Lib/statistics.py

"""
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.
fmean               Fast, floating point arithmetic mean.
geometric_mean      Geometric mean of data.
harmonic_mean       Harmonic mean 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.
multimode           List of modes (most common values of data).
quantiles           Divide data into intervals with equal probability.
==================  ==================================================

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


Statistics for relations between two inputs
-------------------------------------------

==================  ====================================================
Function            Description
==================  ====================================================
covariance          Sample covariance for two variables.
correlation         Pearson's correlation coefficient for two variables.
linear_regression   Intercept and slope for simple linear regression.
==================  ====================================================

Calculate covariance, Pearson's correlation, and simple linear regression
for two inputs:

>>> x = [1, 2, 3, 4, 5, 6, 7, 8, 9]
>>> y = [1, 2, 3, 1, 2, 3, 1, 2, 3]
>>> covariance(x, y)
0.75
>>> correlation(x, y)  #doctest: +ELLIPSIS
0.31622776601...
>>> linear_regression(x, y)  #doctest:
LinearRegression(slope=0.1, intercept=1.5)


Exceptions
----------

A single exception is defined: StatisticsError is a subclass of ValueError.

"""

__all__ = [
    'NormalDist',
    'StatisticsError',
    'correlation',
    'covariance',
    'fmean',
    'geometric_mean',
    'harmonic_mean',
    'linear_regression',
    'mean',
    'median',
    'median_grouped',
    'median_high',
    'median_low',
    'mode',
    'multimode',
    'pstdev',
    'pvariance',
    'quantiles',
    'stdev',
    'variance',
]

import math
import numbers
import random

from fractions import Fraction
from decimal import Decimal
from itertools import groupby, repeat
from bisect import bisect_left, bisect_right
from math import hypot, sqrt, fabs, exp, erf, tau, log, fsum
from operator import itemgetter
from collections import Counter, namedtuple

# === Exceptions ===

class StatisticsError(ValueError):
    pass


# === Private utilities ===

def _sum(data):
    """_sum(data) -> (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.

    Examples
    --------

    >>> _sum([3, 2.25, 4.5, -0.5, 0.25])
    (<class 'float'>, Fraction(19, 2), 5)

    Some sources of round-off error will be avoided:

    # Built-in sum returns zero.
    >>> _sum([1e50, 1, -1e50] * 1000)
    (<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
    partials = {}
    partials_get = partials.get
    T = int
    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.
        total = sum(Fraction(n, d) for d, n in 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:
        return x.as_integer_ratio()
    except AttributeError:
        pass
    except (OverflowError, ValueError):
        # float NAN or INF.
        assert not _isfinite(x)
        return (x, None)
    try:
        # x may be an Integral ABC.
        return (x.numerator, x.denominator)
    except AttributeError:
        msg = f"can't convert type '{type(x).__name__}' to numerator/denominator"
        raise TypeError(msg)


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 _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


def _fail_neg(values, errmsg='negative value'):
    """Iterate over values, failing if any are less than zero."""
    for x in values:
        if x < 0:
            raise StatisticsError(errmsg)
        yield x


# === 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)


def fmean(data):
    """Convert data to floats and compute the arithmetic mean.

    This runs faster than the mean() function and it always returns a float.
    If the input dataset is empty, it raises a StatisticsError.

    >>> fmean([3.5, 4.0, 5.25])
    4.25
    """
    try:
        n = len(data)
    except TypeError:
        # Handle iterators that do not define __len__().
        n = 0
        def count(iterable):
            nonlocal n
            for n, x in enumerate(iterable, start=1):
                yield x
        total = fsum(count(data))
    else:
        total = fsum(data)
    try:
        return total / n
    except ZeroDivisionError:
        raise StatisticsError('fmean requires at least one data point') from None


def geometric_mean(data):
    """Convert data to floats and compute the geometric mean.

    Raises a StatisticsError if the input dataset is empty,
    if it contains a zero, or if it contains a negative value.

    No special efforts are made to achieve exact results.
    (However, this may change in the future.)

    >>> round(geometric_mean([54, 24, 36]), 9)
    36.0
    """
    try:
        return exp(fmean(map(log, data)))
    except ValueError:
        raise StatisticsError('geometric mean requires a non-empty dataset '
                              ' containing positive numbers') from None


def harmonic_mean(data, weights=None):
    """Return the harmonic mean of data.

    The harmonic mean is the reciprocal of the arithmetic mean of the
    reciprocals of the data.  It can be used for averaging ratios or
    rates, for example speeds.

    Suppose a car travels 40 km/hr for 5 km and then speeds-up to
    60 km/hr for another 5 km. What is the average speed?

        >>> harmonic_mean([40, 60])
        48.0

    Suppose a car travels 40 km/hr for 5 km, and when traffic clears,
    speeds-up to 60 km/hr for the remaining 30 km of the journey. What
    is the average speed?

        >>> harmonic_mean([40, 60], weights=[5, 30])
        56.0

    If ``data`` is empty, or any element is less than zero,
    ``harmonic_mean`` will raise ``StatisticsError``.
    """
    if iter(data) is data:
        data = list(data)
    errmsg = 'harmonic mean does not support negative values'
    n = len(data)
    if n < 1:
        raise StatisticsError('harmonic_mean requires at least one data point')
    elif n == 1 and weights is None:
        x = data[0]
        if isinstance(x, (numbers.Real, Decimal)):
            if x < 0:
                raise StatisticsError(errmsg)
            return x
        else:
            raise TypeError('unsupported type')
    if weights is None:
        weights = repeat(1, n)
        sum_weights = n
    else:
        if iter(weights) is weights:
            weights = list(weights)
        if len(weights) != n:
            raise StatisticsError('Number of weights does not match data size')
        _, sum_weights, _ = _sum(w for w in _fail_neg(weights, errmsg))
    try:
        data = _fail_neg(data, errmsg)
        T, total, count = _sum(w / x if w else 0 for w, x in zip(weights, data))
    except ZeroDivisionError:
        return 0
    if total <= 0:
        raise StatisticsError('Weighted sum must be positive')
    return _convert(sum_weights / total, 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 are multiple modes with same frequency, return the first one
    encountered:

        >>> mode(['red', 'red', 'green', 'blue', 'blue'])
        'red'

    If *data* is empty, ``mode``, raises StatisticsError.

    """
    pairs = Counter(iter(data)).most_common(1)
    try:
        return pairs[0][0]
    except IndexError:
        raise StatisticsError('no mode for empty data') from None


def multimode(data):
    """Return a list of the most frequently occurring values.

    Will return more than one result if there are multiple modes
    or an empty list if *data* is empty.

    >>> multimode('aabbbbbbbbcc')
    ['b']
    >>> multimode('aabbbbccddddeeffffgg')
    ['b', 'd', 'f']
    >>> multimode('')
    []
    """
    counts = Counter(iter(data)).most_common()
    maxcount, mode_items = next(groupby(counts, key=itemgetter(1)), (0, []))
    return list(map(itemgetter(0), mode_items))


# Notes on methods for computing quantiles
# ----------------------------------------
#
# There is no one perfect way to compute quantiles.  Here we offer
# two methods that serve common needs.  Most other packages
# surveyed offered at least one or both of these two, making them
# "standard" in the sense of "widely-adopted and reproducible".
# They are also easy to explain, easy to compute manually, and have
# straight-forward interpretations that aren't surprising.

# The default method is known as "R6", "PERCENTILE.EXC", or "expected
# value of rank order statistics". The alternative method is known as
# "R7", "PERCENTILE.INC", or "mode of rank order statistics".

# For sample data where there is a positive probability for values
# beyond the range of the data, the R6 exclusive method is a
# reasonable choice.  Consider a random sample of nine values from a
# population with a uniform distribution from 0.0 to 1.0.  The
# distribution of the third ranked sample point is described by
# betavariate(alpha=3, beta=7) which has mode=0.250, median=0.286, and
# mean=0.300.  Only the latter (which corresponds with R6) gives the
# desired cut point with 30% of the population falling below that
# value, making it comparable to a result from an inv_cdf() function.
# The R6 exclusive method is also idempotent.

# For describing population data where the end points are known to
# be included in the data, the R7 inclusive method is a reasonable
# choice.  Instead of the mean, it uses the mode of the beta
# distribution for the interior points.  Per Hyndman & Fan, "One nice
# property is that the vertices of Q7(p) divide the range into n - 1
# intervals, and exactly 100p% of the intervals lie to the left of
# Q7(p) and 100(1 - p)% of the intervals lie to the right of Q7(p)."

# If needed, other methods could be added.  However, for now, the
# position is that fewer options make for easier choices and that
# external packages can be used for anything more advanced.

def quantiles(data, *, n=4, method='exclusive'):
    """Divide *data* into *n* continuous intervals with equal probability.

    Returns a list of (n - 1) cut points separating the intervals.

    Set *n* to 4 for quartiles (the default).  Set *n* to 10 for deciles.
    Set *n* to 100 for percentiles which gives the 99 cuts points that
    separate *data* in to 100 equal sized groups.

    The *data* can be any iterable containing sample.
    The cut points are linearly interpolated between data points.

    If *method* is set to *inclusive*, *data* is treated as population
    data.  The minimum value is treated as the 0th percentile and the
    maximum value is treated as the 100th percentile.
    """
    if n < 1:
        raise StatisticsError('n must be at least 1')
    data = sorted(data)
    ld = len(data)
    if ld < 2:
        raise StatisticsError('must have at least two data points')
    if method == 'inclusive':
        m = ld - 1
        result = []
        for i in range(1, n):
            j, delta = divmod(i * m, n)
            interpolated = (data[j] * (n - delta) + data[j + 1] * delta) / n
            result.append(interpolated)
        return result
    if method == 'exclusive':
        m = ld + 1
        result = []
        for i in range(1, n):
            j = i * m // n                               # rescale i to m/n
            j = 1 if j < 1 else ld-1 if j > ld-1 else j  # clamp to 1 .. ld-1
            delta = i*m - j*n                            # exact integer math
            interpolated = (data[j - 1] * (n - delta) + data[j] * delta) / n
            result.append(interpolated)
        return result
    raise ValueError(f'Unknown method: {method!r}')


# === 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 not None:
        T, total, count = _sum((x-c)**2 for x in data)
        return (T, total)
    T, total, count = _sum(data)
    mean_n, mean_d = (total / count).as_integer_ratio()
    partials = Counter()
    for n, d in map(_exact_ratio, data):
        diff_n = n * mean_d - d * mean_n
        diff_d = d * mean_d
        partials[diff_d * diff_d] += diff_n * diff_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:
        total = sum(Fraction(n, d) for d, n in partials.items())
    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 a sequence or 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

    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

    """
    # Fixme: Despite the exact sum of squared deviations, some inaccuracy
    # remain because there are two rounding steps.  The first occurs in
    # the _convert() step for variance(), the second occurs in math.sqrt().
    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

    """
    # Fixme: Despite the exact sum of squared deviations, some inaccuracy
    # remain because there are two rounding steps.  The first occurs in
    # the _convert() step for pvariance(), the second occurs in math.sqrt().
    var = pvariance(data, mu)
    try:
        return var.sqrt()
    except AttributeError:
        return math.sqrt(var)


# === Statistics for relations between two inputs ===

# See https://en.wikipedia.org/wiki/Covariance
#     https://en.wikipedia.org/wiki/Pearson_correlation_coefficient
#     https://en.wikipedia.org/wiki/Simple_linear_regression


def covariance(x, y, /):
    """Covariance

    Return the sample covariance of two inputs *x* and *y*. Covariance
    is a measure of the joint variability of two inputs.

    >>> x = [1, 2, 3, 4, 5, 6, 7, 8, 9]
    >>> y = [1, 2, 3, 1, 2, 3, 1, 2, 3]
    >>> covariance(x, y)
    0.75
    >>> z = [9, 8, 7, 6, 5, 4, 3, 2, 1]
    >>> covariance(x, z)
    -7.5
    >>> covariance(z, x)
    -7.5

    """
    n = len(x)
    if len(y) != n:
        raise StatisticsError('covariance requires that both inputs have same number of data points')
    if n < 2:
        raise StatisticsError('covariance requires at least two data points')
    xbar = fsum(x) / n
    ybar = fsum(y) / n
    sxy = fsum((xi - xbar) * (yi - ybar) for xi, yi in zip(x, y))
    return sxy / (n - 1)


def correlation(x, y, /):
    """Pearson's correlation coefficient

    Return the Pearson's correlation coefficient for two inputs. Pearson's
    correlation coefficient *r* takes values between -1 and +1. It measures the
    strength and direction of the linear relationship, where +1 means very
    strong, positive linear relationship, -1 very strong, negative linear
    relationship, and 0 no linear relationship.

    >>> x = [1, 2, 3, 4, 5, 6, 7, 8, 9]
    >>> y = [9, 8, 7, 6, 5, 4, 3, 2, 1]
    >>> correlation(x, x)
    1.0
    >>> correlation(x, y)
    -1.0

    """
    n = len(x)
    if len(y) != n:
        raise StatisticsError('correlation requires that both inputs have same number of data points')
    if n < 2:
        raise StatisticsError('correlation requires at least two data points')
    xbar = fsum(x) / n
    ybar = fsum(y) / n
    sxy = fsum((xi - xbar) * (yi - ybar) for xi, yi in zip(x, y))
    sxx = fsum((xi - xbar) ** 2.0 for xi in x)
    syy = fsum((yi - ybar) ** 2.0 for yi in y)
    try:
        return sxy / sqrt(sxx * syy)
    except ZeroDivisionError:
        raise StatisticsError('at least one of the inputs is constant')


LinearRegression = namedtuple('LinearRegression', ('slope', 'intercept'))


def linear_regression(x, y, /):
    """Slope and intercept for simple linear regression.

    Return the slope and intercept of simple linear regression
    parameters estimated using ordinary least squares. Simple linear
    regression describes relationship between an independent variable
    *x* and a dependent variable *y* in terms of linear function:

        y = slope * x + intercept + noise

    where *slope* and *intercept* are the regression parameters that are
    estimated, and noise represents the variability of the data that was
    not explained by the linear regression (it is equal to the
    difference between predicted and actual values of the dependent
    variable).

    The parameters are returned as a named tuple.

    >>> x = [1, 2, 3, 4, 5]
    >>> noise = NormalDist().samples(5, seed=42)
    >>> y = [3 * x[i] + 2 + noise[i] for i in range(5)]
    >>> linear_regression(x, y)  #doctest: +ELLIPSIS
    LinearRegression(slope=3.09078914170..., intercept=1.75684970486...)

    """
    n = len(x)
    if len(y) != n:
        raise StatisticsError('linear regression requires that both inputs have same number of data points')
    if n < 2:
        raise StatisticsError('linear regression requires at least two data points')
    xbar = fsum(x) / n
    ybar = fsum(y) / n
    sxy = fsum((xi - xbar) * (yi - ybar) for xi, yi in zip(x, y))
    sxx = fsum((xi - xbar) ** 2.0 for xi in x)
    try:
        slope = sxy / sxx   # equivalent to:  covariance(x, y) / variance(x)
    except ZeroDivisionError:
        raise StatisticsError('x is constant')
    intercept = ybar - slope * xbar
    return LinearRegression(slope=slope, intercept=intercept)


## Normal Distribution #####################################################


def _normal_dist_inv_cdf(p, mu, sigma):
    # There is no closed-form solution to the inverse CDF for the normal
    # distribution, so we use a rational approximation instead:
    # Wichura, M.J. (1988). "Algorithm AS241: The Percentage Points of the
    # Normal Distribution".  Applied Statistics. Blackwell Publishing. 37
    # (3): 477–484. doi:10.2307/2347330. JSTOR 2347330.
    q = p - 0.5
    if fabs(q) <= 0.425:
        r = 0.180625 - q * q
        # Hash sum: 55.88319_28806_14901_4439
        num = (((((((2.50908_09287_30122_6727e+3 * r +
                     3.34305_75583_58812_8105e+4) * r +
                     6.72657_70927_00870_0853e+4) * r +
                     4.59219_53931_54987_1457e+4) * r +
                     1.37316_93765_50946_1125e+4) * r +
                     1.97159_09503_06551_4427e+3) * r +
                     1.33141_66789_17843_7745e+2) * r +
                     3.38713_28727_96366_6080e+0) * q
        den = (((((((5.22649_52788_52854_5610e+3 * r +
                     2.87290_85735_72194_2674e+4) * r +
                     3.93078_95800_09271_0610e+4) * r +
                     2.12137_94301_58659_5867e+4) * r +
                     5.39419_60214_24751_1077e+3) * r +
                     6.87187_00749_20579_0830e+2) * r +
                     4.23133_30701_60091_1252e+1) * r +
                     1.0)
        x = num / den
        return mu + (x * sigma)
    r = p if q <= 0.0 else 1.0 - p
    r = sqrt(-log(r))
    if r <= 5.0:
        r = r - 1.6
        # Hash sum: 49.33206_50330_16102_89036
        num = (((((((7.74545_01427_83414_07640e-4 * r +
                     2.27238_44989_26918_45833e-2) * r +
                     2.41780_72517_74506_11770e-1) * r +
                     1.27045_82524_52368_38258e+0) * r +
                     3.64784_83247_63204_60504e+0) * r +
                     5.76949_72214_60691_40550e+0) * r +
                     4.63033_78461_56545_29590e+0) * r +
                     1.42343_71107_49683_57734e+0)
        den = (((((((1.05075_00716_44416_84324e-9 * r +
                     5.47593_80849_95344_94600e-4) * r +
                     1.51986_66563_61645_71966e-2) * r +
                     1.48103_97642_74800_74590e-1) * r +
                     6.89767_33498_51000_04550e-1) * r +
                     1.67638_48301_83803_84940e+0) * r +
                     2.05319_16266_37758_82187e+0) * r +
                     1.0)
    else:
        r = r - 5.0
        # Hash sum: 47.52583_31754_92896_71629
        num = (((((((2.01033_43992_92288_13265e-7 * r +
                     2.71155_55687_43487_57815e-5) * r +
                     1.24266_09473_88078_43860e-3) * r +
                     2.65321_89526_57612_30930e-2) * r +
                     2.96560_57182_85048_91230e-1) * r +
                     1.78482_65399_17291_33580e+0) * r +
                     5.46378_49111_64114_36990e+0) * r +
                     6.65790_46435_01103_77720e+0)
        den = (((((((2.04426_31033_89939_78564e-15 * r +
                     1.42151_17583_16445_88870e-7) * r +
                     1.84631_83175_10054_68180e-5) * r +
                     7.86869_13114_56132_59100e-4) * r +
                     1.48753_61290_85061_48525e-2) * r +
                     1.36929_88092_27358_05310e-1) * r +
                     5.99832_20655_58879_37690e-1) * r +
                     1.0)
    x = num / den
    if q < 0.0:
        x = -x
    return mu + (x * sigma)


# If available, use C implementation
try:
    from _statistics import _normal_dist_inv_cdf
except ImportError:
    pass


class NormalDist:
    "Normal distribution of a random variable"
    # https://en.wikipedia.org/wiki/Normal_distribution
    # https://en.wikipedia.org/wiki/Variance#Properties

    __slots__ = {
        '_mu': 'Arithmetic mean of a normal distribution',
        '_sigma': 'Standard deviation of a normal distribution',
    }

    def __init__(self, mu=0.0, sigma=1.0):
        "NormalDist where mu is the mean and sigma is the standard deviation."
        if sigma < 0.0:
            raise StatisticsError('sigma must be non-negative')
        self._mu = float(mu)
        self._sigma = float(sigma)

    @classmethod
    def from_samples(cls, data):
        "Make a normal distribution instance from sample data."
        if not isinstance(data, (list, tuple)):
            data = list(data)
        xbar = fmean(data)
        return cls(xbar, stdev(data, xbar))

    def samples(self, n, *, seed=None):
        "Generate *n* samples for a given mean and standard deviation."
        gauss = random.gauss if seed is None else random.Random(seed).gauss
        mu, sigma = self._mu, self._sigma
        return [gauss(mu, sigma) for i in range(n)]

    def pdf(self, x):
        "Probability density function.  P(x <= X < x+dx) / dx"
        variance = self._sigma ** 2.0
        if not variance:
            raise StatisticsError('pdf() not defined when sigma is zero')
        return exp((x - self._mu)**2.0 / (-2.0*variance)) / sqrt(tau*variance)

    def cdf(self, x):
        "Cumulative distribution function.  P(X <= x)"
        if not self._sigma:
            raise StatisticsError('cdf() not defined when sigma is zero')
        return 0.5 * (1.0 + erf((x - self._mu) / (self._sigma * sqrt(2.0))))

    def inv_cdf(self, p):
        """Inverse cumulative distribution function.  x : P(X <= x) = p

        Finds the value of the random variable such that the probability of
        the variable being less than or equal to that value equals the given
        probability.

        This function is also called the percent point function or quantile
        function.
        """
        if p <= 0.0 or p >= 1.0:
            raise StatisticsError('p must be in the range 0.0 < p < 1.0')
        if self._sigma <= 0.0:
            raise StatisticsError('cdf() not defined when sigma at or below zero')
        return _normal_dist_inv_cdf(p, self._mu, self._sigma)

    def quantiles(self, n=4):
        """Divide into *n* continuous intervals with equal probability.

        Returns a list of (n - 1) cut points separating the intervals.

        Set *n* to 4 for quartiles (the default).  Set *n* to 10 for deciles.
        Set *n* to 100 for percentiles which gives the 99 cuts points that
        separate the normal distribution in to 100 equal sized groups.
        """
        return [self.inv_cdf(i / n) for i in range(1, n)]

    def overlap(self, other):
        """Compute the overlapping coefficient (OVL) between two normal distributions.

        Measures the agreement between two normal probability distributions.
        Returns a value between 0.0 and 1.0 giving the overlapping area in
        the two underlying probability density functions.

            >>> N1 = NormalDist(2.4, 1.6)
            >>> N2 = NormalDist(3.2, 2.0)
            >>> N1.overlap(N2)
            0.8035050657330205
        """
        # See: "The overlapping coefficient as a measure of agreement between
        # probability distributions and point estimation of the overlap of two
        # normal densities" -- Henry F. Inman and Edwin L. Bradley Jr
        # http://dx.doi.org/10.1080/03610928908830127
        if not isinstance(other, NormalDist):
            raise TypeError('Expected another NormalDist instance')
        X, Y = self, other
        if (Y._sigma, Y._mu) < (X._sigma, X._mu):  # sort to assure commutativity
            X, Y = Y, X
        X_var, Y_var = X.variance, Y.variance
        if not X_var or not Y_var:
            raise StatisticsError('overlap() not defined when sigma is zero')
        dv = Y_var - X_var
        dm = fabs(Y._mu - X._mu)
        if not dv:
            return 1.0 - erf(dm / (2.0 * X._sigma * sqrt(2.0)))
        a = X._mu * Y_var - Y._mu * X_var
        b = X._sigma * Y._sigma * sqrt(dm**2.0 + dv * log(Y_var / X_var))
        x1 = (a + b) / dv
        x2 = (a - b) / dv
        return 1.0 - (fabs(Y.cdf(x1) - X.cdf(x1)) + fabs(Y.cdf(x2) - X.cdf(x2)))

    def zscore(self, x):
        """Compute the Standard Score.  (x - mean) / stdev

        Describes *x* in terms of the number of standard deviations
        above or below the mean of the normal distribution.
        """
        # https://www.statisticshowto.com/probability-and-statistics/z-score/
        if not self._sigma:
            raise StatisticsError('zscore() not defined when sigma is zero')
        return (x - self._mu) / self._sigma

    @property
    def mean(self):
        "Arithmetic mean of the normal distribution."
        return self._mu

    @property
    def median(self):
        "Return the median of the normal distribution"
        return self._mu

    @property
    def mode(self):
        """Return the mode of the normal distribution

        The mode is the value x where which the probability density
        function (pdf) takes its maximum value.
        """
        return self._mu

    @property
    def stdev(self):
        "Standard deviation of the normal distribution."
        return self._sigma

    @property
    def variance(self):
        "Square of the standard deviation."
        return self._sigma ** 2.0

    def __add__(x1, x2):
        """Add a constant or another NormalDist instance.

        If *other* is a constant, translate mu by the constant,
        leaving sigma unchanged.

        If *other* is a NormalDist, add both the means and the variances.
        Mathematically, this works only if the two distributions are
        independent or if they are jointly normally distributed.
        """
        if isinstance(x2, NormalDist):
            return NormalDist(x1._mu + x2._mu, hypot(x1._sigma, x2._sigma))
        return NormalDist(x1._mu + x2, x1._sigma)

    def __sub__(x1, x2):
        """Subtract a constant or another NormalDist instance.

        If *other* is a constant, translate by the constant mu,
        leaving sigma unchanged.

        If *other* is a NormalDist, subtract the means and add the variances.
        Mathematically, this works only if the two distributions are
        independent or if they are jointly normally distributed.
        """
        if isinstance(x2, NormalDist):
            return NormalDist(x1._mu - x2._mu, hypot(x1._sigma, x2._sigma))
        return NormalDist(x1._mu - x2, x1._sigma)

    def __mul__(x1, x2):
        """Multiply both mu and sigma by a constant.

        Used for rescaling, perhaps to change measurement units.
        Sigma is scaled with the absolute value of the constant.
        """
        return NormalDist(x1._mu * x2, x1._sigma * fabs(x2))

    def __truediv__(x1, x2):
        """Divide both mu and sigma by a constant.

        Used for rescaling, perhaps to change measurement units.
        Sigma is scaled with the absolute value of the constant.
        """
        return NormalDist(x1._mu / x2, x1._sigma / fabs(x2))

    def __pos__(x1):
        "Return a copy of the instance."
        return NormalDist(x1._mu, x1._sigma)

    def __neg__(x1):
        "Negates mu while keeping sigma the same."
        return NormalDist(-x1._mu, x1._sigma)

    __radd__ = __add__

    def __rsub__(x1, x2):
        "Subtract a NormalDist from a constant or another NormalDist."
        return -(x1 - x2)

    __rmul__ = __mul__

    def __eq__(x1, x2):
        "Two NormalDist objects are equal if their mu and sigma are both equal."
        if not isinstance(x2, NormalDist):
            return NotImplemented
        return x1._mu == x2._mu and x1._sigma == x2._sigma

    def __hash__(self):
        "NormalDist objects hash equal if their mu and sigma are both equal."
        return hash((self._mu, self._sigma))

    def __repr__(self):
        return f'{type(self).__name__}(mu={self._mu!r}, sigma={self._sigma!r})'