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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
+
+
+Exceptions
+----------
+
+A single exception is defined: StatisticsError is a subclass of ValueError.
+
+"""
+
+__all__ = [
+    'NormalDist',
+    'StatisticsError',
+    'fmean',
+    'geometric_mean',
+    'harmonic_mean',
+    '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
+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
+
+# === Exceptions ===
+
+class StatisticsError(ValueError):
+    pass
+
+
+# === Private utilities ===
+
+def _sum(data, start=0):
+    """_sum(data [, start]) -> (type, sum, count)
+
+    Return a high-precision sum of the given numeric data as a fraction,
+    together with the type to be converted to and the count of items.
+
+    If optional argument ``start`` is given, it is added to the total.
+    If ``data`` is empty, ``start`` (defaulting to 0) is returned.
+
+
+    Examples
+    --------
+
+    >>> _sum([3, 2.25, 4.5, -0.5, 1.0], 0.75)
+    (<class 'float'>, Fraction(11, 1), 5)
+
+    Some sources of round-off error will be avoided:
+
+    # 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
+    n, d = _exact_ratio(start)
+    partials = {d: n}
+    partials_get = partials.get
+    T = _coerce(int, type(start))
+    for typ, values in groupby(data, type):
+        T = _coerce(T, typ)  # or raise TypeError
+        for n, d in map(_exact_ratio, values):
+            count += 1
+            partials[d] = partials_get(d, 0) + n
+    if None in partials:
+        # The sum will be a NAN or INF. We can ignore all the finite
+        # partials, and just look at this special one.
+        total = partials[None]
+        assert not _isfinite(total)
+    else:
+        # Sum all the partial sums using builtin sum.
+        # FIXME is this faster if we sum them in order of the denominator?
+        total = sum(Fraction(n, d) for d, n in sorted(partials.items()))
+    return (T, total, count)
+
+
+def _isfinite(x):
+    try:
+        return x.is_finite()  # Likely a Decimal.
+    except AttributeError:
+        return math.isfinite(x)  # Coerces to float first.
+
+
+def _coerce(T, S):
+    """Coerce types T and S to a common type, or raise TypeError.
+
+    Coercion rules are currently an implementation detail. See the CoerceTest
+    test class in test_statistics for details.
+    """
+    # See http://bugs.python.org/issue24068.
+    assert T is not bool, "initial type T is bool"
+    # If the types are the same, no need to coerce anything. Put this
+    # first, so that the usual case (no coercion needed) happens as soon
+    # as possible.
+    if T is S:  return T
+    # Mixed int & other coerce to the other type.
+    if S is int or S is bool:  return T
+    if T is int:  return S
+    # If one is a (strict) subclass of the other, coerce to the subclass.
+    if issubclass(S, T):  return S
+    if issubclass(T, S):  return T
+    # Ints coerce to the other type.
+    if issubclass(T, int):  return S
+    if issubclass(S, int):  return T
+    # Mixed fraction & float coerces to float (or float subclass).
+    if issubclass(T, Fraction) and issubclass(S, float):
+        return S
+    if issubclass(T, float) and issubclass(S, Fraction):
+        return T
+    # Any other combination is disallowed.
+    msg = "don't know how to coerce %s and %s"
+    raise TypeError(msg % (T.__name__, S.__name__))
+
+
+def _exact_ratio(x):
+    """Return Real number x to exact (numerator, denominator) pair.
+
+    >>> _exact_ratio(0.25)
+    (1, 4)
+
+    x is expected to be an int, Fraction, Decimal or float.
+    """
+    try:
+        # Optimise the common case of floats. We expect that the most often
+        # used numeric type will be builtin floats, so try to make this as
+        # fast as possible.
+        if type(x) is float or type(x) is Decimal:
+            return x.as_integer_ratio()
+        try:
+            # x may be an int, Fraction, or Integral ABC.
+            return (x.numerator, x.denominator)
+        except AttributeError:
+            try:
+                # x may be a float or Decimal subclass.
+                return x.as_integer_ratio()
+            except AttributeError:
+                # Just give up?
+                pass
+    except (OverflowError, ValueError):
+        # float NAN or INF.
+        assert not _isfinite(x)
+        return (x, None)
+    msg = "can't convert type '{}' to numerator/denominator"
+    raise TypeError(msg.format(type(x).__name__))
+
+
+def _convert(value, T):
+    """Convert value to given numeric type T."""
+    if type(value) is T:
+        # This covers the cases where T is Fraction, or where value is
+        # a NAN or INF (Decimal or float).
+        return value
+    if issubclass(T, int) and value.denominator != 1:
+        T = float
+    try:
+        # FIXME: what do we do if this overflows?
+        return T(value)
+    except TypeError:
+        if issubclass(T, Decimal):
+            return T(value.numerator) / T(value.denominator)
+        else:
+            raise
+
+
+def _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):
+    """Return the harmonic mean of data.
+
+    The harmonic mean, sometimes called the subcontrary mean, is the
+    reciprocal of the arithmetic mean of the reciprocals of the data,
+    and is often appropriate when averaging quantities which are rates
+    or ratios, for example speeds. Example:
+
+    Suppose an investor purchases an equal value of shares in each of
+    three companies, with P/E (price/earning) ratios of 2.5, 3 and 10.
+    What is the average P/E ratio for the investor's portfolio?
+
+    >>> harmonic_mean([2.5, 3, 10])  # For an equal investment portfolio.
+    3.6
+
+    Using the arithmetic mean would give an average of about 5.167, which
+    is too high.
+
+    If ``data`` is empty, or any element is less than zero,
+    ``harmonic_mean`` will raise ``StatisticsError``.
+    """
+    # For a justification for using harmonic mean for P/E ratios, see
+    # http://fixthepitch.pellucid.com/comps-analysis-the-missing-harmony-of-summary-statistics/
+    # http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2621087
+    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:
+        x = data[0]
+        if isinstance(x, (numbers.Real, Decimal)):
+            if x < 0:
+                raise StatisticsError(errmsg)
+            return x
+        else:
+            raise TypeError('unsupported type')
+    try:
+        T, total, count = _sum(1 / x for x in _fail_neg(data, errmsg))
+    except ZeroDivisionError:
+        return 0
+    assert count == n
+    return _convert(n / 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)
+    c = mean(data)
+    T, total, count = _sum((x-c)**2 for x in data)
+    # The following sum should mathematically equal zero, but due to rounding
+    # error may not.
+    U, total2, count2 = _sum((x - c) for x in data)
+    assert T == U and count == count2
+    total -= total2 ** 2 / len(data)
+    assert not total < 0, 'negative sum of square deviations: %f' % total
+    return (T, total)
+
+
+def variance(data, xbar=None):
+    """Return the sample variance of data.
+
+    data should be an iterable of Real-valued numbers, with at least two
+    values. The optional argument xbar, if given, should be the mean of
+    the data. If it is missing or None, the mean is automatically calculated.
+
+    Use this function when your data is a sample from a population. To
+    calculate the variance from the entire population, see ``pvariance``.
+
+    Examples:
+
+    >>> data = [2.75, 1.75, 1.25, 0.25, 0.5, 1.25, 3.5]
+    >>> variance(data)
+    1.3720238095238095
+
+    If you have already calculated the mean of your data, you can pass it as
+    the optional second argument ``xbar`` to avoid recalculating it:
+
+    >>> m = mean(data)
+    >>> variance(data, m)
+    1.3720238095238095
+
+    This function does not check that ``xbar`` is actually the mean of
+    ``data``. Giving arbitrary values for ``xbar`` may lead to invalid or
+    impossible results.
+
+    Decimals and Fractions are supported:
+
+    >>> from decimal import Decimal as D
+    >>> variance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")])
+    Decimal('31.01875')
+
+    >>> from fractions import Fraction as F
+    >>> variance([F(1, 6), F(1, 2), F(5, 3)])
+    Fraction(67, 108)
+
+    """
+    if iter(data) is data:
+        data = list(data)
+    n = len(data)
+    if n < 2:
+        raise StatisticsError('variance requires at least two data points')
+    T, ss = _ss(data, xbar)
+    return _convert(ss / (n - 1), T)
+
+
+def pvariance(data, mu=None):
+    """Return the population variance of ``data``.
+
+    data should be 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
+
+    """
+    var = variance(data, xbar)
+    try:
+        return var.sqrt()
+    except AttributeError:
+        return math.sqrt(var)
+
+
+def pstdev(data, mu=None):
+    """Return the square root of the population variance.
+
+    See ``pvariance`` for arguments and other details.
+
+    >>> pstdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
+    0.986893273527251
+
+    """
+    var = pvariance(data, mu)
+    try:
+        return var.sqrt()
+    except AttributeError:
+        return math.sqrt(var)
+
+
+## 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})'