AlkantarClanX12

Your IP : 18.222.56.71


Current Path : /opt/alt/python312/lib64/python3.12/__pycache__/
Upload File :
Current File : //opt/alt/python312/lib64/python3.12/__pycache__/statistics.cpython-312.opt-1.pyc

�

�Q�f3����UdZgd�ZddlZddlZddlZddlZddlmZddlm	Z	ddl
mZmZm
Z
ddlmZmZddlmZmZmZmZmZmZmZmZmZdd	lmZdd
lmZddlmZm Z m!Z!ed�Z"Gd
�de#�Z$d�Z%dCd�Z&d�Z'd�Z(d�Z)d�Z*dDd�Z+ddddd�de,e-fd�Z.de/de/de/fd�Z0dejbjdzd zZ3e/e4d!<de/de/de-fd"�Z5de/de/de	fd#�Z6d$�Z7dCd%�Z8d&�Z9dCd'�Z:d(�Z;d)�Z<d*�Z=dEd+�Z>d,�Z?d-�Z@d.d/d0�d1�ZAdCd2�ZBdCd3�ZCdCd4�ZDdCd5�ZEd6�ZFd7�ZGd8d9�d:�ZHe d;d<�ZIdd=�d>�ZJd?�ZK	dd@lLmKZKGdA�dB�ZNy#eM$rY�wxYw)Fa�

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.

)�
NormalDist�StatisticsError�correlation�
covariance�fmean�geometric_mean�
harmonic_mean�linear_regression�mean�median�median_grouped�median_high�
median_low�mode�	multimode�pstdev�	pvariance�	quantiles�stdev�variance�N��Fraction)�Decimal)�count�groupby�repeat)�bisect_left�bisect_right)	�hypot�sqrt�fabs�exp�erf�tau�log�fsum�sumprod)�reduce)�
itemgetter)�Counter�
namedtuple�defaultdict�@c��eZdZy)rN)�__name__�
__module__�__qualname__���1/opt/alt/python312/lib64/python3.12/statistics.pyrr�s��r3rc�n�d}t�}|j}i}|j}t|t�D]9\}}||�tt|�D]\}}	|dz
}||	d�|z||	<��;d|vr|d}
n td�|j�D��}
tt|t�}||
|fS)a�_sum(data) -> (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.

    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.
    r�Nc3�:K�|]\}}t||����y�w�Nr��.0�d�ns   r4�	<genexpr>z_sum.<locals>.<genexpr>�s����@�/?�t�q�!�H�Q��N�/?���)�set�add�getr�type�map�_exact_ratio�sum�itemsr(�_coerce�int)�datar�types�	types_add�partials�partials_get�typ�valuesr<r;�total�Ts            r4�_sumrR�s���@
�E��E�E��	�	�I��H��<�<�L��t�T�*���V��#����f�-�D�A�q��Q�J�E�&�q�!�,�q�0�H�Q�K�.�+�
�x�������@�x�~�~�/?�@�@���w��s�#�A�
�u�e��r3c�|�����t��fd�|D��\}}}||�|fSd}t�}|j}tt�}tt�}t|t�D]G\}	}
||	�tt|
�D]'\}�|dz
}|�xx|z
cc<|�xx||zz
cc<�)�I|std�x}�n_d|vr|dx}�nStd�|j�D��}td�|j�D��}
||
z||zz
|z}||z�tt|t�}||�|fS)a3Return the exact mean and sum of square deviations of sequence data.

    Calculations are done in a single pass, allowing the input to be an iterator.

    If given *c* is used the mean; otherwise, it is calculated from the data.
    Use the *c* argument with care, as it can lead to garbage results.

    Nc3�2�K�|]}|�z
x��z���y�wr8r2)r:�x�cr;s  ��r4r=z_ss.<locals>.<genexpr>�s�����<�t�!�1�q�5�j�a�A�-�t�s�rr6c3�:K�|]\}}t||����y�wr8rr9s   r4r=z_ss.<locals>.<genexpr>�s����@�,?�D�A�q��!�Q��,?�r>c3�@K�|]\}}t|||z����y�wr8rr9s   r4r=z_ss.<locals>.<genexpr>�s"����D�/C�t�q�!�(�1�a��c�"�/C�s�)rRr?r@r,rHrrBrCrDrrErFr(rG)rIrVrQ�ssdrrJrK�sx_partials�sxx_partialsrNrOr<�sx�sxxr;s `            @r4�_ssr^�sQ���	�}��<�t�<�<�
��3���3��5�!�!�
�E��E�E��	�	�I��c�"�K��s�#�L��t�T�*���V��#����f�-�D�A�q��Q�J�E���N�a��N���O�q�1�u�$�O�.�+���1�+���a�	
��	��d�#�#��a��@�K�,=�,=�,?�@�
@���D�|�/A�/A�/C�D�D���s�{�R�"�W�$��-����J���w��s�#�A�
�s�A�u��r3c�l�	|j�S#t$rtj|�cYSwxYwr8)�	is_finite�AttributeError�math�isfinite)rUs r4�	_isfiniterd�s1�� ��{�{�}���� ��}�}�Q��� �s��3�3c��||ur|S|tus|tur|S|tur|St||�r|St||�r|St|t�r|St|t�r|St|t�rt|t�r|St|t�rt|t�r|Sd}t||j|jfz��)z�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.
    z"don't know how to coerce %s and %s)rH�bool�
issubclassr�float�	TypeErrorr/)rQ�S�msgs   r4rGrGs���	�A�v�q���C�x�1��9�a�x��C�x��(��!�Q���(��!�Q���(��!�S��1�H��!�S��1�H��!�X��:�a��#7����!�U��
�1�h� 7���
.�C�
�C�1�:�:�q�z�z�2�2�
3�3r3c��	|j�S#t$rYnttf$r|dfcYSwxYw	|j|j
fS#t$r%dt
|�j�d�}t|��wxYw)z�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.
    Nzcan't convert type 'z' to numerator/denominator)	�as_integer_ratiora�
OverflowError�
ValueError�	numerator�denominatorrBr/ri)rUrks  r4rDrDs���<��!�!�#�#���
���:�&���4�y��������Q�]�]�+�+����$�T�!�W�%5�%5�$6�6P�Q����n���s��	2�2�2�A�.A<c��t|�|ur|St|t�r|jdk7rt}	||�S#t
$r9t|t�r'||j�||j�zcYS�wxYw)z&Convert value to given numeric type T.r6)rBrgrHrqrhrirrp)�valuerQs  r4�_convertrtMsz���E�{�a�����!�S��e�/�/�1�4������x������a��!��U�_�_�%��%�*;�*;�(<�<�<��	�s�>�>B�>Bc#�BK�|D]}|dkrt|��|���y�w)z7Iterate over values, failing if any are less than zero.rN)r)rO�errmsgrUs   r4�	_fail_negrw_s'����
���q�5�!�&�)�)����s�F�averager6)�key�reverse�ties�start�returnc�T�|dk7rtd|����|�t||�}tt|t	��|��}|dz
}dgt|�z}t
|td���D]:\}}	t|	�}
t|
�}||dzdzz}|
D]
\}
}|||<�||z
}�<|S)a
Rank order a dataset. The lowest value has rank 1.

    Ties are averaged so that equal values receive the same rank:

        >>> data = [31, 56, 31, 25, 75, 18]
        >>> _rank(data)
        [3.5, 5.0, 3.5, 2.0, 6.0, 1.0]

    The operation is idempotent:

        >>> _rank([3.5, 5.0, 3.5, 2.0, 6.0, 1.0])
        [3.5, 5.0, 3.5, 2.0, 6.0, 1.0]

    It is possible to rank the data in reverse order so that the
    highest value has rank 1.  Also, a key-function can extract
    the field to be ranked:

        >>> goals = [('eagles', 45), ('bears', 48), ('lions', 44)]
        >>> _rank(goals, key=itemgetter(1), reverse=True)
        [2.0, 1.0, 3.0]

    Ranks are conventionally numbered starting from one; however,
    setting *start* to zero allows the ranks to be used as array indices:

        >>> prize = ['Gold', 'Silver', 'Bronze', 'Certificate']
        >>> scores = [8.1, 7.3, 9.4, 8.3]
        >>> [prize[int(i)] for i in _rank(scores, start=0, reverse=True)]
        ['Bronze', 'Certificate', 'Gold', 'Silver']

    rxzUnknown tie resolution method: )rzr6r)ry�)	rorC�sorted�zipr�lenrr)�list)rIryrzr{r|�val_pos�i�result�_�g�group�size�rankrs�orig_poss               r4�_rankr�gs���J�y���:�4�(�C�D�D�
���3��~���S��u�w�'��9�G�
��	�A��S�3�w�<�
�F���Z��]�3���1��Q����5�z���D�1�H��>�!��$�O�E�8�#�F�8�� %�	�T�	��
4��Mr3r<�mc�N�tj||z�}|||z|z|k7zS)zFSquare root of n/m, rounded to the nearest integer using round-to-odd.)rb�isqrt)r<r��as   r4�_integer_sqrt_of_frac_rtor��s-��	
�
�
�1��6��A���!��A���
��r3r��_sqrt_bit_widthc���|j�|j�z
tz
dz}|dk\rt||d|zz�|z}d}||zSt|d|zz|�}d|z}||zS)z1Square root of n/m as a float, correctly rounded.rrr6���)�
bit_lengthr�r�)r<r��qrprqs     r4�_float_sqrt_of_fracr��s���
����!�,�,�.�	(�?�	:�q�@�A��A�v�-�a��a�!�e��<��A�	����{�"�"�.�a�2��6�k�1�=�	��A�2�g���{�"�"r3c��|dkr|std�S||}}t|�t|�zj�}|j�\}}|j�}|j�\}}d|z||zdzz|||z||zzdzzkDr|S|j	�}|j�\}	}
d|z||
zdzz|||	z|
|zzdzzkr|S|S)z3Square root of n/m as a Decimal, correctly rounded.rz0.0�r)rr rm�	next_plus�
next_minus)r<r��root�nr�dr�plus�np�dp�minus�nm�dms           r4�_decimal_sqrt_of_fracr��s���
	�A�v���5�>�!��r�A�2�1���A�J����#�)�)�+�D�
�
"�
"�
$�F�B���>�>��D�
�
"�
"�
$�F�B���1�u��2���z��A��B���B���� 2�2�2����O�O��E�
�
#�
#�
%�F�B���1�u��2���z��A��B���B���� 2�2�2����Kr3c�^�t|�\}}}|dkrtd��t||z|�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.
    r6z%mean requires at least one data point)rRrrt)rIrQrPr<s    r4r
r
�s7�� �t�*�K�A�u�a��1�u��E�F�F��E�A�I�q�!�!r3c�\��|�)	t|��t|�}�std��|�zSt	|t
tf�st|�}	t||�}t|�}|std��||zS#t$rd��fd�}||�}Y��wxYw#t$rtd��wxYw)z�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
    rc3�@�K�t|d��D]	\�}|���y�w)Nr6�r|)�	enumerate)�iterablerUr<s  �r4rzfmean.<locals>.count�s!�����%�h�a�8�D�A�q��G�9�s�z&fmean requires at least one data pointz(data and weights must be the same lengthzsum of weights must be non-zero)	r�rir&r�
isinstancer��tupler'ro)rI�weightsrrP�num�denr<s      @r4rr�s������		��D�	�A��T�
���!�"J�K�K��q�y���g��e�}�-��w�-��J��d�G�$���w�-�C���?�@�@���9���+�	��A�
���;�D�	�� �J��H�I�I�J�s�A8�B�8B�B�B+c�z�	tttt|���S#t$r
td�d�wxYw)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
    zGgeometric mean requires a non-empty dataset containing positive numbersN)r"rrCr%ror)rIs r4rrsE��G��5��S�$��(�)�)���G��<�=�BF�	G�G�s�!$�:c�x�t|�|urt|�}d}t|�}|dkrtd��|dk(rD|�B|d}t	|t
jtf�r|dkrt|��|Std��|�td|�}|}nQt|�|urt|�}t|�|k7rtd��td�t||�D��\}}}	t||�}td�t||�D��\}}}	|dkrtd	��t||z|�S#t$rYywxYw)
a�Return 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 valuesr6z.harmonic_mean requires at least one data pointrzunsupported typez*Number of weights does not match data sizec3� K�|]}|���y�wr8r2)r:�ws  r4r=z harmonic_mean.<locals>.<genexpr>Ns���� G�,F�q��,F�s�c3�4K�|]\}}|r||znd���y�w)rNr2)r:r�rUs   r4r=z harmonic_mean.<locals>.<genexpr>Qs"����P�=O�T�Q���q�1�u�q�0�=O�s�zWeighted sum must be positive)�iterr�r�rr��numbers�RealrrirrRrwr��ZeroDivisionErrorrt)
rIr�rvr<rU�sum_weightsr�rQrPrs
          r4rr!sM��.�D�z�T���D�z��
=�F��D�	�A��1�u��N�O�O�	
�a��G�O���G���a�'�,�,��0�1��1�u�%�f�-�-��H��.�/�/�����A�,������=�G�#��7�m�G��w�<�1��!�"N�O�O� � G�I�g�v�,F� G�G���;�����v�&���P�S��$�=O�P�P���5�%�
��z��=�>�>��K�%�'��+�+��	����s�",D-�-	D9�8D9c��t|�}t|�}|dk(rtd��|dzdk(r||dzS|dz}||dz
||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
    taking the average of the two middle values:

    >>> median([1, 3, 5])
    3
    >>> median([1, 3, 5, 7])
    4.0

    r�no median for empty datarr6�r�r�r)rIr<r�s   r4rrYsg���$�<�D��D�	�A��A�v��8�9�9��1�u��z��A��F�|��
��F���Q��U��d�1�g�%��*�*r3c��t|�}t|�}|dk(rtd��|dzdk(r||dzS||dzdz
S)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

    rr�rr6r��rIr<s  r4rrqsU���$�<�D��D�	�A��A�v��8�9�9��1�u��z��A��F�|���A��F�Q�J��r3c�^�t|�}t|�}|dk(rtd��||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

    rr�rr�r�s  r4r
r
�s7���$�<�D��D�	�A��A�v��8�9�9���Q��<�r3c�*�t|�}t|�}|std��||dz}t||�}t	|||��}	t|�}t|�}||dzz
}|}||z
}|||dz|z
z|zzS#t$rtd��wxYw)a�Estimates the median for numeric data binned around the midpoints
    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.

    r�r)�loz$Value cannot be converted to a floatr-)r�r�rrrrhrori)	rI�intervalr<rUr��j�L�cf�fs	         r4rr�s���V�$�<�D��D�	�A���8�9�9�	
�Q�!�V��A�	�D�!��A��T�1��#�A�A���?���!�H��	
�H�s�N��A�	
�B�	�A��A��x�1�q�5�2�:�&��*�*�*���A��>�@�@�A�s�A=�=Bc��tt|��jd�}	|ddS#t$r
t	d�d�wxYw)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.

    r6rzno mode for empty dataN)r*r��most_common�
IndexErrorr)rI�pairss  r4rr�sP��.
�D��J��+�+�A�.�E�B��Q�x��{����B��6�7�T�A�B�s	�-�Ac���tt|��}|sgSt|j��}|j	�D��cgc]
\}}||k(s�|��c}}Scc}}w)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*r��maxrOrF)rI�counts�maxcountrsrs     r4rrsS���T�$�Z�
 �F���	��6�=�=�?�#�H�&,�l�l�n�J�n�l�e�U���8I�E�n�J�J��Js�
A�Ar��	exclusive)r<�methodc�(�|dkrtd��t|�}t|�}|dkrtd��|dk(rW|dz
}g}td|�D]?}t	||z|�\}}||||z
z||dz|zz|z}	|j|	��A|S|dk(rn|dz}g}td|�D]V}||z|z}|dkrdn||dz
kDr|dz
n|}||z||zz
}||dz
||z
z|||zz|z}	|j|	��X|St
d|����)a�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.
    r6zn must be at least 1rz"must have at least two data points�	inclusiver��Unknown method: )rr�r��range�divmod�appendro)
rIr<r��ldr�r�r�r��delta�interpolateds
          r4rr9sa�� 	�1�u��4�5�5��$�<�D�	�T��B�	�A�v��B�C�C�
�����F�����q�!��A��a�!�e�Q�'�H�A�u� ��G�q�5�y�1�D��Q��K�%�4G�G�1�L�L��M�M�,�'���
�
�����F�����q�!��A��A���
�A���U���B�q�D���1��a�A��a�C�!�A�#�I�E� ��Q��K�1�u�9�5��Q��%��G�1�L�L��M�M�,�'���
�
�'��z�2�
3�3r3c�h�t||�\}}}}|dkrtd��t||dz
z|�S)a�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)

    rz*variance requires at least two data pointsr6�r^rrt)rI�xbarrQ�ssrVr<s      r4rrjs@��L�d�D�/�K�A�r�1�a��1�u��J�K�K��B�!�a�%�L�!�$�$r3c�b�t||�\}}}}|dkrtd��t||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)

    r6z*pvariance requires at least one data pointr�)rI�murQr�rVr<s      r4rr�s<��F�d�B�-�K�A�r�1�a��1�u��J�K�K��B��F�A��r3c���t||�\}}}}|dkrtd��||dz
z}t|t�r t	|j
|j�St|j
|j�S)z�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

    r�'stdev requires at least two data pointsr6�r^rrgrr�rprqr�)rIr�rQr�rVr<�msss       r4rr�sk���d�D�/�K�A�r�1�a��1�u��G�H�H�
��A��,�C��!�W��$�S�]�]�C�O�O�D�D��s�}�}�c�o�o�>�>r3c���t||�\}}}}|dkrtd��||z}t|t�r t	|j
|j�St|j
|j�S)z�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

    r6z'pstdev requires at least one data pointr�)rIr�rQr�rVr<r�s       r4rr�sg���d�B�-�K�A�r�1�a��1�u��G�H�H�
�q�&�C��!�W��$�S�]�]�C�O�O�D�D��s�}�}�c�o�o�>�>r3c��t|�\}}}}|dkrtd��||dz
z}	t|�t|j|j
�fS#t$r%t|�t|�t|�zfcYSwxYw)zFIn one pass, compute the mean and sample standard deviation as floats.rr�r6)r^rrhr�rprqra)rIrQr�r�r<r�s      r4�_mean_stdevr��s�����Y�N�A�r�4���1�u��G�H�H�
��A��,�C�4��T�{�/��
�
�s���O�O�O���4��T�{�E�$�K�%��)�3�3�3�4�s�*A�+B�Bc�����t|�}t|�|k7rtd��|dkrtd��t|�|z�t|�|z�t�fd�|D��fd�|D��}||dz
zS)apCovariance

    Return the sample covariance of two inputs *x* and *y*. Covariance
    is a measure of the joint variability of 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
    >>> 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 pointsrz,covariance requires at least two data pointsc3�(�K�|]	}|�z
���y�wr8r2)r:�xir�s  �r4r=zcovariance.<locals>.<genexpr>s�����)�q��2��9�q���c3�(�K�|]	}|�z
���y�wr8r2�r:�yi�ybars  �r4r=zcovariance.<locals>.<genexpr>s�����+B��"�B��I��r�r6)r�rr&r')rU�yr<�sxyr�r�s    @@r4rr�sx���"	�A��A�
�1�v��{��d�e�e��1�u��L�M�M���7�Q�;�D���7�Q�;�D�
�)�q�)�+B��+B�
C�C��!�a�%�=�r3�linear)r�c��t|�}t|�|k7rtd��|dkrtd��|dvrtd|����|dk(r#|dz
dz}t||�	�}t||�	�}n@t	|�|z}t	|�|z}|D�cgc]}||z
��	}}|D�cgc]}||z
��	}}t||�}	t||�}
t||�}	|	t
|
|z�zScc}wcc}w#t$rtd
��wxYw)alPearson's correlation coefficient

    Return the Pearson's correlation coefficient for two inputs. Pearson's
    correlation coefficient *r* takes values between -1 and +1. It measures
    the strength and direction of a 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

    If *method* is "ranked", computes Spearman's rank correlation coefficient
    for two inputs.  The data is replaced by ranks.  Ties are averaged
    so that equal values receive the same rank.  The resulting coefficient
    measures the strength of a monotonic relationship.

    Spearman's rank correlation coefficient is appropriate for ordinal
    data or for continuous data that doesn't meet the linear proportion
    requirement for Pearson's correlation coefficient.
    zEcorrelation requires that both inputs have same number of data pointsrz-correlation requires at least two data points>r��rankedr�r�r6r�r�z&at least one of the inputs is constant)r�rror�r&r'r r�)rUr�r�r<r|r�r�r�r�r�r]�syys            r4rrs+��.	�A��A�
�1�v��{��e�f�f��1�u��M�N�N�
�)�)��+�F�:�6�7�7�
����Q��"����!�5�!���!�5�!���A�w��{���A�w��{��!"�#��2�R�$�Y���#�!"�#��2�R�$�Y���#�
�!�Q�-�C�
�!�Q�-�C�
�!�Q�-�C�H��T�#��)�_�$�$��

$��#���H��F�G�G�H�s�C%�!C*�C/�/D�LinearRegression��slope�	intercept)�proportionalc��
�t|�}t|�|k7rtd��|dkrtd��|s9t|�|z}t|�|z�
|D�cgc]}||z
��	}}�
fd�|D�}t||�dz}t||�}	||z}|rdn�
|zz
}	t||	��Scc}w#t$rtd��wxYw)a�Slope and intercept for simple linear regression.

    Return the slope and intercept of simple linear regression
    parameters estimated using ordinary least squares. Simple linear
    regression describes relationship between an independent variable
    *x* and a dependent variable *y* in terms of a linear function:

        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...)

    If *proportional* is true, the independent variable *x* and the
    dependent variable *y* are assumed to be directly proportional.
    The data is fit to a line passing through the origin.

    Since the *intercept* will always be 0.0, the underlying linear
    function simplifies to:

        y = slope * x + noise

    >>> y = [3 * x[i] + noise[i] for i in range(5)]
    >>> linear_regression(x, y, proportional=True)  #doctest: +ELLIPSIS
    LinearRegression(slope=3.02447542484..., intercept=0.0)

    zKlinear regression requires that both inputs have same number of data pointsrz3linear regression requires at least two data pointsc3�(�K�|]	}|�z
���y�wr8r2r�s  �r4r=z$linear_regression.<locals>.<genexpr>us�����#��2�R�$�Y��r��z
x is constantr�)r�rr&r'r�r�)rUr�r�r<r�r�r�r]r�r�r�s          @r4r	r	Fs����L	�A��A�
�1�v��{��k�l�l��1�u��S�T�T���A�w��{���A�w��{��!"�#��2�R�$�Y���#�#��#��
�!�Q�-�#�
�C�
�!�Q�-�C�/��c�	��$������)<�I��%�9�=�=��
$���/��o�.�.�/�s�B+�
B0�0Cc���|dz
}t|�dkrpd||zz
}d|zdz|zdz|zdz|zdz|zd	z|zd
z|zdz|z}d|zd
z|zdz|zdz|zdz|zdz|zdz|zdz}||z}|||zzS|dkr|nd|z
}tt|��}|dkr^|dz
}d|zdz|zdz|zdz|zdz|zdz|zdz|zdz}d|zd z|zd!z|zd"z|zd#z|zd$z|zd%z|zdz}n]|dz
}d&|zd'z|zd(z|zd)z|zd*z|zd+z|zd,z|zd-z}d.|zd/z|zd0z|zd1z|zd2z|zd3z|zd4z|zdz}||z}|dkr|}|||zzS)5N��?g333333�?g��Q��?g^�}o)��@g�E.k�R�@g ��Ul�@g*u��>l�@g�N����@g�"]Ξ@gnC���`@gu��@giK��~j�@gv��|E�@g��d�|1�@gfR��r��@g��u.2�@g���~y�@g�n8(E@��?r�g@g�������?g鬷�ZaI?gg�El�D�?g7\�����?g�uS�S�?g�=�.
@gj%b�@g���Hw�@gjR�e�?g�9dh?
>g('߿��A?g��~z �?g@�3��?gɅ3��?g3fR�x�?gI�F��l@g����t��>g*�Y��n�>gESB\T?g�N;A+�?g�UR1��?gE�F���?gP�n��@g&�>���@g����i�<g�@�F�>g�tcI,\�>g�ŝ���I?g*F2�v�?g�C4�?g��O�1�?)r!r r%)�pr��sigmar��rr�r�rUs        r4�_normal_dist_inv_cdfr�s���	
�C��A��A�w�%���q�1�u���0�1�4�0�1�45�6�0�1�45�6�1�1�56�6�1�	1�56�	6�
1�1�
56�6�1�
1�56�
6�1�1�56�6��1�1�4�0�1�45�6�0�1�45�6�1�1�56�6�1�	1�56�	6�
1�1�
56�6�1�
1�56�
6����
�#�I���Q��Y���
�#�X��3��7�A��c�!�f�W�
�A��C�x�
��G��1�A�5�1�2�56�7�1�2�56�7�2�2�67�7�2�	2�67�	7�
2�2�
67�7�2�
2�67�
7�2�2��2�A�5�1�2�56�7�1�2�56�7�2�2�67�7�2�	2�67�	7�
2�2�
67�7�2�
2�67�
7����
��G��1�A�5�1�2�56�7�1�2�56�7�2�2�67�7�2�	2�67�	7�
2�2�
67�7�2�
2�67�
7�2�2��3�Q�6�1�2�56�7�1�2�56�7�2�2�67�7�2�	2�67�	7�
2�2�
67�7�2�
2�67�
7����	�c�	�A��3�w�
�B��
��U���r3)rc��eZdZdZddd�Zd!d�Zed��Zdd�d	�Zd
�Z	d�Z
d�Zd"d
�Zd�Z
d�Zed��Zed��Zed��Zed��Zed��Zd�Zd�Zd�Zd�Zd�Zd�ZeZd�ZeZd�Zd�Zd�Z d�Z!d �Z"y)#rz(Normal distribution of a random variablez(Arithmetic mean of a normal distributionz+Standard deviation of a normal distribution��_mu�_sigmac�d�|dkrtd��t|�|_t|�|_y)zDNormalDist where mu is the mean and sigma is the standard deviation.r�zsigma must be non-negativeN)rrhr	r
)�selfr�rs   r4�__init__zNormalDist.__init__�s+���3�;�!�">�?�?���9����E�l��r3c��|t|��S)z5Make a normal distribution instance from sample data.)r�)�clsrIs  r4�from_sampleszNormalDist.from_samples�s���K��%�&�&r3N)�seedc���|�tjntj|�j}|j|j}}td|�D�cgc]}|||���
c}Scc}w)z=Generate *n* samples for a given mean and standard deviation.N)�random�gauss�Randomr	r
r)rr<rrr�rr�s       r4�sampleszNormalDist.samples�sV�� $�����&�-�-��2E�2K�2K���H�H�d�k�k�E��*0��q�/�:�/�Q��b�%� �/�:�:��:s�A+c��|j|jz}|std��||jz
}t||zd|zz�t	t
|z�zS)z4Probability density function.  P(x <= X < x+dx) / dxz$pdf() not defined when sigma is zerog�)r
rr	r"r r$)rrUr�diffs    r4�pdfzNormalDist.pdf�sV���;�;����,���!�"H�I�I��4�8�8�|���4�$�;�$��/�2�3�d�3��>�6J�J�Jr3c��|jstd��ddt||jz
|jtzz�zzS)z,Cumulative distribution function.  P(X <= x)z$cdf() not defined when sigma is zerorr)r
rr#r	�_SQRT2�rrUs  r4�cdfzNormalDist.cdf�s@���{�{�!�"H�I�I��c�C��T�X�X��$�+�+��2F� G�H�H�I�Ir3c�n�|dks|dk\rtd��t||j|j�S)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.
        r�rz$p must be in the range 0.0 < p < 1.0)rrr	r
)rrs  r4�inv_cdfzNormalDist.inv_cdf�s4��
��8�q�C�x�!�"H�I�I�#�A�t�x�x����=�=r3c�d�td|�D�cgc]}|j||z���c}Scc}w)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.
        r6)r�r)rr<r�s   r4rzNormalDist.quantiless/��.3�1�a�[�9�[�����Q��U�#�[�9�9��9s�-c	��t|t�std��||}}|j|jf|j|jfkr||}}|j
|j
}}|r|st
d��||z
}t|j|jz
�}|s%dt|d|jztzz�z
S|j|z|j|zz
}|j|jzt||z|t||z�zz�z}	||	z|z}
||	z
|z}dt|j|
�|j|
�z
�t|j|�|j|�z
�zz
S)a�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
        z$Expected another NormalDist instancez(overlap() not defined when sigma is zerorr-)
r�rrir
r	rrr!r#rr r%r)r�other�X�Y�X_var�Y_var�dvr�r��b�x1�x2s            r4�overlapzNormalDist.overlaps_�� �%��,��B�C�C��U�1��
�H�H�a�e�e�����!�%�%�0�0��a�q�A��z�z�1�:�:�u���E�!�"L�M�M�
�U�]��
�!�%�%�!�%�%�-�
 �����R�3����>�F�#:�;�<�<�<�
�E�E�E�M�A�E�E�E�M�)��
�H�H�q�x�x��$�r�B�w��c�%�%�-�6H�1H�'H�"I�I���!�e�r�\���!�e�r�\���d�1�5�5��9�q�u�u�R�y�0�1�D����r��Q�U�U�2�Y�9N�4O�O�P�Pr3c�h�|jstd��||jz
|jzS)z�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.
        z'zscore() not defined when sigma is zero)r
rr	rs  r4�zscorezNormalDist.zscore9s.���{�{�!�"K�L�L��D�H�H�����+�+r3c��|jS)z+Arithmetic mean of the normal distribution.�r	�rs r4r
zNormalDist.meanD�
���x�x�r3c��|jS)z,Return the median of the normal distributionr/r0s r4rzNormalDist.medianIr1r3c��|jS)z�Return the mode of the normal distribution

        The mode is the value x where which the probability density
        function (pdf) takes its maximum value.
        r/r0s r4rzNormalDist.modeNs
���x�x�r3c��|jS)z.Standard deviation of the normal distribution.�r
r0s r4rzNormalDist.stdevWs���{�{�r3c�4�|j|jzS)z!Square of the standard deviation.r5r0s r4rzNormalDist.variance\s���{�{�T�[�[�(�(r3c���t|t�rAt|j|jzt|j|j��St|j|z|j�S)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.
        �r�rr	rr
�r)r*s  r4�__add__zNormalDist.__add__a�O���b�*�%��b�f�f�r�v�v�o�u�R�Y�Y��	�	�/J�K�K��"�&�&�2�+�r�y�y�1�1r3c���t|t�rAt|j|jz
t|j|j��St|j|z
|j�S)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.
        r8r9s  r4�__sub__zNormalDist.__sub__or;r3c�`�t|j|z|jt|�z�S)z�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.
        �rr	r
r!r9s  r4�__mul__zNormalDist.__mul__}�&���"�&�&�2�+�r�y�y�4��8�';�<�<r3c�`�t|j|z|jt|�z�S)z�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.
        r?r9s  r4�__truediv__zNormalDist.__truediv__�rAr3c�B�t|j|j�S)zReturn a copy of the instance.�rr	r
�r)s r4�__pos__zNormalDist.__pos__�s���"�&�&�"�)�)�,�,r3c�D�t|j|j�S)z(Negates mu while keeping sigma the same.rErFs r4�__neg__zNormalDist.__neg__�s���2�6�6�'�2�9�9�-�-r3c��||z
S)z<Subtract a NormalDist from a constant or another NormalDist.r2r9s  r4�__rsub__zNormalDist.__rsub__�s���b��z�r3c��t|t�stS|j|jk(xr|j|jk(S)zFTwo NormalDist objects are equal if their mu and sigma are both equal.)r�r�NotImplementedr	r
r9s  r4�__eq__zNormalDist.__eq__�s7���"�j�)�!�!��v�v�����:�B�I�I����$:�:r3c�D�t|j|jf�S)zCNormalDist objects hash equal if their mu and sigma are both equal.)�hashr	r
r0s r4�__hash__zNormalDist.__hash__�s���T�X�X�t�{�{�+�,�,r3c�f�t|�j�d|j�d|j�d�S)Nz(mu=z, sigma=�))rBr/r	r
r0s r4�__repr__zNormalDist.__repr__�s.���t�*�%�%�&�d�4�8�8�,�h�t�{�{�o�Q�O�Or3c�2�|j|jfSr8rr0s r4�__getstate__zNormalDist.__getstate__�s���x�x����$�$r3c�"�|\|_|_yr8r)r�states  r4�__setstate__zNormalDist.__setstate__�s�� %����$�+r3)r�r)r�)#r/r0r1�__doc__�	__slots__r
�classmethodrrrrrrr+r-�propertyr
rrrrr:r=r@rCrGrI�__radd__rK�__rmul__rNrQrTrVrYr2r3r4rr�s��.�
:�?��I�
#��'��'�"&�;�K�J�>�	:� Q�D	,������������������)��)�2�2�=�=�-�.��H���H�;�-�P�%�&r3rr8)znegative value)r)OrZ�__all__rbr�r�sys�	fractionsr�decimalr�	itertoolsrrr�bisectrrrr r!r"r#r$r%r&r'�	functoolsr(�operatorr)�collectionsr*r+r,rrorrRr^rdrGrDrtrwr�rhr�rHr��
float_info�mant_digr��__annotations__r�r�r
rrrrrr
rrrrrrrrr�rrr�r	r�_statistics�ImportErrorrr2r3r4�<module>rns���h�T��.��
�
���,�,�,�E�E�E���8�8�	
�c���	�j�	�3�l&�R �4�>+�\�$���I�Q�3�4�PU�;�3�l��������3�>�>�2�2�2�Q�6���6�
#�3�
#�3�
#�5�
#��S��S��W��<"�,!�HG�&5,�p+�0 �,�&E+�PB�<K�r�;�(4�b)%�X&�R?�$?�$
4�(�8$,�-H�`�0�2H�I��05�7>�zG�V	�0�
Z&�Z&��	�	��	�s�1E�E
�	E