scipy.stats.cramervonmises(rvs, cdf, args=(), *, axis=0, nan_policy='propagate', keepdims=False)[source]#

Perform the one-sample Cramér-von Mises test for goodness of fit.

This performs a test of the goodness of fit of a cumulative distribution function (cdf) \(F\) compared to the empirical distribution function \(F_n\) of observed random variates \(X_1, ..., X_n\) that are assumed to be independent and identically distributed ([1]). The null hypothesis is that the \(X_i\) have cumulative distribution \(F\).


A 1-D array of observed values of the random variables \(X_i\). The sample must contain at least two observations.

cdfstr or callable

The cumulative distribution function \(F\) to test the observations against. If a string, it should be the name of a distribution in scipy.stats. If a callable, that callable is used to calculate the cdf: cdf(x, *args) -> float.

argstuple, optional

Distribution parameters. These are assumed to be known; see Notes.

axisint or None, default: 0

If an int, the axis of the input along which to compute the statistic. The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If None, the input will be raveled before computing the statistic.

nan_policy{‘propagate’, ‘omit’, ‘raise’}

Defines how to handle input NaNs.

  • propagate: if a NaN is present in the axis slice (e.g. row) along which the statistic is computed, the corresponding entry of the output will be NaN.

  • omit: NaNs will be omitted when performing the calculation. If insufficient data remains in the axis slice along which the statistic is computed, the corresponding entry of the output will be NaN.

  • raise: if a NaN is present, a ValueError will be raised.

keepdimsbool, default: False

If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array.

resobject with attributes

Cramér-von Mises statistic.


The p-value.


Added in version 1.6.0.

The p-value relies on the approximation given by equation 1.8 in [2]. It is important to keep in mind that the p-value is only accurate if one tests a simple hypothesis, i.e. the parameters of the reference distribution are known. If the parameters are estimated from the data (composite hypothesis), the computed p-value is not reliable.

Beginning in SciPy 1.9, np.matrix inputs (not recommended for new code) are converted to np.ndarray before the calculation is performed. In this case, the output will be a scalar or np.ndarray of appropriate shape rather than a 2D np.matrix. Similarly, while masked elements of masked arrays are ignored, the output will be a scalar or np.ndarray rather than a masked array with mask=False.



Csörgő, S. and Faraway, J. (1996). The Exact and Asymptotic Distribution of Cramér-von Mises Statistics. Journal of the Royal Statistical Society, pp. 221-234.


Suppose we wish to test whether data generated by scipy.stats.norm.rvs were, in fact, drawn from the standard normal distribution. We choose a significance level of alpha=0.05.

>>> import numpy as np
>>> from scipy import stats
>>> rng = np.random.default_rng()
>>> x = stats.norm.rvs(size=500, random_state=rng)
>>> res = stats.cramervonmises(x, 'norm')
>>> res.statistic, res.pvalue
(0.1072085112565724, 0.5508482238203407)

The p-value exceeds our chosen significance level, so we do not reject the null hypothesis that the observed sample is drawn from the standard normal distribution.

Now suppose we wish to check whether the same samples shifted by 2.1 is consistent with being drawn from a normal distribution with a mean of 2.

>>> y = x + 2.1
>>> res = stats.cramervonmises(y, 'norm', args=(2,))
>>> res.statistic, res.pvalue
(0.8364446265294695, 0.00596286797008283)

Here we have used the args keyword to specify the mean (loc) of the normal distribution to test the data against. This is equivalent to the following, in which we create a frozen normal distribution with mean 2.1, then pass its cdf method as an argument.

>>> frozen_dist = stats.norm(loc=2)
>>> res = stats.cramervonmises(y, frozen_dist.cdf)
>>> res.statistic, res.pvalue
(0.8364446265294695, 0.00596286797008283)

In either case, we would reject the null hypothesis that the observed sample is drawn from a normal distribution with a mean of 2 (and default variance of 1) because the p-value is less than our chosen significance level.