scipy.stats.kstest(rvs, cdf, args=(), N=20, alternative='two_sided', mode='approx', **kwds)

Perform the Kolmogorov-Smirnov test for goodness of fit

This performs a test of the distribution G(x) of an observed random variable against a given distribution F(x). Under the null hypothesis the two distributions are identical, G(x)=F(x). The alternative hypothesis can be either ‘two_sided’ (default), ‘less’ or ‘greater’. The KS test is only valid for continuous distributions.

Parameters :

rvs : string or array or callable

string: name of a distribution in scipy.stats

array: 1-D observations of random variables

callable: function to generate random variables, requires keyword argument size

cdf : string or callable

string: name of a distribution in scipy.stats, if rvs is a string then cdf can evaluate to False or be the same as rvs callable: function to evaluate cdf

args : tuple, sequence

distribution parameters, used if rvs or cdf are strings

N : int

sample size if rvs is string or callable

alternative : ‘two_sided’ (default), ‘less’ or ‘greater’

defines the alternative hypothesis (see explanation)

mode : ‘approx’ (default) or ‘asymp’

defines the distribution used for calculating p-value

‘approx’ : use approximation to exact distribution of test statistic

‘asymp’ : use asymptotic distribution of test statistic

Returns :

D : float

KS test statistic, either D, D+ or D-

p-value : float

one-tailed or two-tailed p-value


In the one-sided test, the alternative is that the empirical cumulative distribution function of the random variable is “less” or “greater” than the cumulative distribution function F(x) of the hypothesis, G(x)<=F(x), resp. G(x)>=F(x).


>>> from scipy import stats
>>> import numpy as np
>>> from scipy.stats import kstest
>>> x = np.linspace(-15,15,9)
>>> kstest(x,'norm')
(0.44435602715924361, 0.038850142705171065)
>>> np.random.seed(987654321) # set random seed to get the same result
>>> kstest('norm','',N=100)
(0.058352892479417884, 0.88531190944151261)

is equivalent to this

>>> np.random.seed(987654321)
>>> kstest(stats.norm.rvs(size=100),'norm')
(0.058352892479417884, 0.88531190944151261)

Test against one-sided alternative hypothesis:

>>> np.random.seed(987654321)

Shift distribution to larger values, so that cdf_dgp(x)< norm.cdf(x):

>>> x = stats.norm.rvs(loc=0.2, size=100)
>>> kstest(x,'norm', alternative = 'less')
(0.12464329735846891, 0.040989164077641749)

Reject equal distribution against alternative hypothesis: less

>>> kstest(x,'norm', alternative = 'greater')
(0.0072115233216311081, 0.98531158590396395)

Don’t reject equal distribution against alternative hypothesis: greater

>>> kstest(x,'norm', mode='asymp')
(0.12464329735846891, 0.08944488871182088)

Testing t distributed random variables against normal distribution:

With 100 degrees of freedom the t distribution looks close to the normal distribution, and the kstest does not reject the hypothesis that the sample came from the normal distribution

>>> np.random.seed(987654321)
>>> stats.kstest(stats.t.rvs(100,size=100),'norm')
(0.072018929165471257, 0.67630062862479168)

With 3 degrees of freedom the t distribution looks sufficiently different from the normal distribution, that we can reject the hypothesis that the sample came from the normal distribution at a alpha=10% level

>>> np.random.seed(987654321)
>>> stats.kstest(stats.t.rvs(3,size=100),'norm')
(0.131016895759829, 0.058826222555312224)

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