Sync contrib/tools/python3 layout with upstream

* Move src/ subdir contents to the top of the layout
* Rename self-written lib -> lib2 to avoid CaseFolding warning from the VCS
* Regenerate contrib/libs/python proxy-headers accordingly
4ccc62ac1511abcf0fed14ccade38e984e088f1e

1 files changed, 1454 insertions, 0 deletions

diff --git a/contrib/tools/python3/Lib/statistics.py b/contrib/tools/python3/Lib/statistics.py
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+++ b/contrib/tools/python3/Lib/statistics.py
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+"""
+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
+import sys
+
+from fractions import Fraction
+from decimal import Decimal
+from itertools import count, groupby, repeat
+from bisect import bisect_left, bisect_right
+from math import hypot, sqrt, fabs, exp, erf, tau, log, fsum, sumprod
+from functools import reduce
+from operator import itemgetter
+from collections import Counter, namedtuple, defaultdict
+
+_SQRT2 = sqrt(2.0)
+
+# === 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
+    types = set()
+    types_add = types.add
+    partials = {}
+    partials_get = partials.get
+    for typ, values in groupby(data, type):
+        types_add(typ)
+        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())
+    T = reduce(_coerce, types, int)  # or raise TypeError
+    return (T, total, count)
+
+
+def _ss(data, c=None):
+    """Return the exact mean and sum of square deviations of sequence data.
+
+    Calculations are done in a single pass, allowing the input to be an iterator.
+
+    If given *c* is used the mean; otherwise, it is calculated from the data.
+    Use the *c* argument with care, as it can lead to garbage results.
+
+    """
+    if c is not None:
+        T, ssd, count = _sum((d := x - c) * d for x in data)
+        return (T, ssd, c, count)
+    count = 0
+    types = set()
+    types_add = types.add
+    sx_partials = defaultdict(int)
+    sxx_partials = defaultdict(int)
+    for typ, values in groupby(data, type):
+        types_add(typ)
+        for n, d in map(_exact_ratio, values):
+            count += 1
+            sx_partials[d] += n
+            sxx_partials[d] += n * n
+    if not count:
+        ssd = c = Fraction(0)
+    elif None in sx_partials:
+        # The sum will be a NAN or INF. We can ignore all the finite
+        # partials, and just look at this special one.
+        ssd = c = sx_partials[None]
+        assert not _isfinite(ssd)
+    else:
+        sx = sum(Fraction(n, d) for d, n in sx_partials.items())
+        sxx = sum(Fraction(n, d*d) for d, n in sxx_partials.items())
+        # This formula has poor numeric properties for floats,
+        # but with fractions it is exact.
+        ssd = (count * sxx - sx * sx) / count
+        c = sx / count
+    T = reduce(_coerce, types, int)  # or raise TypeError
+    return (T, ssd, c, 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.
+    """
+
+    # XXX We should revisit whether using fractions to accumulate exact
+    # ratios is the right way to go.
+
+    # The integer ratios for binary floats can have numerators or
+    # denominators with over 300 decimal digits.  The problem is more
+    # acute with decimal floats where the default decimal context
+    # supports a huge range of exponents from Emin=-999999 to
+    # Emax=999999.  When expanded with as_integer_ratio(), numbers like
+    # Decimal('3.14E+5000') and Decimal('3.14E-5000') have large
+    # numerators or denominators that will slow computation.
+
+    # When the integer ratios are accumulated as fractions, the size
+    # grows to cover the full range from the smallest magnitude to the
+    # largest.  For example, Fraction(3.14E+300) + Fraction(3.14E-300),
+    # has a 616 digit numerator.  Likewise,
+    # Fraction(Decimal('3.14E+5000')) + Fraction(Decimal('3.14E-5000'))
+    # has 10,003 digit numerator.
+
+    # This doesn't seem to have been problem in practice, but it is a
+    # potential pitfall.
+
+    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 _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
+
+
+def _rank(data, /, *, key=None, reverse=False, ties='average', start=1) -> list[float]:
+    """Rank order a dataset. The lowest value has rank 1.
+
+    Ties are averaged so that equal values receive the same rank:
+
+        >>> data = [31, 56, 31, 25, 75, 18]
+        >>> _rank(data)
+        [3.5, 5.0, 3.5, 2.0, 6.0, 1.0]
+
+    The operation is idempotent:
+
+        >>> _rank([3.5, 5.0, 3.5, 2.0, 6.0, 1.0])
+        [3.5, 5.0, 3.5, 2.0, 6.0, 1.0]
+
+    It is possible to rank the data in reverse order so that the
+    highest value has rank 1.  Also, a key-function can extract
+    the field to be ranked:
+
+        >>> goals = [('eagles', 45), ('bears', 48), ('lions', 44)]
+        >>> _rank(goals, key=itemgetter(1), reverse=True)
+        [2.0, 1.0, 3.0]
+
+    Ranks are conventionally numbered starting from one; however,
+    setting *start* to zero allows the ranks to be used as array indices:
+
+        >>> prize = ['Gold', 'Silver', 'Bronze', 'Certificate']
+        >>> scores = [8.1, 7.3, 9.4, 8.3]
+        >>> [prize[int(i)] for i in _rank(scores, start=0, reverse=True)]
+        ['Bronze', 'Certificate', 'Gold', 'Silver']
+
+    """
+    # If this function becomes public at some point, more thought
+    # needs to be given to the signature.  A list of ints is
+    # plausible when ties is "min" or "max".  When ties is "average",
+    # either list[float] or list[Fraction] is plausible.
+
+    # Default handling of ties matches scipy.stats.mstats.spearmanr.
+    if ties != 'average':
+        raise ValueError(f'Unknown tie resolution method: {ties!r}')
+    if key is not None:
+        data = map(key, data)
+    val_pos = sorted(zip(data, count()), reverse=reverse)
+    i = start - 1
+    result = [0] * len(val_pos)
+    for _, g in groupby(val_pos, key=itemgetter(0)):
+        group = list(g)
+        size = len(group)
+        rank = i + (size + 1) / 2
+        for value, orig_pos in group:
+            result[orig_pos] = rank
+        i += size
+    return result
+
+
+def _integer_sqrt_of_frac_rto(n: int, m: int) -> int:
+    """Square root of n/m, rounded to the nearest integer using round-to-odd."""
+    # Reference: https://www.lri.fr/~melquion/doc/05-imacs17_1-expose.pdf
+    a = math.isqrt(n // m)
+    return a | (a*a*m != n)
+
+
+# For 53 bit precision floats, the bit width used in
+# _float_sqrt_of_frac() is 109.
+_sqrt_bit_width: int = 2 * sys.float_info.mant_dig + 3
+
+
+def _float_sqrt_of_frac(n: int, m: int) -> float:
+    """Square root of n/m as a float, correctly rounded."""
+    # See principle and proof sketch at: https://bugs.python.org/msg407078
+    q = (n.bit_length() - m.bit_length() - _sqrt_bit_width) // 2
+    if q >= 0:
+        numerator = _integer_sqrt_of_frac_rto(n, m << 2 * q) << q
+        denominator = 1
+    else:
+        numerator = _integer_sqrt_of_frac_rto(n << -2 * q, m)
+        denominator = 1 << -q
+    return numerator / denominator   # Convert to float
+
+
+def _decimal_sqrt_of_frac(n: int, m: int) -> Decimal:
+    """Square root of n/m as a Decimal, correctly rounded."""
+    # Premise:  For decimal, computing (n/m).sqrt() can be off
+    #           by 1 ulp from the correctly rounded result.
+    # Method:   Check the result, moving up or down a step if needed.
+    if n <= 0:
+        if not n:
+            return Decimal('0.0')
+        n, m = -n, -m
+
+    root = (Decimal(n) / Decimal(m)).sqrt()
+    nr, dr = root.as_integer_ratio()
+
+    plus = root.next_plus()
+    np, dp = plus.as_integer_ratio()
+    # test: n / m > ((root + plus) / 2) ** 2
+    if 4 * n * (dr*dp)**2 > m * (dr*np + dp*nr)**2:
+        return plus
+
+    minus = root.next_minus()
+    nm, dm = minus.as_integer_ratio()
+    # test: n / m < ((root + minus) / 2) ** 2
+    if 4 * n * (dr*dm)**2 < m * (dr*nm + dm*nr)**2:
+        return minus
+
+    return root
+
+
+# === 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.
+    """
+    T, total, n = _sum(data)
+    if n < 1:
+        raise StatisticsError('mean requires at least one data point')
+    return _convert(total / n, T)
+
+
+def fmean(data, weights=None):
+    """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
+    """
+    if weights is None:
+        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
+            data = count(data)
+        total = fsum(data)
+        if not n:
+            raise StatisticsError('fmean requires at least one data point')
+        return total / n
+    if not isinstance(weights, (list, tuple)):
+        weights = list(weights)
+    try:
+        num = sumprod(data, weights)
+    except ValueError:
+        raise StatisticsError('data and weights must be the same length')
+    den = fsum(weights)
+    if not den:
+        raise StatisticsError('sum of weights must be non-zero')
+    return num / den
+
+
+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=1.0):
+    """Estimates the median for numeric data binned around the midpoints
+    of consecutive, fixed-width intervals.
+
+    The *data* can be any iterable of numeric data with each value being
+    exactly the midpoint of a bin.  At least one value must be present.
+
+    The *interval* is width of each bin.
+
+    For example, demographic information may have been summarized into
+    consecutive ten-year age groups with each group being represented
+    by the 5-year midpoints of the intervals:
+
+        >>> demographics = Counter({
+        ...    25: 172,   # 20 to 30 years old
+        ...    35: 484,   # 30 to 40 years old
+        ...    45: 387,   # 40 to 50 years old
+        ...    55:  22,   # 50 to 60 years old
+        ...    65:   6,   # 60 to 70 years old
+        ... })
+
+    The 50th percentile (median) is the 536th person out of the 1071
+    member cohort.  That person is in the 30 to 40 year old age group.
+
+    The regular median() function would assume that everyone in the
+    tricenarian age group was exactly 35 years old.  A more tenable
+    assumption is that the 484 members of that age group are evenly
+    distributed between 30 and 40.  For that, we use median_grouped().
+
+        >>> data = list(demographics.elements())
+        >>> median(data)
+        35
+        >>> round(median_grouped(data, interval=10), 1)
+        37.5
+
+    The caller is responsible for making sure the data points are separated
+    by exact multiples of *interval*.  This is essential for getting a
+    correct result.  The function does not check this precondition.
+
+    Inputs may be any numeric type that can be coerced to a float during
+    the interpolation step.
+
+    """
+    data = sorted(data)
+    n = len(data)
+    if not n:
+        raise StatisticsError("no median for empty data")
+
+    # Find the value at the midpoint. Remember this corresponds to the
+    # midpoint of the class interval.
+    x = data[n // 2]
+
+    # Using O(log n) bisection, find where all the x values occur in the data.
+    # All x will lie within data[i:j].
+    i = bisect_left(data, x)
+    j = bisect_right(data, x, lo=i)
+
+    # Coerce to floats, raising a TypeError if not possible
+    try:
+        interval = float(interval)
+        x = float(x)
+    except ValueError:
+        raise TypeError(f'Value cannot be converted to a float')
+
+    # Interpolate the median using the formula found at:
+    # https://www.cuemath.com/data/median-of-grouped-data/
+    L = x - interval / 2.0    # Lower limit of the median interval
+    cf = i                    # Cumulative frequency of the preceding interval
+    f = j - i                 # Number of elements in the median internal
+    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))
+    if not counts:
+        return []
+    maxcount = max(counts.values())
+    return [value for value, count in counts.items() if count == maxcount]
+
+
+# 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
+
+
+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)
+
+    """
+    T, ss, c, n = _ss(data, xbar)
+    if n < 2:
+        raise StatisticsError('variance requires at least two data points')
+    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)
+
+    """
+    T, ss, c, n = _ss(data, mu)
+    if n < 1:
+        raise StatisticsError('pvariance requires at least one data point')
+    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
+
+    """
+    T, ss, c, n = _ss(data, xbar)
+    if n < 2:
+        raise StatisticsError('stdev requires at least two data points')
+    mss = ss / (n - 1)
+    if issubclass(T, Decimal):
+        return _decimal_sqrt_of_frac(mss.numerator, mss.denominator)
+    return _float_sqrt_of_frac(mss.numerator, mss.denominator)
+
+
+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
+
+    """
+    T, ss, c, n = _ss(data, mu)
+    if n < 1:
+        raise StatisticsError('pstdev requires at least one data point')
+    mss = ss / n
+    if issubclass(T, Decimal):
+        return _decimal_sqrt_of_frac(mss.numerator, mss.denominator)
+    return _float_sqrt_of_frac(mss.numerator, mss.denominator)
+
+
+def _mean_stdev(data):
+    """In one pass, compute the mean and sample standard deviation as floats."""
+    T, ss, xbar, n = _ss(data)
+    if n < 2:
+        raise StatisticsError('stdev requires at least two data points')
+    mss = ss / (n - 1)
+    try:
+        return float(xbar), _float_sqrt_of_frac(mss.numerator, mss.denominator)
+    except AttributeError:
+        # Handle Nans and Infs gracefully
+        return float(xbar), float(xbar) / float(ss)
+
+
+# === 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 = sumprod((xi - xbar for xi in x), (yi - ybar for yi in y))
+    return sxy / (n - 1)
+
+
+def correlation(x, y, /, *, method='linear'):
+    """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 a 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
+
+    If *method* is "ranked", computes Spearman's rank correlation coefficient
+    for two inputs.  The data is replaced by ranks.  Ties are averaged
+    so that equal values receive the same rank.  The resulting coefficient
+    measures the strength of a monotonic relationship.
+
+    Spearman's rank correlation coefficient is appropriate for ordinal
+    data or for continuous data that doesn't meet the linear proportion
+    requirement for Pearson's correlation coefficient.
+    """
+    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')
+    if method not in {'linear', 'ranked'}:
+        raise ValueError(f'Unknown method: {method!r}')
+    if method == 'ranked':
+        start = (n - 1) / -2            # Center rankings around zero
+        x = _rank(x, start=start)
+        y = _rank(y, start=start)
+    else:
+        xbar = fsum(x) / n
+        ybar = fsum(y) / n
+        x = [xi - xbar for xi in x]
+        y = [yi - ybar for yi in y]
+    sxy = sumprod(x, y)
+    sxx = sumprod(x, x)
+    syy = sumprod(y, 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, /, *, proportional=False):
+    """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 a 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...)
+
+    If *proportional* is true, the independent variable *x* and the
+    dependent variable *y* are assumed to be directly proportional.
+    The data is fit to a line passing through the origin.
+
+    Since the *intercept* will always be 0.0, the underlying linear
+    function simplifies to:
+
+        y = slope * x + noise
+
+    >>> y = [3 * x[i] + noise[i] for i in range(5)]
+    >>> linear_regression(x, y, proportional=True)  #doctest: +ELLIPSIS
+    LinearRegression(slope=3.02447542484..., intercept=0.0)
+
+    """
+    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')
+    if not proportional:
+        xbar = fsum(x) / n
+        ybar = fsum(y) / n
+        x = [xi - xbar for xi in x]  # List because used three times below
+        y = (yi - ybar for yi in y)  # Generator because only used once below
+    sxy = sumprod(x, y) + 0.0        # Add zero to coerce result to a float
+    sxx = sumprod(x, x)
+    try:
+        slope = sxy / sxx   # equivalent to:  covariance(x, y) / variance(x)
+    except ZeroDivisionError:
+        raise StatisticsError('x is constant')
+    intercept = 0.0 if proportional else 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."
+        return cls(*_mean_stdev(data))
+
+    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 _ in repeat(None, n)]
+
+    def pdf(self, x):
+        "Probability density function.  P(x <= X < x+dx) / dx"
+        variance = self._sigma * self._sigma
+        if not variance:
+            raise StatisticsError('pdf() not defined when sigma is zero')
+        diff = x - self._mu
+        return exp(diff * diff / (-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 * _SQRT2)))
+
+    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')
+        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 * _SQRT2))
+        a = X._mu * Y_var - Y._mu * X_var
+        b = X._sigma * Y._sigma * sqrt(dm * dm + 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 * self._sigma
+
+    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})'
+
+    def __getstate__(self):
+        return self._mu, self._sigma
+
+    def __setstate__(self, state):
+        self._mu, self._sigma = state