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

by Steven D'Aprano.

1 files changed, 612 insertions, 0 deletions

diff --git a/Lib/statistics.py b/Lib/statistics.py
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+##  Module statistics.py
+##
+##  Copyright (c) 2013 Steven D'Aprano <steve+python@pearwood.info>.
+##
+##  Licensed under the Apache License, Version 2.0 (the "License");
+##  you may not use this file except in compliance with the License.
+##  You may obtain a copy of the License at
+##
+##  http://www.apache.org/licenses/LICENSE-2.0
+##
+##  Unless required by applicable law or agreed to in writing, software
+##  distributed under the License is distributed on an "AS IS" BASIS,
+##  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+##  See the License for the specific language governing permissions and
+##  limitations under the License.
+
+
+"""
+Basic statistics module.
+
+This module provides functions for calculating statistics of data, including
+averages, variance, and standard deviation.
+
+Calculating averages
+--------------------
+
+==================  =============================================
+Function            Description
+==================  =============================================
+mean                Arithmetic mean (average) of data.
+median              Median (middle value) of data.
+median_low          Low median of data.
+median_high         High median of data.
+median_grouped      Median, or 50th percentile, of grouped data.
+mode                Mode (most common value) of data.
+==================  =============================================
+
+Calculate the arithmetic mean ("the average") of data:
+
+>>> mean([-1.0, 2.5, 3.25, 5.75])
+2.625
+
+
+Calculate the standard median of discrete data:
+
+>>> median([2, 3, 4, 5])
+3.5
+
+
+Calculate the median, or 50th percentile, of data grouped into class intervals
+centred on the data values provided. E.g. if your data points are rounded to
+the nearest whole number:
+
+>>> median_grouped([2, 2, 3, 3, 3, 4])  #doctest: +ELLIPSIS
+2.8333333333...
+
+This should be interpreted in this way: you have two data points in the class
+interval 1.5-2.5, three data points in the class interval 2.5-3.5, and one in
+the class interval 3.5-4.5. The median of these data points is 2.8333...
+
+
+Calculating variability or spread
+---------------------------------
+
+==================  =============================================
+Function            Description
+==================  =============================================
+pvariance           Population variance of data.
+variance            Sample variance of data.
+pstdev              Population standard deviation of data.
+stdev               Sample standard deviation of data.
+==================  =============================================
+
+Calculate the standard deviation of sample data:
+
+>>> stdev([2.5, 3.25, 5.5, 11.25, 11.75])  #doctest: +ELLIPSIS
+4.38961843444...
+
+If you have previously calculated the mean, you can pass it as the optional
+second argument to the four "spread" functions to avoid recalculating it:
+
+>>> data = [1, 2, 2, 4, 4, 4, 5, 6]
+>>> mu = mean(data)
+>>> pvariance(data, mu)
+2.5
+
+
+Exceptions
+----------
+
+A single exception is defined: StatisticsError is a subclass of ValueError.
+
+"""
+
+__all__ = [ 'StatisticsError',
+            'pstdev', 'pvariance', 'stdev', 'variance',
+            'median',  'median_low', 'median_high', 'median_grouped',
+            'mean', 'mode',
+          ]
+
+
+import collections
+import math
+import numbers
+import operator
+
+from fractions import Fraction
+from decimal import Decimal
+
+
+# === Exceptions ===
+
+class StatisticsError(ValueError):
+    pass
+
+
+# === Private utilities ===
+
+def _sum(data, start=0):
+    """_sum(data [, start]) -> value
+
+    Return a high-precision sum of the given numeric data. If optional
+    argument ``start`` is given, it is added to the total. If ``data`` is
+    empty, ``start`` (defaulting to 0) is returned.
+
+
+    Examples
+    --------
+
+    >>> _sum([3, 2.25, 4.5, -0.5, 1.0], 0.75)
+    11.0
+
+    Some sources of round-off error will be avoided:
+
+    >>> _sum([1e50, 1, -1e50] * 1000)  # Built-in sum returns zero.
+    1000.0
+
+    Fractions and Decimals are also supported:
+
+    >>> from fractions import Fraction as F
+    >>> _sum([F(2, 3), F(7, 5), F(1, 4), F(5, 6)])
+    Fraction(63, 20)
+
+    >>> from decimal import Decimal as D
+    >>> data = [D("0.1375"), D("0.2108"), D("0.3061"), D("0.0419")]
+    >>> _sum(data)
+    Decimal('0.6963')
+
+    """
+    n, d = _exact_ratio(start)
+    T = type(start)
+    partials = {d: n}  # map {denominator: sum of numerators}
+    # Micro-optimizations.
+    coerce_types = _coerce_types
+    exact_ratio = _exact_ratio
+    partials_get = partials.get
+    # Add numerators for each denominator, and track the "current" type.
+    for x in data:
+        T = _coerce_types(T, type(x))
+        n, d = exact_ratio(x)
+        partials[d] = partials_get(d, 0) + n
+    if None in partials:
+        assert issubclass(T, (float, Decimal))
+        assert not math.isfinite(partials[None])
+        return T(partials[None])
+    total = Fraction()
+    for d, n in sorted(partials.items()):
+        total += Fraction(n, d)
+    if issubclass(T, int):
+        assert total.denominator == 1
+        return T(total.numerator)
+    if issubclass(T, Decimal):
+        return T(total.numerator)/total.denominator
+    return T(total)
+
+
+def _exact_ratio(x):
+    """Convert Real number x exactly to (numerator, denominator) pair.
+
+    >>> _exact_ratio(0.25)
+    (1, 4)
+
+    x is expected to be an int, Fraction, Decimal or float.
+    """
+    try:
+        try:
+            # int, Fraction
+            return (x.numerator, x.denominator)
+        except AttributeError:
+            # float
+            try:
+                return x.as_integer_ratio()
+            except AttributeError:
+                # Decimal
+                try:
+                    return _decimal_to_ratio(x)
+                except AttributeError:
+                    msg = "can't convert type '{}' to numerator/denominator"
+                    raise TypeError(msg.format(type(x).__name__)) from None
+    except (OverflowError, ValueError):
+        # INF or NAN
+        if __debug__:
+            # Decimal signalling NANs cannot be converted to float :-(
+            if isinstance(x, Decimal):
+                assert not x.is_finite()
+            else:
+                assert not math.isfinite(x)
+        return (x, None)
+
+
+# FIXME This is faster than Fraction.from_decimal, but still too slow.
+def _decimal_to_ratio(d):
+    """Convert Decimal d to exact integer ratio (numerator, denominator).
+
+    >>> from decimal import Decimal
+    >>> _decimal_to_ratio(Decimal("2.6"))
+    (26, 10)
+
+    """
+    sign, digits, exp = d.as_tuple()
+    if exp in ('F', 'n', 'N'):  # INF, NAN, sNAN
+        assert not d.is_finite()
+        raise ValueError
+    num = 0
+    for digit in digits:
+        num = num*10 + digit
+    if sign:
+        num = -num
+    den = 10**-exp
+    return (num, den)
+
+
+def _coerce_types(T1, T2):
+    """Coerce types T1 and T2 to a common type.
+
+    >>> _coerce_types(int, float)
+    <class 'float'>
+
+    Coercion is performed according to this table, where "N/A" means
+    that a TypeError exception is raised.
+
+    +----------+-----------+-----------+-----------+----------+
+    |          | int       | Fraction  | Decimal   | float    |
+    +----------+-----------+-----------+-----------+----------+
+    | int      | int       | Fraction  | Decimal   | float    |
+    | Fraction | Fraction  | Fraction  | N/A       | float    |
+    | Decimal  | Decimal   | N/A       | Decimal   | float    |
+    | float    | float     | float     | float     | float    |
+    +----------+-----------+-----------+-----------+----------+
+
+    Subclasses trump their parent class; two subclasses of the same
+    base class will be coerced to the second of the two.
+
+    """
+    # Get the common/fast cases out of the way first.
+    if T1 is T2: return T1
+    if T1 is int: return T2
+    if T2 is int: return T1
+    # Subclasses trump their parent class.
+    if issubclass(T2, T1): return T2
+    if issubclass(T1, T2): return T1
+    # Floats trump everything else.
+    if issubclass(T2, float): return T2
+    if issubclass(T1, float): return T1
+    # Subclasses of the same base class give priority to the second.
+    if T1.__base__ is T2.__base__: return T2
+    # Otherwise, just give up.
+    raise TypeError('cannot coerce types %r and %r' % (T1, T2))
+
+
+def _counts(data):
+    # Generate a table of sorted (value, frequency) pairs.
+    if data is None:
+        raise TypeError('None is not iterable')
+    table = collections.Counter(data).most_common()
+    if not table:
+        return table
+    # Extract the values with the highest frequency.
+    maxfreq = table[0][1]
+    for i in range(1, len(table)):
+        if table[i][1] != maxfreq:
+            table = table[:i]
+            break
+    return table
+
+
+# === Measures of central tendency (averages) ===
+
+def mean(data):
+    """Return the sample arithmetic mean of data.
+
+    >>> mean([1, 2, 3, 4, 4])
+    2.8
+
+    >>> from fractions import Fraction as F
+    >>> mean([F(3, 7), F(1, 21), F(5, 3), F(1, 3)])
+    Fraction(13, 21)
+
+    >>> from decimal import Decimal as D
+    >>> mean([D("0.5"), D("0.75"), D("0.625"), D("0.375")])
+    Decimal('0.5625')
+
+    If ``data`` is empty, StatisticsError will be raised.
+    """
+    if iter(data) is data:
+        data = list(data)
+    n = len(data)
+    if n < 1:
+        raise StatisticsError('mean requires at least one data point')
+    return _sum(data)/n
+
+
+# FIXME: investigate ways to calculate medians without sorting? Quickselect?
+def median(data):
+    """Return the median (middle value) of numeric data.
+
+    When the number of data points is odd, return the middle data point.
+    When the number of data points is even, the median is interpolated by
+    taking the average of the two middle values:
+
+    >>> median([1, 3, 5])
+    3
+    >>> median([1, 3, 5, 7])
+    4.0
+
+    """
+    data = sorted(data)
+    n = len(data)
+    if n == 0:
+        raise StatisticsError("no median for empty data")
+    if n%2 == 1:
+        return data[n//2]
+    else:
+        i = n//2
+        return (data[i - 1] + data[i])/2
+
+
+def median_low(data):
+    """Return the low median of numeric data.
+
+    When the number of data points is odd, the middle value is returned.
+    When it is even, the smaller of the two middle values is returned.
+
+    >>> median_low([1, 3, 5])
+    3
+    >>> median_low([1, 3, 5, 7])
+    3
+
+    """
+    data = sorted(data)
+    n = len(data)
+    if n == 0:
+        raise StatisticsError("no median for empty data")
+    if n%2 == 1:
+        return data[n//2]
+    else:
+        return data[n//2 - 1]
+
+
+def median_high(data):
+    """Return the high median of data.
+
+    When the number of data points is odd, the middle value is returned.
+    When it is even, the larger of the two middle values is returned.
+
+    >>> median_high([1, 3, 5])
+    3
+    >>> median_high([1, 3, 5, 7])
+    5
+
+    """
+    data = sorted(data)
+    n = len(data)
+    if n == 0:
+        raise StatisticsError("no median for empty data")
+    return data[n//2]
+
+
+def median_grouped(data, interval=1):
+    """"Return the 50th percentile (median) of grouped continuous data.
+
+    >>> median_grouped([1, 2, 2, 3, 4, 4, 4, 4, 4, 5])
+    3.7
+    >>> median_grouped([52, 52, 53, 54])
+    52.5
+
+    This calculates the median as the 50th percentile, and should be
+    used when your data is continuous and grouped. In the above example,
+    the values 1, 2, 3, etc. actually represent the midpoint of classes
+    0.5-1.5, 1.5-2.5, 2.5-3.5, etc. The middle value falls somewhere in
+    class 3.5-4.5, and interpolation is used to estimate it.
+
+    Optional argument ``interval`` represents the class interval, and
+    defaults to 1. Changing the class interval naturally will change the
+    interpolated 50th percentile value:
+
+    >>> median_grouped([1, 3, 3, 5, 7], interval=1)
+    3.25
+    >>> median_grouped([1, 3, 3, 5, 7], interval=2)
+    3.5
+
+    This function does not check whether the data points are at least
+    ``interval`` apart.
+    """
+    data = sorted(data)
+    n = len(data)
+    if n == 0:
+        raise StatisticsError("no median for empty data")
+    elif n == 1:
+        return data[0]
+    # Find the value at the midpoint. Remember this corresponds to the
+    # centre of the class interval.
+    x = data[n//2]
+    for obj in (x, interval):
+        if isinstance(obj, (str, bytes)):
+            raise TypeError('expected number but got %r' % obj)
+    try:
+        L = x - interval/2  # The lower limit of the median interval.
+    except TypeError:
+        # Mixed type. For now we just coerce to float.
+        L = float(x) - float(interval)/2
+    cf = data.index(x)  # Number of values below the median interval.
+    # FIXME The following line could be more efficient for big lists.
+    f = data.count(x)  # Number of data points in the median interval.
+    return L + interval*(n/2 - cf)/f
+
+
+def mode(data):
+    """Return the most common data point from discrete or nominal data.
+
+    ``mode`` assumes discrete data, and returns a single value. This is the
+    standard treatment of the mode as commonly taught in schools:
+
+    >>> mode([1, 1, 2, 3, 3, 3, 3, 4])
+    3
+
+    This also works with nominal (non-numeric) data:
+
+    >>> mode(["red", "blue", "blue", "red", "green", "red", "red"])
+    'red'
+
+    If there is not exactly one most common value, ``mode`` will raise
+    StatisticsError.
+    """
+    # Generate a table of sorted (value, frequency) pairs.
+    table = _counts(data)
+    if len(table) == 1:
+        return table[0][0]
+    elif table:
+        raise StatisticsError(
+                'no unique mode; found %d equally common values' % len(table)
+                )
+    else:
+        raise StatisticsError('no mode for empty data')
+
+
+# === Measures of spread ===
+
+# See http://mathworld.wolfram.com/Variance.html
+#     http://mathworld.wolfram.com/SampleVariance.html
+#     http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance
+#
+# Under no circumstances use the so-called "computational formula for
+# variance", as that is only suitable for hand calculations with a small
+# amount of low-precision data. It has terrible numeric properties.
+#
+# See a comparison of three computational methods here:
+# http://www.johndcook.com/blog/2008/09/26/comparing-three-methods-of-computing-standard-deviation/
+
+def _ss(data, c=None):
+    """Return sum of square deviations of sequence data.
+
+    If ``c`` is None, the mean is calculated in one pass, and the deviations
+    from the mean are calculated in a second pass. Otherwise, deviations are
+    calculated from ``c`` as given. Use the second case with care, as it can
+    lead to garbage results.
+    """
+    if c is None:
+        c = mean(data)
+    ss = _sum((x-c)**2 for x in data)
+    # The following sum should mathematically equal zero, but due to rounding
+    # error may not.
+    ss -= _sum((x-c) for x in data)**2/len(data)
+    assert not ss < 0, 'negative sum of square deviations: %f' % ss
+    return ss
+
+
+def variance(data, xbar=None):
+    """Return the sample variance of data.
+
+    data should be an iterable of Real-valued numbers, with at least two
+    values. The optional argument xbar, if given, should be the mean of
+    the data. If it is missing or None, the mean is automatically calculated.
+
+    Use this function when your data is a sample from a population. To
+    calculate the variance from the entire population, see ``pvariance``.
+
+    Examples:
+
+    >>> data = [2.75, 1.75, 1.25, 0.25, 0.5, 1.25, 3.5]
+    >>> variance(data)
+    1.3720238095238095
+
+    If you have already calculated the mean of your data, you can pass it as
+    the optional second argument ``xbar`` to avoid recalculating it:
+
+    >>> m = mean(data)
+    >>> variance(data, m)
+    1.3720238095238095
+
+    This function does not check that ``xbar`` is actually the mean of
+    ``data``. Giving arbitrary values for ``xbar`` may lead to invalid or
+    impossible results.
+
+    Decimals and Fractions are supported:
+
+    >>> from decimal import Decimal as D
+    >>> variance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")])
+    Decimal('31.01875')
+
+    >>> from fractions import Fraction as F
+    >>> variance([F(1, 6), F(1, 2), F(5, 3)])
+    Fraction(67, 108)
+
+    """
+    if iter(data) is data:
+        data = list(data)
+    n = len(data)
+    if n < 2:
+        raise StatisticsError('variance requires at least two data points')
+    ss = _ss(data, xbar)
+    return ss/(n-1)
+
+
+def pvariance(data, mu=None):
+    """Return the population variance of ``data``.
+
+    data should be an iterable of Real-valued numbers, with at least one
+    value. The optional argument mu, if given, should be the mean of
+    the data. If it is missing or None, the mean is automatically calculated.
+
+    Use this function to calculate the variance from the entire population.
+    To estimate the variance from a sample, the ``variance`` function is
+    usually a better choice.
+
+    Examples:
+
+    >>> data = [0.0, 0.25, 0.25, 1.25, 1.5, 1.75, 2.75, 3.25]
+    >>> pvariance(data)
+    1.25
+
+    If you have already calculated the mean of the data, you can pass it as
+    the optional second argument to avoid recalculating it:
+
+    >>> mu = mean(data)
+    >>> pvariance(data, mu)
+    1.25
+
+    This function does not check that ``mu`` is actually the mean of ``data``.
+    Giving arbitrary values for ``mu`` may lead to invalid or impossible
+    results.
+
+    Decimals and Fractions are supported:
+
+    >>> from decimal import Decimal as D
+    >>> pvariance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")])
+    Decimal('24.815')
+
+    >>> from fractions import Fraction as F
+    >>> pvariance([F(1, 4), F(5, 4), F(1, 2)])
+    Fraction(13, 72)
+
+    """
+    if iter(data) is data:
+        data = list(data)
+    n = len(data)
+    if n < 1:
+        raise StatisticsError('pvariance requires at least one data point')
+    ss = _ss(data, mu)
+    return ss/n
+
+
+def stdev(data, xbar=None):
+    """Return the square root of the sample variance.
+
+    See ``variance`` for arguments and other details.
+
+    >>> stdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
+    1.0810874155219827
+
+    """
+    var = variance(data, xbar)
+    try:
+        return var.sqrt()
+    except AttributeError:
+        return math.sqrt(var)
+
+
+def pstdev(data, mu=None):
+    """Return the square root of the population variance.
+
+    See ``pvariance`` for arguments and other details.
+
+    >>> pstdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
+    0.986893273527251
+
+    """
+    var = pvariance(data, mu)
+    try:
+        return var.sqrt()
+    except AttributeError:
+        return math.sqrt(var)