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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.

)
NormalDistStatisticsErrorcorrelation
covariancefmeangeometric_mean
harmonic_meanlinear_regressionmeanmedianmedian_groupedmedian_high
median_lowmode	multimodepstdev	pvariance	quantilesstdevvarianceNFraction)Decimal)groupbyrepeat)bisect_leftbisect_right)hypotsqrtfabsexperftaulogfsum)reduce)mul)Counter
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    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.
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%FB1u2zABB 2222Kr1cxt|\}}}|dkrtdt||z|S)aReturn 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.
    r4z%mean requires at least one data point)rPrrr)rFrOrNr:s    r2r
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    This runs faster than the mean() function and it always returns a float.
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    >>> fmean([3.5, 4.0, 5.25])
    4.25
    rc3BKt|dD]	\}|V
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r1Nz&fmean requires at least one data pointz(data and weights must be the same lengthzsum of weights must be non-zero)lenrgr%rlistr@r')rFweightsrGrNnum_weightsnumdenr:s       @r2rrs>	II
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    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
    zGgeometric mean requires a non-empty dataset containing positive numbersN)r!rr@r$rmr)rFs r2rrs`G5S$(()))GGG<==BF	GGs	.1Ac,t||urt|}d}t|}|dkrtd|dkrQ|O|d}t	|t
jtfr|dkrt||Std|td|}|}nmt||urt|}t||krtdtdt||D\}}}	t||}td	t||D\}}}	n#t$rYdSwxYw|dkrtd
t||z|S)aReturn 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``.
    z.harmonic mean does not support negative valuesr4z.harmonic_mean requires at least one data pointNrzunsupported typez*Number of weights does not match data sizec3K|]}|VdSr6r0)r8ws  r2r;z harmonic_mean.<locals>.<genexpr>s" G Gq G G G G G Gr1c3.K|]\}}|r||zndVdS)rNr0)r8rrSs   r2r;z harmonic_mean.<locals>.<genexpr>s3PPTQ0q1uuqPPPPPPr1zWeighted sum must be positive)iterrrr
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||zdzS)aBReturn 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
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    >>> median([1, 3, 5])
    3
    >>> median([1, 3, 5, 7])
    4.0

    rno median for empty datar}r4sortedrr)rFr:is   r2rr%sr$<<DD		AAvv89991uzzAF|
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    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

    rrr}r4rrFr:s  r2rr=s`$<<DD		AAvv89991uzzAF|AFQJr1c~t|}t|}|dkrtd||dzS)aReturn 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

    rrr}rrs  r2r
r
Ss@$<<DD		AAvv8999Q<r1?ctt|}t|}|std||dz}t||}t	|||}	t|}t|}n#t$rtdwxYw||dzz
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    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.

    rr})loz$Value cannot be converted to a floatr+)rrrrrrfrmrg)	rFintervalr:rSrjLcffs	         r2rrfsV$<<DD		A:8999	
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B	AAx1q52:&***sA==Bctt|d}	|ddS#t$rt	ddwxYw)axReturn 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.

    r4rzno mode for empty dataN)r(rmost_common
IndexErrorr)rFpairss  r2rrsi.
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?Actt|}|sgSt|fd|DS)a.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('')
    []
    c&g|]
\}}|k|Sr0r0)r8rqrGmaxcounts   r2
<listcomp>zmultimode.<locals>.<listcomp>s'JJJleU8I8IE8I8I8Ir1)r(rmaxrMrC)rFcountsrs  @r2rrs\T$ZZ
 
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}g}td|D]M}t	||z|\}}||||z
z||dz|zz|z}	||	N|S|dkr||dz}g}td|D]b}||z|z}|dkrdn||dz
kr|dz
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d|)aDivide *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.
    r4zn must be at least 1r}z"must have at least two data points	inclusiverzUnknown method: )rrrrangedivmodappendrm)
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Fq!	(	(Aa!eQ''HAu Gq5y1DQK%4GG1LLMM,''''

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33r1ct||\}}}}|dkrtdt||dz
z|S)aReturn 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)

    r}z*variance requires at least two data pointsr4r\rrr)rFxbarrOssrTr:s      r2rr6sJLdD//KAr1a1uuJKKKB!a%L!$$$r1c|t||\}}}}|dkrtdt||z|S)a,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)

    r4z*pvariance requires at least one data pointr)rFmurOrrTr:s      r2rrbsFFdB--KAr1a1uuJKKKBFAr1ct||\}}}}|dkrtd||dz
z}t|trt	|j|jSt|j|jS)zReturn 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

    r}'stdev requires at least two data pointsr4r\rrerrrnror)rFrrOrrTr:msss       r2rrsydD//KAr1a1uuGHHH
A,C!WE$S]CODDDs}co>>>r1ct||\}}}}|dkrtd||z}t|trt	|j|jSt|j|jS)zReturn 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

    r4z'pstdev requires at least one data pointr)rFrrOrrTr:rs       r2rrsudB--KAr1a1uuGHHH
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    >>> y = [1, 2, 3, 1, 2, 3, 1, 2, 3]
    >>> covariance(x, y)
    0.75
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    Return the Pearson's correlation coefficient for two inputs. Pearson's
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    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]
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        y = slope * x + intercept + noise

    where *slope* and *intercept* are the regression parameters that are
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    The parameters are returned as a named tuple.

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    Since the *intercept* will always be 0.0, the underlying linear
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    >>> y = [3 * x[i] + noise[i] for i in range(5)]
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