# scipy.stats.kstest¶

scipy.stats.kstest(rvs, cdf, args=(), N=20, alternative='two-sided', mode='approx')[source]

Perform the Kolmogorov-Smirnov test for goodness of fit.

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

Parameters
rvsstr, array or callable

If a string, it should be the name of a distribution in scipy.stats. If an array, it should be a 1-D array of observations of random variables. If a callable, it should be a function to generate random variables; it is required to have a keyword argument size.

cdfstr or callable

If a string, it should be the name of a distribution in scipy.stats. If rvs is a string then cdf can be False or the same as rvs. If a callable, that callable is used to calculate the cdf.

argstuple, sequence, optional

Distribution parameters, used if rvs or cdf are strings.

Nint, optional

Sample size if rvs is string or callable. Default is 20.

alternative{‘two-sided’, ‘less’,’greater’}, optional

Defines the alternative hypothesis (see explanation above). Default is ‘two-sided’.

mode‘approx’ (default) or ‘asymp’, optional

Defines the distribution used for calculating the p-value.

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

• ‘asymp’ : use asymptotic distribution of test statistic

Returns
statisticfloat

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

pvaluefloat

One-tailed or two-tailed p-value.

Notes

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 G(x) of the hypothesis, F(x)<=G(x), resp. F(x)>=G(x).

Examples

>>> from scipy import stats

>>> x = np.linspace(-15, 15, 9)
>>> stats.kstest(x, 'norm')
(0.44435602715924361, 0.038850142705171065)

>>> np.random.seed(987654321) # set random seed to get the same result
>>> stats.kstest('norm', False, N=100)
(0.058352892479417884, 0.88531190944151261)


The above lines are equivalent to:

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


Test against one-sided alternative hypothesis

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

>>> np.random.seed(987654321)
>>> x = stats.norm.rvs(loc=0.2, size=100)
>>> stats.kstest(x,'norm', alternative = 'less')
(0.12464329735846891, 0.040989164077641749)


Reject equal distribution against alternative hypothesis: less

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


Don’t reject equal distribution against alternative hypothesis: greater

>>> stats.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 K-S test 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 the 10% level:

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


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