Lib/statistics.py - platform/external/python/cpython3 - Gitiles

 ##  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


 # === 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:
             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 subclass.
                 return x.as_integer_ratio()
             except AttributeError:
                 try:
                     # x may be a Decimal.
                     return _decimal_to_ratio(x)
                 except AttributeError:
                     # Just give up?
                     pass
     except (OverflowError, ValueError):
         # float NAN or INF.
         assert not math.isfinite(x)
         return (x, None)
     msg = "can't convert type '{}' to numerator/denominator"
     raise TypeError(msg.format(type(x).__name__))


 # 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()
         return (d, None)
     num = 0
     for digit in digits:
         num = num*10 + digit
     if exp < 0:
         den = 10**-exp
     else:
         num *= 10**exp
         den = 1
     if sign:
         num = -num
     return (num, den)


 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


 # === 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
     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)
     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')
     ss = _ss(data, mu)
     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)
	## 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



	# === 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:
	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 subclass.
	return x.as_integer_ratio()
	except AttributeError:
	try:
	# x may be a Decimal.
	return _decimal_to_ratio(x)
	except AttributeError:
	# Just give up?
	pass
	except (OverflowError, ValueError):
	# float NAN or INF.
	assert not math.isfinite(x)
	return (x, None)
	msg = "can't convert type '{}' to numerator/denominator"
	raise TypeError(msg.format(type(x).__name__))


	# 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()
	return (d, None)
	num = 0
	for digit in digits:
	num = num*10 + digit
	if exp < 0:
	den = 10**-exp
	else:
	num = 10*exp
	den = 1
	if sign:
	num = -num
	return (num, den)


	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


	# === 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
	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)
	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')
	ss = _ss(data, mu)
	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)