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d� Z dd� Z dd� Z dd� Z!dd� Z"dd� Z#dd� Z$dOd d �Z%d d � Z&d d!� Z'd"d#� Z(dPd$d%�Z)d&d'� Z*d(d)� Z+d*d+� Z,dQd-d.�Z-d/d0� Z.d1d2� Z/d3d4d5�d6d7�Z0dPd8d9�Z1dPd:d;�Z2dPd<d=�Z3dPd>d?�Z4dPd@dA�Z5dBdC� Z6dDdE� Z7edFdG�Z8dHdI� Z9dJdK� Z:zddLl;m:Z: W n e<y� Y nw G dMdN� dN�Z=dS )Ra�
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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NormalDist�StatisticsError� correlation�
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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.
r � Nc s � � | ]
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rF c C s( z| � � W S ty t�| � Y S w r/ )Z is_finite�AttributeError�mathZisfinite)�xr+ r+ r, r< � s
�r< c C s� | t usJ d��| |u r| S |tu s|t u r| S | tu r |S t|| �r%|S t| |�r,| S t| t�r3|S t|t�r:| S t| t�rFt|t�rF|S t| t�rRt|t�rR| S d}t|| j|jf ��)z�Coerce types T and S to a common type, or raise TypeError.
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test class in test_statistics for details.
zinitial type T is boolz"don't know how to coerce %s and %s)�boolr7 �
issubclassr �float� TypeErrorr( )rB �S�msgr+ r+ r, r9 � s r9 c C s~ z| � � W S ty Y n ttfy" t| �r J �| df Y S w z| j| jfW S ty> dt| �j� d�}t |��w )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_ratiorG �
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ValueErrorr< � numerator� denominatorr8 r( rM )rI rO r+ r+ r, r; � s
� �r; c C sf t | �|u r| S t|t�r| jdkrt}z|| �W S ty2 t|t�r1|| j�|| j� Y S � w )z&Convert value to given numeric type T.r- )r8 rK r7 rT rL rM r rS )�valuerB r+ r+ r, �_convert s
�rV c C s* t | |�}|t| �kr| | |kr|S t�)z,Locate the leftmost value exactly equal to x)r �lenrR )�arI �ir+ r+ r, �
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rZ c C s: t | ||d�}|t| �d kr| |d |kr|d S t�)z-Locate the rightmost value exactly equal to x)�lor- )r rW rR )rX �lrI rY r+ r+ r, �
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t| �} t| �}|dk rtd��t| �\}}}||ks#J �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�listrW r rF rV )r? r3 rB rE r@ r+ r+ r, r 4 s r c sh zt | �� W n ty d� � fdd�}t|| ��}Y nw t| �}z|� W S ty3 td�d�w )z�Convert data to floats and compute the arithmetic mean.
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>>> fmean([3.5, 4.0, 5.25])
4.25
r c 3 s" � t | dd�D ]\� }|V qd S )Nr- )�start)� enumerate)�iterablerI �r3 r+ r, r@ \ s ��zfmean.<locals>.countz&fmean requires at least one data pointN)rW rM r$ �ZeroDivisionErrorr )r? r@ rE r+ rf r, r N s �
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t ttt| ���W S ty td�d�w )aY Convert 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 r r: r# rR r )r? r+ r+ r, r i s ��r c
C s2 t | �| u r
t| �} d}t| �}|dk rtd��|dkr:|du r:| d }t|tjtf�r6|dk r4t|��|S td��|du rFt d|�}|}n#t |�|u rPt|�}t|�|krZtd��t
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y� Y dS w |dkr�td ��t|| |�S ) 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 valuesr- z.harmonic_mean requires at least one data pointNr zunsupported typez*Number of weights does not match data sizec s s � | ]}|V qd S r/ r+ )r1 �wr+ r+ r, r4 � s � z harmonic_mean.<locals>.<genexpr>c s s$ � | ]
\}}|r || nd V qdS )r Nr+ )r1 rh rI r+ r+ r, r4 � � �" z Weighted sum must be positive)ra rb rW r �
isinstance�numbersZRealr rM r rF r` �ziprg rV )
r? Zweightsr_ r3 rI Z sum_weights�_rB rE r@ r+ r+ r, r | s<
" �r c C sX t | �} t| �}|dkrtd��|d dkr | |d S |d }| |d | | d S )aB Return the median (middle value) of numeric data.
When the number of data points is odd, return the middle data point.
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taking the average of the two middle values:
>>> median([1, 3, 5])
3
>>> median([1, 3, 5, 7])
4.0
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c C sH t | �} t| �}|dkrtd��|d dkr | |d S | |d d S )a Return the low median of numeric data.
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>>> median_low([1, 3, 5])
3
>>> median_low([1, 3, 5, 7])
3
r rn ro r- rp �r? r3 r+ r+ r, r
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c C s, t | �} t| �}|dkrtd��| |d S )a Return the high median of data.
When the number of data points is odd, the middle value is returned.
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>>> median_high([1, 3, 5])
3
>>> median_high([1, 3, 5, 7])
5
r rn ro rp rr r+ r+ r, r � s
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C s� t | �} t| �}|dkrtd��|dkr| d S | |d }||fD ]}t|ttf�r1td| ��q"z||d }W n tyM t|�t|�d }Y nw t| |�}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])
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.
r rn r- ro zexpected number but got %r)
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r c C s: t t| ���d�}z|d d W S ty td�d�w )ax Return 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- r zno mode for empty dataN)r& ra � most_common�
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�r c C s@ t t| ���� }tt|td�d�dg f�\}}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& ra rz �nextr r% rb r: )r? ZcountsZmaxcountZ
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r � � exclusive)r3 �methodc
C s< |dk rt d��t| �} t| �}|dk rt d��|dkrL|d }g }td|�D ]"}t|| |�\}}| | || | |d | | } |�| � q'|S |dkr�|d }g }td|�D ]9}|| | }|dk ridn ||d krs|d n|}|| || }| |d || | | | | } |�| � q[|S td|����)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.
r- zn must be at least 1ro z"must have at least two data pointsZ inclusiver zUnknown method: )r rq rW �range�divmod�appendrR )
r? r3 r� Zld�m�resultrY �jZdeltaZ interpolatedr+ r+ r, r � s2 $ $$ r c s� � durt � fdd�| D ��\}}}||fS t | �\}}}|| �� \}}t� }tt| �D ] \}} || | | }
| | } || | |
|
7 < q-d|v r\|d }t|�rXJ �||fS tdd� |�� D ��}||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.
Nc 3 � � | ] }|� d V qdS )ro Nr+ )r1 rI ��cr+ r, r4 � � � z_ss.<locals>.<genexpr>c s r. r/ r r0 r+ r+ r, r4 � r5 )rF rP r& r: r; r< r= r> ) r? r� rB rE r@ Zmean_nZmean_drA r3 r2 Zdiff_nZdiff_dr+ r� r, �_ss� s �r� c C sL t | �| u r
t| �} t| �}|dk rtd��t| |�\}}t||d |�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)
ro z*variance requires at least two data pointsr- �ra rb rW r r� rV )r? �xbarr3 rB �ssr+ r+ r, r � s &r c C sH t | �| u r
t| �} t| �}|dk rtd��t| |�\}}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� )r? �mur3 rB r� r+ r+ r, r s #r c C �2 t | |�}z|�� W S ty t�|� Y S w )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 r rG rH )r? r� �varr+ r+ r, r 0 �
�r c C r� )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
)r r rG rH )r? r� r� r+ r+ r, r C r� r c sn t | �}t |�|krtd��|dk rtd��t| �| � t|�| �t� �fdd�t| |�D ��}||d S )ap Covariance
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 pointsro z,covariance requires at least two data pointsc 3 �$ � | ]
\}}|� |� V qd S r/ r+ �r1 �xi�yi�r� �ybarr+ r, r4 u ri z covariance.<locals>.<genexpr>r- )rW r r$ rl )rI �yr3 �sxyr+ r� r, r ] s r c s� t | �}t |�|krtd��|dk rtd��t| �| � t|�| �t� �fdd�t| |�D ��}t� fdd�| D ��}t�fdd�|D ��}z |t|| � W S ty[ td��w ) a Pearson'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 the linear relationship, where +1 means very
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 pointsro z-correlation requires at least two data pointsc 3 r� r/ r+ r� r� r+ r, r4 � ri z correlation.<locals>.<genexpr>c 3 r� �� @Nr+ �r1 r� �r� r+ r, r4 � r� c 3 r� r� r+ )r1 r� )r� r+ r, r4 � r� z&at least one of the inputs is constant)rW r r$ rl r rg )rI r� r3 r� �sxxZsyyr+ r� r, r y s �r �LinearRegression��slope� interceptc s� t | �}t |�|krtd��|dk rtd��t| �| � t|�| �t� �fdd�t| |�D ��}t� fdd�| D ��}z|| }W n tyM td��w �|� }t||d�S ) 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 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...)
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&zNormalDist.pdfc C s2 | j std��ddt|| j | j td� � S )z,Cumulative distribution function. P(X <= x)z$cdf() not defined when sigma is zeror� r� r� )r� r r! r� r �r� rI r+ r+ r, �cdfA s $zNormalDist.cdfc C s: |dks|dkr t d��| jdkrt d��t|| j| j�S )aS Inverse 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.
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function.
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zNormalDist.inv_cdfr~ c s � �fdd�t d� �D �S )an Divide 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.
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� �t|� | �|� | � � 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
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� r� zNormalDist.medianc C r� )z�Return the mode of the normal distribution
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function (pdf) takes its maximum value.
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leaving sigma unchanged.
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independent or if they are jointly normally distributed.
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zNormalDist.__add__c C s8 t |t�rt| j|j t| j|j��S t| j| | j�S )as Subtract a constant or another NormalDist instance.
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leaving sigma unchanged.
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Mathematically, this works only if the two distributions are
independent or if they are jointly normally distributed.
r� r� r+ r+ r, �__sub__� r� zNormalDist.__sub__c C s t | j| | jt|� �S )z�Multiply both mu and sigma by a constant.
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r� r� r+ r+ r, � __truediv__� r� zNormalDist.__truediv__c C s t | j| j�S )z Return a copy of the instance.�r r� r� �r� r+ r+ r, �__pos__� s zNormalDist.__pos__c C s t | j | j�S )z(Negates mu while keeping sigma the same.r� r� r+ r+ r, �__neg__� � zNormalDist.__neg__c C s
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