
if S is int or S is bool: return T
if T is int: return S
# If one is a (strict) subclass of the other, coerce to the subclass.
if issubclass(S, T): return S
if issubclass(T, S): return T
# Ints coerce to the other type.
if issubclass(T, int): return S
if issubclass(S, int): return T
# Mixed fraction & float coerces to float (or float subclass).
if issubclass(T, Fraction) and issubclass(S, float):
return S
if issubclass(T, float) and issubclass(S, Fraction):
return T
# Any other combination is disallowed.
msg = "don't know how to coerce %s and %s"
raise TypeError(msg % (T.__name__, S.__name__))
def _exact_ratio(x):
"""Return Real number x to exact (numerator, denominator) pair.
>>> _exact_ratio(0.25)
(1, 4)
x is expected to be an int, Fraction, Decimal or float.
"""
try:
# Optimise the common case of floats. We expect that the most often
# used numeric type will be builtin floats, so try to make this as
# fast as possible.
if type(x) is float or type(x) is Decimal:
return x.as_integer_ratio()
try:
# x may be an int, Fraction, or Integral ABC.
return (x.numerator, x.denominator)
except AttributeError:
try:
# x may be a float or Decimal subclass.
return x.as_integer_ratio()
except AttributeError:
# Just give up?
pass
except (OverflowError, ValueError):
# float NAN or INF.
assert not _isfinite(x)
return (x, None)
msg = "can't convert type '{}' to numerator/denominator"
raise TypeError(msg.format(type(x).__name__))
def _convert(value, T):
"""Convert value to given numeric type T."""
if type(value) is T:
# This covers the cases where T is Fraction, or where value is
# a NAN or INF (Decimal or float).
return value
if issubclass(T, int) and value.denominator != 1:
T = float
try:
# FIXME: what do we do if this overflows?
return T(value)
except TypeError:
if issubclass(T, Decimal):
return T(value.numerator)/T(value.denominator)
else:
raise
def _counts(data):
# Generate a table of sorted (value, frequency) pairs.
table = collections.Counter(iter(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
def _find_lteq(a, x):
'Locate the leftmost value exactly equal to x'
i = bisect_left(a, x)
if i != len(a) and a[i] == x:
return i
raise ValueError
def _find_rteq(a, l, x):
'Locate the rightmost value exactly equal to x'
i = bisect_right(a, x, lo=l)
if i != (len(a)+1) and a[i-1] == x:
return i-1
raise ValueError
# === 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')
T, total, count = _sum(data)
assert count == n
return _convert(total/n, T)
# 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
# Uses bisection search to search for x in data with log(n) time complexity
# Find the position of leftmost occurrence of x in data
l1 = _find_lteq(data, x)
# Find the position of rightmost occurrence of x in data[l1...len(data)]
# Assuming always l1 <= l2
l2 = _find_rteq(data, l1, x)
cf = l1
f = l2 - l1 + 1
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)
T, total, count = _sum((x-c)**2 for x in data)
# The following sum should mathematically equal zero, but due to rounding
# error may not.
U, total2, count2 = _sum((x-c) for x in data)
assert T == U and count == count2
total -= total2**2/len(data)
assert not total < 0, 'negative sum of square deviations: %f' % total
return (T, total)
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')
T, ss = _ss(data, xbar)
return _convert(ss/(n-1), T)
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')
T, ss = _ss(data, mu)
return _convert(ss/n, T)
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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