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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.


NormalDistStatisticsErrorfmeangeometric_mean
harmonic_meanmeanmedianmedian_groupedmedian_high
median_lowmode	multimodepstdev	pvariance	quantilesstdevvarianceNFraction)Decimal)groupby)bisect_leftbisect_right)hypotsqrtfabsexperftaulogfsum)
itemgetter)Counterc@seZdZdS)rN)__name__
__module____qualname__r&r&&/usr/local/lib/python3.8/statistics.pyruscCsd}t|\}}||i}|j}ttt|}t|tD]@\}}	t||}tt|	D]"\}}|d7}||d|||<qRq6d|kr|d}
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||
|fS)aC_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.
    rNcss|]\}}t||VqdSNr).0dnr&r&r'	<genexpr>sz_sum.<locals>.<genexpr>)_exact_ratioget_coerceinttypermap	_isfiniteAssertionErrorsumsorteditems)datastartcountr,r+ZpartialsZpartials_getTtypvaluestotalr&r&r'_sum{s$
r@cCs.z
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r(t|YSXdSr))Z	is_finiteAttributeErrormathisfinite)xr&r&r'r4s
r4cCs|tk	std||kr|S|tks,|tkr0|S|tkr<|St||rJ|St||rX|St|trf|St|trt|St|trt|tr|St|trt|tr|Sd}t||j|jfdS)zCoerce 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.
    zinitial type T is boolz"don't know how to coerce %s and %sN)boolr5r1
issubclassrfloat	TypeErrorr#)r<Smsgr&r&r'r0s*



r0cCszrt|tkst|tkr$|WSz|j|jfWWStk
rnz|WYWStk
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dS)zReturn Real number x to exact (numerator, denominator) pair.

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    (1, 4)

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    Nz0can't convert type '{}' to numerator/denominator)r2rGras_integer_ratio	numeratordenominatorrA
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r.cCspt||kr|St|tr(|jdkr(t}z
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rjt|trd||j||jYSYnXdS)z&Convert value to given numeric type T.r(N)r2rFr1rMrGrHrrL)valuer<r&r&r'_converts

rRcCs.t||}|t|kr&|||kr&|StdS)z,Locate the leftmost value exactly equal to xN)rlenrO)arDir&r&r'
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rVcCs>t|||d}|t|dkr6||d|kr6|dStdS)z-Locate the rightmost value exactly equal to x)lor(N)rrSrO)rTlrDrUr&r&r'
_find_rteqs rYnegative valueccs$|D]}|dkrt||VqdS)z7Iterate over values, failing if any are less than zero.rN)r)r>errmsgrDr&r&r'	_fail_negsr\cCsTt||krt|}t|}|dkr,tdt|\}}}||ksFtt|||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.
    r(z%mean requires at least one data point)iterlistrSrr@r5rR)r9r,r<r?r;r&r&r'r'scstzt|Wn0tk
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rntddYnXdS)zConvert 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
    rc3s t|ddD]\}|VqdS)Nr()r:)	enumerate)iterablerDr,r&r'r;Oszfmean.<locals>.countz&fmean requires at least one data pointN)rSrHr ZeroDivisionErrorr)r9r;r?r&rar'rAs	
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r2tddYnXdS)aYConvert 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
    zHgeometric mean requires a non-empty dataset  containing positive numbersN)rrr3rrOr)r9r&r&r'r\scCst||krt|}d}t|}|dkr2tdn<|dkrn|d}t|tjtfrf|dkrbt||Stdz"t	ddt
||D\}}}Wntk
rYdSX||kstt
|||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,
    ``harmonic_mean`` will raise ``StatisticsError``.
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"cCs\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.
    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

    rno median for empty datar(Nr7rSr)r9r,rUr&r&r'rs
cCsLt|}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.
    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

    rrfrgr(Nrhr9r,r&r&r'r
scCs,t|}t|}|dkr td||dS)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

    rrfrgrhrir&r&r'r	s
r(c
Cst|}t|}|dkr"tdn|dkr2|dS||d}||fD]}t|ttfrFtd|qFz||d}Wn(tk
rt|t|d}YnXt||}t	|||}|}||d}	|||d||	S)aReturn 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.
    rrfr(rgzexpected number but got %r)
r7rSrrdstrbytesrHrGrVrY)
r9intervalr,rDobjLl1l2Zcffr&r&r'rs&

cCsHt|}t|d}z|ddWStk
rBtddYnXdS)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.

    r(rzno mode for empty dataN)r]r"most_common
IndexErrorr)r9Zpairsr&r&r'rscCs@tt|}tt|tdddgf\}}tttd|S)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('')
    []
    r()keyr)r"r]rrnextrr!r^r3)r9ZcountsZmaxcountZ
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	exclusive)r,methodc
CsL|dkrtdt|}t|}|dkr0td|dkr|d}g}td|D]N}|||}||||}||||||d||}	||	qN|S|dkr:|d}g}td|D]r}|||}|dkrdn||dkr|dn|}||||}||d||||||}	||	q|Std|dS)	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.
    r(zn must be at least 1rgz"must have at least two data pointsZ	inclusiverwzUnknown method: N)rr7rSrangeappendrO)
r9r,rxZldmresultrUjZdeltaZinterpolatedr&r&r'rls4$
$$csdk	r,tfdd|D\}}}||fSt|tfdd|D\}}}tfdd|D\}}}||kr|||kst||dt|8}|dkrtd|||fS)	a;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.
    Nc3s|]}|dVqdSrgNr&rccr&r'r-sz_ss.<locals>.<genexpr>c3s|]}|dVqdSr~r&rcrr&r'r-sc3s|]}|VqdSr)r&rcrr&r'r-srgrz%negative sum of square deviations: %f)r@rr5rS)r9rr<r?r;UZtotal2Zcount2r&rr'_sssrcCsLt||krt|}t|}|dkr,tdt||\}}t||d|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)

    rgz*variance requires at least two data pointsr(r]r^rSrrrR)r9xbarr,r<ssr&r&r'rs&cCsHt||krt|}t|}|dkr,tdt||\}}t|||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)

    r(z*pvariance requires at least one data pointr)r9mur,r<rr&r&r'rs#cCs8t||}z
|WStk
r2t|YSXdS)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

    N)rrrArB)r9rvarr&r&r'rs
	

cCs8t||}z
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r2t|YSXdS)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

    N)rrrArB)r9rrr&r&r'r
&s
	

cCs|d}t|dkrd||}d|d|d|d|d|d	|d
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>g('߿A?g~z ?g@3?gɅ3?g3fRx?gIFl@gt>g*Yn>gESB\T?gN;A+?gUR1?gEF?gPn@g&>@gi<g@F>gtcI,\>gŝI?g*F2v?gC4?gO1?)rrr)prsigmaqrnumZdenrDr&r&r'_normal_dist_inv_cdf9sd
	
rc@seZdZdZdddZd8ddZed	d
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eddZeddZeddZed d!Zed"d#Zd$d%Zd&d'Zd(d)Zd*d+Zd,d-Zd.d/ZeZd0d1ZeZd2d3Zd4d5Zd6d7ZdS):rz(Normal distribution of a random variablez(Arithmetic mean of a normal distributionz+Standard deviation of a normal distribution)_mu_sigmarrcCs(|dkrtdt||_t||_dS)zDNormalDist where mu is the mean and sigma is the standard deviation.rzsigma must be non-negativeN)rrGrr)selfrrr&r&r'__init__s
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<listcomp>sz&NormalDist.samples.<locals>.<listcomp>)randomrRandomrrry)rr,rr&rr'samplesszNormalDist.samplescCs<|jd}|stdt||jdd|tt|S)z4Probability density function.  P(x <= X < x+dx) / dx@z$pdf() not defined when sigma is zerog)rrrrrr)rrDrr&r&r'pdfs
zNormalDist.pdfcCs2|jstdddt||j|jtdS)z,Cumulative distribution function.  P(X <= x)z$cdf() not defined when sigma is zerorrr)rrrrr)rrDr&r&r'cdfszNormalDist.cdfcCs:|dks|dkrtd|jdkr*tdt||j|jS)aSInverse 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.
        rrz$p must be in the range 0.0 < p < 1.0z-cdf() not defined when sigma at or below zero)rrrr)rrr&r&r'inv_cdfs


zNormalDist.inv_cdfrvcsfddtdDS)anDivide 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.
        csg|]}|qSr&)rrr,rr&r'rsz(NormalDist.quantiles.<locals>.<listcomp>r()ry)rr,r&rr'rs	zNormalDist.quantilescCst|tstd||}}|j|jf|j|jfkr>||}}|j|j}}|rT|s\td||}t|j|j}|sdt|d|jt	dS|j||j|}|j|jt	|d|t
||}	||	|}
||	|}dt||
||
t||||S)aCompute 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
        z$Expected another NormalDist instancez(overlap() not defined when sigma is zerorr)rdrrHrrrrrrrrr)rotherXYZX_varZY_varZdvZdmrTbx1x2r&r&r'overlaps"


(zNormalDist.overlapcCs|jS)z+Arithmetic mean of the normal distribution.rrr&r&r'rszNormalDist.meancCs|jS)z,Return the median of the normal distributionrrr&r&r'rszNormalDist.mediancCs|jS)zReturn the mode of the normal distribution

        The mode is the value x where which the probability density
        function (pdf) takes its maximum value.
        rrr&r&r'rszNormalDist.modecCs|jS)z.Standard deviation of the normal distribution.rrr&r&r'rszNormalDist.stdevcCs
|jdS)z!Square of the standard deviation.rrrr&r&r'rszNormalDist.variancecCs8t|tr&t|j|jt|j|jSt|j||jS)ajAdd 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.
        rdrrrrrrr&r&r'__add__	s

zNormalDist.__add__cCs8t|tr&t|j|jt|j|jSt|j||jS)asSubtract 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.
        rrr&r&r'__sub__s

zNormalDist.__sub__cCst|j||jt|S)zMultiply 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.
        rrrrrr&r&r'__mul__%szNormalDist.__mul__cCst|j||jt|S)zDivide 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.
        rrr&r&r'__truediv__-szNormalDist.__truediv__cCst|j|jS)zReturn a copy of the instance.rrrrr&r&r'__pos__5szNormalDist.__pos__cCst|j|jS)z(Negates mu while keeping sigma the same.rrr&r&r'__neg__9szNormalDist.__neg__cCs
||S)z<Subtract a NormalDist from a constant or another NormalDist.r&rr&r&r'__rsub__?szNormalDist.__rsub__cCs&t|tstS|j|jko$|j|jkS)zFTwo NormalDist objects are equal if their mu and sigma are both equal.)rdrNotImplementedrrrr&r&r'__eq__Es
zNormalDist.__eq__cCst|j|jfS)zCNormalDist objects hash equal if their mu and sigma are both equal.)hashrrrr&r&r'__hash__KszNormalDist.__hash__cCs t|jd|jd|jdS)Nz(mu=z, sigma=))r2r#rrrr&r&r'__repr__OszNormalDist.__repr__)rr)rv) r#r$r%__doc__	__slots__rclassmethodrrrrrrrpropertyrrrrrrrrrrr__radd__r__rmul__rrrr&r&r&r'rsF


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Zerion Mini Shell 1.0