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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.

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    (<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")]
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    (<class 'decimal.Decimal'>, Fraction(6963, 10000), 4)

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t|�}t|�}|dkrtd��t|�\}}}t|||�S)a�Return the sample arithmetic mean of data.

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    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�listrWrrErV)r>r3rArDr?r+r+r,r	4sr	cshzt|��Wntyd��fdd�}t||��}Ynwt|�}z|�WSty3td�d�w)z�Convert data to floats and compute the arithmetic mean.

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    zGgeometric mean requires a non-empty dataset containing positive numbersN)r rr:r#rRr)r>r+r+r,ris��rc
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    rates, for example speeds.

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    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,
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    3

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    5

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    52.5

    This calculates the median as the 50th percentile, and should be
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    Optional argument ``interval`` represents the class interval, and
    defaults to 1. Changing the class interval naturally will change the
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    >>> median_grouped([1, 3, 3, 5, 7], interval=1)
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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
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    separate *data* in to 100 equal sized groups.

    The *data* can be any iterable containing sample.
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7<q-d|vrV|d}||fStdd�|��D��}||fS)a;Return sum of square deviations of sequence data.

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    Nc3��|]	}|�dVqdS)roNr+)r1rH��cr+r,r4���z_ss.<locals>.<genexpr>csr.r/rr0r+r+r,r4�r5)rErPr&r:r;r<r=)r>r�rArDr?Zmean_nZmean_dr@r3r2Zdiff_nZdiff_dr+r�r,�_ss�s�r�cCsLt|�|ur
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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
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    Fraction(67, 108)

    roz*variance requires at least two data pointsr-�rarbrWrr�rV)r>�xbarr3rA�ssr+r+r,r�s&rcCsHt|�|ur
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    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
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    Fraction(13, 72)

    r-z*pvariance requires at least one data pointr�)r>�mur3rAr�r+r+r,rs#rcC�2t||�}z|��WStyt�|�YSw)z�Return the square root of the sample variance.

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�rcCr�)z�Return the square root of the population variance.

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    0.986893273527251

    )rrrFrG)r>r�r�r+r+r,rCr�rcsnt|�}t|�|krtd��|dkrtd��t|�|�t|�|�t��fdd�t||�D��}||dS)apCovariance

    Return the sample covariance of two inputs *x* and *y*. Covariance
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    >>> 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

    zDcovariance requires that both inputs have same number of data pointsroz,covariance requires at least two data pointsc3�$�|]
\}}|�|�VqdSr/r+�r1�xi�yi�r��ybarr+r,r4urizcovariance.<locals>.<genexpr>r-)rWrr$rl)rH�yr3�sxyr+r�r,r]srcs�t|�}t|�|krtd��|dkrtd��t|�|�t|�|�t��fdd�t||�D��}t�fdd�|D��}t�fdd�|D��}z	|t||�WSty[td��w)	aPearson's correlation coefficient

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

    zEcorrelation requires that both inputs have same number of data pointsroz-correlation requires at least two data pointsc3r�r/r+r�r�r+r,r4�rizcorrelation.<locals>.<genexpr>c3r���@Nr+�r1r��r�r+r,r4�r�c3r�r�r+)r1r�)r�r+r,r4�r�z&at least one of the inputs is constant)rWrr$rlrrg)rHr�r3r��sxxZsyyr+r�r,rys�r�LinearRegression��slope�	interceptcs�t|�}t|�|krtd��|dkrtd��t|�|�t|�|�t��fdd�t||�D��}t�fdd�|D��}z||}WntyMtd��w�|�}t||d�S)	a�Slope and intercept for simple linear regression.

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        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...)

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