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r Y nX e8dFkrddGlm9Z9 ddHlm:Z:m;Z;mZ> ddl?Z?e5dJdKZ@e5dLdMZAe@dN dN j$e@j$ksLtBe@dN dN j2e@j2ksftBdOZCe@DeCZEeADeCZFe:e;fD ]eK qdTZKe:e;eeKeE q$dWdX ZLe5dYdZZMe5d[d\ZNd]ZOdOZCe5Id^d_ eMDeCD ZPeLeMeO eP e5Id`d_ eMDeCD ZPeLeMeO eP e5Idad_ eMDeCD ZPeLeMeO eP e5Idbd_ eMDeCD ZPeLeMeO eP e5Idcd_ eQeMDeCeNDeCD ZPeLeMeN eP e5Iddd_ eQeMDeCeNDeCD ZPeLeMeN eP eHe?R dS )mam
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_grouped median_high
median_lowmode multimodepstdev pvariance quantilesstdevvariance NFraction)Decimal)groupby) bisect_left bisect_right)hypotsqrtfabsexperftaulogfsum)
itemgetter)Counterc @ s e Zd ZdS )r N)__name__
__module__ __qualname__ r& r& //opt/alt/python38/lib64/python3.8/statistics.pyr u s c C s d}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)
(, Fraction(11, 1), 5)
Some sources of round-off error will be avoided:
# Built-in sum returns zero.
>>> _sum([1e50, 1, -1e50] * 1000)
(, 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)])
(, 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)
(, Fraction(6963, 10000), 4)
Mixed types are currently treated as an error, except that int is
allowed.
r Nc s s | ]\}}t ||V qd S Nr ).0dnr& r& r' s z_sum..) _exact_ratioget_coerceinttyper map _isfiniteAssertionErrorsumsorteditems) datastartcountr, r+ ZpartialsZ partials_getTtypvaluestotalr& r& r' _sum{ s $
r@ c C s. z
| W S tk
r( t| Y S X d S r) )Z is_finiteAttributeErrormathZisfinite)xr& r& r' r4 s
r4 c C s | t k std| |kr | S |tks,|t kr0| S | tkr<|S t|| rJ|S t| |rX| S t| trf|S t|trt| S t| trt|tr|S t| trt|tr| S d}t|| j|jf dS )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)boolr5 r1
issubclassr float TypeErrorr# )r< Smsgr& r& r' r0 s*
r0 c C s zrt | tkst | tkr$| W S z| j| jfW W S tk
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r t | rt
| df Y S X d}t | t | j
dS )zReturn Real number x to exact (numerator, denominator) pair.
>>> _exact_ratio(0.25)
(1, 4)
x is expected to be an int, Fraction, Decimal or float.
Nz0can't convert type '{}' to numerator/denominator)r2 rF r as_integer_ratio numerator denominatorrA
OverflowError
ValueErrorr4 r5 rG formatr# )rC rI r& r& r' r. s
r. c C sp t | |kr| S t|tr(| jdkr(t}z
|| W S tk
rj t|trd|| j|| j Y S Y nX dS )z&Convert value to given numeric type T.r( N)r2 rE r1 rL rF rG r rK )valuer< r& r& r' _convert s
rQ c C s. t | |}|t| kr&| | |kr&|S tdS )z,Locate the leftmost value exactly equal to xN)r lenrN )arC ir& r& r'
_find_lteq
s
rU c C s> t | ||d}|t| d kr6| |d |kr6|d S tdS )z-Locate the rightmost value exactly equal to x)lor( N)r rR rN )rS lrC rT r& r& r'
_find_rteq s rX negative valuec c s$ | D ]}|dk rt ||V qdS )z7Iterate over values, failing if any are less than zero.r N)r )r> errmsgrC r& r& r' _fail_neg s r[ c C sT t | | krt| } t| }|dk r,tdt| \}}}||ksFtt|| |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)iterlistrR r r@ r5 rQ )r9 r, r< r? r; r& r& r' r ' s c st z t | W n0 tk
r< d fdd}t|| }Y n
X t| }z
| W S tk
rn tddY nX dS )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
r c 3 s t | ddD ]\ }|V q d S )Nr( )r: ) enumerate)iterablerC r, r& r' r; O s zfmean..countz&fmean requires at least one data pointN)rR rG r ZeroDivisionErrorr )r9 r; r? r& r` r' r A s
c C s8 zt ttt| W S tk
r2 tddY nX dS )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
zHgeometric mean requires a non-empty dataset containing positive numbersN)r r r3 r rN r )r9 r& r& r' r \ s c C s t | | krt| } d}t| }|dk r2tdn<|dkrn| d }t|tjtfrf|dk rbt||S tdz"t dd t
| |D \}}}W n t k
r Y dS X ||kst t
|| |S )a Return 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``.
z.harmonic mean does not support negative valuesr( z.harmonic_mean requires at least one data pointr zunsupported typec s s | ]}d | V qdS )r( Nr& r* rC r& r& r' r- s z harmonic_mean..)r\ r] rR r
isinstancenumbersZRealr rG r@ r[ ra r5 rQ )r9 rZ r, rC r< r? r; r& r& r' r o s&
" c C s\ t | } t| }|dkr td|d dkr8| |d S |d }| |d | | d S dS )aB Return 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 data r( Nr7 rR r )r9 r, rT r& r& r' r s
c C sL t | } t| }|dkr td|d dkr8| |d S | |d d S dS )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
r re rf r( Nrg r9 r, r& r& r' r
s c C s, t | } t| }|dkr td| |d S )a Return 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
r re rf rg rh r& r& r' r s
r( c
C s t | } t| }|dkr"tdn|dkr2| d S | |d }||fD ] }t|ttfrFtd| qFz||d }W n( tk
r t|t|d }Y nX 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 re r( rf zexpected number but got %r)
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r9 Zintervalr, rC objLl1l2Zcffr& r& r' r s&
c C sH t | } t| d}z|d d W S tk
rB tddY nX dS )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\ r" most_common
IndexErrorr )r9 Zpairsr& r& r' r s c C s@ t t| }tt|tdddg 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" r\ rp nextr r! r] r3 )r9 ZcountsZmaxcountZ
mode_itemsr& r& r' r 5 s
exclusive)r, methodc
C sL |dk rt dt| } t| }|dk r0t d|dkr|d }g }td|D ]N}|| | }|| || }| | || | |d | | } || qN|S |dkr:|d }g }td|D ]r}|| | }|dk rdn||d kr|d n|}|| || }| |d || | | | | } || q|S td|dS ) 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 1rf z"must have at least two data pointsZ inclusiveru zUnknown method: N)r r7 rR rangeappendrN )
r9 r, rv ZldmresultrT jZdeltaZ interpolatedr& r& r' r l s4 $
$$ c s dk r,t fdd| D \}}}||fS t| t fdd| D \}}}t fdd| D \}}}||kr|||kst||d t| 8 }|dk rtd| ||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 s | ]}| d V qdS rf Nr& rb cr& r' r- s z_ss..c 3 s | ]}| d V qdS r| r& rb r} r& r' r- s c 3 s | ]}| V qd S r) r& rb r} r& r' r- s rf r z%negative sum of square deviations: %f)r@ r r5 rR )r9 r~ r< r? r; UZtotal2Zcount2r& r} r' _ss s r c C sL t | | krt| } t| }|dk r,tdt| |\}}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)
rf z*variance requires at least two data pointsr( r\ r] rR r r rQ )r9 xbarr, r< ssr& r& r' r s & c C sH t | | krt| } t| }|dk r,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 )r9 mur, r< r r& r& r' r s # c C s8 t | |}z
| W S tk
r2 t| Y S X dS )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)r r rA rB )r9 r varr& r& r' r s
c C s8 t | |}z
| W S tk
r2 t| Y S X dS )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)r r rA rB )r9 r r r& r& r' r
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| d | d | d | d | d | d }|| }||| S |dkr| nd| }tt| }|dkr^|d }d| d | d | d | d | d | d | d }d | d | d! | d" | d# | d$ | d% | d }n|d }d&| d' | d( | d) | d* | d+ | d, | d- }d.| d/ | d0 | d1 | d2 | d3 | d4 | d }|| }|dk r| }||| S )5N ?g333333?g Q ?g^}o)@gE.kR@g U l@g*u>l@g N@g"]Ξ@gnC`@gu @giK~j@gv|E@gd|1@gfRr@gu.2@g~y@gn8(E@ ? g @g?g鬷ZaI?ggElD?g7\?guSS?g=.
@gj%b@gHw@gjRe?g9dh?
>g('߿A?g~z ?g@3?gɅ3?g3fRx?gIFl @gt>g*Yn>gESB\T?gN;A+?gUR1?gE F?gPn@g&>@gigtcI,\>gŝI?g*F2v?gC4?gO1?)r r r )pr sigmaqrZnumZdenrC r& r& r' _normal_dist_inv_cdf9 sd
r c @ s e Zd ZdZdddZd8ddZed d
Zd d d
dZdd Z dd Z
dd Z d9ddZ dd Z
edd Zed d Zed d Zed d! Zed"d# Zd$d% Zd&d' Zd(d) Zd*d+ Zd,d- Zd.d/ ZeZd0d1 ZeZ d2d3 Z d4d5 Z d6d7 Z d S ):r z(Normal distribution of a random variablez(Arithmetic mean of a normal distributionz+Standard deviation of a normal distribution)_mu_sigmar r c C s( |dk rt dt|| _t|| _dS )zDNormalDist where mu is the mean and sigma is the standard deviation.r zsigma must be non-negativeN)r rF r r )selfr r r& r& r' __init__ s
zNormalDist.__init__c C s. t |ttfst|}t|}| |t||S )z5Make a normal distribution instance from sample data.)rc r] tupler r )clsr9 r r& r& r' from_samples s zNormalDist.from_samplesN)seedc sB |dkrt jn
t |j | j| j fddt|D S )z=Generate *n* samples for a given mean and standard deviation.Nc s g | ]} qS r& r& r* rT gaussr r r& r'
s z&NormalDist.samples..)randomr ZRandomr r rw )r r, r r& r r' samples s zNormalDist.samplesc C s< | j d }|stdt|| j d d| tt| S )z4Probability density function. P(x <= X < x+dx) / dx @z$pdf() not defined when sigma is zerog )r r r r r r )r rC r r& r& r' pdf s
zNormalDist.pdfc C s2 | j stdddt|| j | j td S )z,Cumulative distribution function. P(X <= x)z$cdf() not defined when sigma is zeror r r )r r r r r )r rC r& r& r' cdf s zNormalDist.cdfc C s: |dks|dkrt d| jdkr*t dt|| j| jS )aS Inverse cumulative distribution function. x : P(X <= x) = p
Finds the value of the random variable such that the probability of
the variable being less than or equal to that value equals the given
probability.
This function is also called the percent point function or quantile
function.
r r z$p must be in the range 0.0 < p < 1.0z-cdf() not defined when sigma at or below zero)r r r r )r r r& r& r' inv_cdf s
zNormalDist.inv_cdfrt c s fddt 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.
c s g | ]} | qS r& )r r r, r r& r' r s z(NormalDist.quantiles..r( )rw )r r, r& r r' r s zNormalDist.quantilesc C s t |tstd| | }}|j|jf|j|jfk r>|| }}|j|j }}|rT|s\td|| }t|j|j }|sdt|d|j t d S |j| |j| }|j|j t |d |t
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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
z$Expected another NormalDist instancez(overlap() not defined when sigma is zeror r ) rc r rG r r r r r r r r r ) r otherXYZX_varZY_varZdvZdmrS bx1x2r& r& r' overlap s"
( zNormalDist.overlapc C s | j S )z+Arithmetic mean of the normal distribution.r r r& r& r' r s zNormalDist.meanc C s | j S )z,Return the median of the normal distributionr r r& r& r' r s zNormalDist.medianc C s | j S )zReturn the mode of the normal distribution
The mode is the value x where which the probability density
function (pdf) takes its maximum value.
r r r& r& r' r s zNormalDist.modec C s | j S )z.Standard deviation of the normal distribution.r r r& r& r' r s zNormalDist.stdevc C s
| j d S )z!Square of the standard deviation.r r r r& r& r' r s zNormalDist.variancec C s8 t |tr&t| j|j t| j|jS t| j| | jS )aj Add 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.
rc r r r r r r r& r& r' __add__ s
zNormalDist.__add__c C s8 t |tr&t| j|j t| j|jS t| j| | jS )as Subtract 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.
r r r& r& r' __sub__ s
zNormalDist.__sub__c C s t | 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.
r r r r r r& r& r' __mul__% s zNormalDist.__mul__c C s t | 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.
r r r& r& r' __truediv__- s zNormalDist.__truediv__c C s t | j| jS )z Return a copy of the instance.r r r r r& r& r' __pos__5 s zNormalDist.__pos__c C s t | j | jS )z(Negates mu while keeping sigma the same.r r r& r& r' __neg__9 s zNormalDist.__neg__c C s
| | S )z s S (
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