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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.

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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.

    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.
    r�Ncss|]\}}t||�VqdS�Nr)�.0�d�nr&r&r'�	<genexpr>��z_sum.<locals>.<genexpr>)
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rCcCs�||ur|S|tus|tur |S|tur,|St||�r:|St||�rH|St|t�rV|St|t�rd|St|t�r|t|t�r||St|t�r�t|t�r�|Sd}t||j|jf��dS)z�Coerce types T and S to a common type, or raise TypeError.

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    z"don't know how to coerce %s and %sN)r2�bool�
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t|�j���dS)z�Return Real number x to exact (numerator, denominator) pair.

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rQcCs.t||�}|t|�kr&|||kr&|St�dS)z,Locate the leftmost value exactly equal to xN)r�lenrN)�arB�ir&r&r'�
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rUcCs>t|||d�}|t|�dkr6||d|kr6|dSt�dS)z-Locate the rightmost value exactly equal to x)�lor(N)rrRrN)rS�lrBrTr&r&r'�
_find_rteqs rX�negative valueccs$|D]}|dkrt|��|VqdS)z7Iterate over values, failing if any are less than zero.rN)r)r=�errmsgrBr&r&r'�	_fail_negsr[cCsHt|�|urt|�}t|�}|dkr,td��t|�\}}}t|||�S)a�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.
    r(z%mean requires at least one data point)�iter�listrRrr?rQ)r8r,r;r>r:r&r&r'r'srcspzt|��Wn.ty:d��fdd�}t||��}Yn
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|�WStyjtd�d�Yn0dS)z�Convert data to floats and compute the arithmetic mean.

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    >>> fmean([3.5, 4.0, 5.25])
    4.25
    rc3s t|dd�D]\�}|VqdS)Nr()r9)�	enumerate)�iterablerB�r,r&r'r:Oszfmean.<locals>.countz&fmean requires at least one data pointN)rRrGr �ZeroDivisionErrorr)r8r:r>r&r`r'rAs	
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    if it contains a zero, or if it contains a negative value.

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    (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)rrr4rrNr)r8r&r&r'r\s�rcCs�t|�|urt|�}d}t|�}|dkr2td��n<|dkrn|d}t|tjtf�rf|dkrbt|��|Std��z"t	dd�t
||�D��\}}}Wnty�YdS0t|||�S)aReturn 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,
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    z.harmonic mean does not support negative valuesr(z.harmonic_mean requires at least one data pointrzunsupported typecss|]}d|VqdS)r(Nr&�r*rBr&r&r'r-�r.z harmonic_mean.<locals>.<genexpr>)
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"rcCs\t|�}t|�}|dkr td��|ddkr8||dS|d}||d||dSdS)aBReturn the median (middle value) of numeric data.

    When the number of data points is odd, return the middle data point.
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    3
    >>> median([1, 3, 5, 7])
    4.0

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rcCsLt|�}t|�}|dkr td��|ddkr8||dS||ddSdS)a	Return the low median of numeric data.

    When the number of data points is odd, the middle value is returned.
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    3

    rrerfr(Nrg�r8r,r&r&r'r
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    >>> median_high([1, 3, 5])
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    5

    rrerfrgrhr&r&r'r	�s
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Cs�t|�}t|�}|dkr"td��n|dkr2|dS||d}||fD]}t|ttf�rFtd|��qFz||d}Wn&ty�t|�t|�d}Yn0t||�}t	|||�}|}||d}	|||d||	S)a�Return the 50th percentile (median) of grouped continuous data.

    >>> median_grouped([1, 2, 2, 3, 4, 4, 4, 4, 4, 5])
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    >>> 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

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rcCsBtt|���d�}z|ddWSty<td�d�Yn0dS)axReturn the most common data point from discrete or nominal data.

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        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:

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        'red'

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    ['b']
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    ['b', 'd', 'f']
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    []
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CsB|dkrtd��t|�}t|�}|dkr0td��|dkr�|d}g}td|�D]D}t|||�\}}||||||d||}	|�|	�qN|S|dk�r0|d}g}td|�D]r}|||}|dkr�dn||dkr�|dn|}||||}||d||||||}	|�|	�q�|Std|����dS)	a�Divide *data* into *n* continuous intervals with equal probability.

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    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.

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    Nc3s|]}|�dVqdS�rfNr&rb��cr&r'r-�r.z_ss.<locals>.<genexpr>c3s|]}|�dVqdSr}r&rbr~r&r'r-�r.c3s|]}|�VqdSr)r&rbr~r&r'r-�r.rf)r?rrR)r8rr;r>r:�UZtotal2Zcount2r&r~r'�_ss�sr�cCsLt|�|urt|�}t|�}|dkr,td��t||�\}}t||d|�S)a�Return the sample variance of data.

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    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)

    rfz*variance requires at least two data pointsr(�r\r]rRrr�rQ)r8�xbarr,r;�ssr&r&r'r�s&rcCsHt|�|urt|�}t|�}|dkr,td��t||�\}}t|||�S)a,Return the population variance of ``data``.

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    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)

    r(z*pvariance requires at least one data pointr�)r8�mur,r;r�r&r&r'r�s#rcCs6t||�}z
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            0.8035050657330205
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