scipy.stats.chisquare(f_obs, f_exp=None, ddof=0, axis=0)[source]#

Calculate a one-way chi-square test.

The chi-square test tests the null hypothesis that the categorical data has the given frequencies.


Observed frequencies in each category.

f_exparray_like, optional

Expected frequencies in each category. By default the categories are assumed to be equally likely.

ddofint, optional

“Delta degrees of freedom”: adjustment to the degrees of freedom for the p-value. The p-value is computed using a chi-squared distribution with k - 1 - ddof degrees of freedom, where k is the number of observed frequencies. The default value of ddof is 0.

axisint or None, optional

The axis of the broadcast result of f_obs and f_exp along which to apply the test. If axis is None, all values in f_obs are treated as a single data set. Default is 0.

res: Power_divergenceResult

An object containing attributes:

statisticfloat or ndarray

The chi-squared test statistic. The value is a float if axis is None or f_obs and f_exp are 1-D.

pvaluefloat or ndarray

The p-value of the test. The value is a float if ddof and the result attribute statistic are scalars.

See also


Fisher exact test on a 2x2 contingency table.


An unconditional exact test. An alternative to chi-squared test for small sample sizes.

Chi-square test


This test is invalid when the observed or expected frequencies in each category are too small. A typical rule is that all of the observed and expected frequencies should be at least 5. According to [3], the total number of samples is recommended to be greater than 13, otherwise exact tests (such as Barnard’s Exact test) should be used because they do not overreject.

Also, the sum of the observed and expected frequencies must be the same for the test to be valid; chisquare raises an error if the sums do not agree within a relative tolerance of 1e-8.

The default degrees of freedom, k-1, are for the case when no parameters of the distribution are estimated. If p parameters are estimated by efficient maximum likelihood then the correct degrees of freedom are k-1-p. If the parameters are estimated in a different way, then the dof can be between k-1-p and k-1. However, it is also possible that the asymptotic distribution is not chi-square, in which case this test is not appropriate.



Lowry, Richard. “Concepts and Applications of Inferential Statistics”. Chapter 8.


Pearson, Karl. “On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling”, Philosophical Magazine. Series 5. 50 (1900), pp. 157-175.


When only the mandatory f_obs argument is given, it is assumed that the expected frequencies are uniform and given by the mean of the observed frequencies:

>>> import numpy as np
>>> from scipy.stats import chisquare
>>> chisquare([16, 18, 16, 14, 12, 12])
Power_divergenceResult(statistic=2.0, pvalue=0.84914503608460956)

The optional f_exp argument gives the expected frequencies.

>>> chisquare([16, 18, 16, 14, 12, 12], f_exp=[16, 16, 16, 16, 16, 8])
Power_divergenceResult(statistic=3.5, pvalue=0.62338762774958223)

When f_obs is 2-D, by default the test is applied to each column.

>>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T
>>> obs.shape
(6, 2)
>>> chisquare(obs)
Power_divergenceResult(statistic=array([2.        , 6.66666667]), pvalue=array([0.84914504, 0.24663415]))

By setting axis=None, the test is applied to all data in the array, which is equivalent to applying the test to the flattened array.

>>> chisquare(obs, axis=None)
Power_divergenceResult(statistic=23.31034482758621, pvalue=0.015975692534127565)
>>> chisquare(obs.ravel())
Power_divergenceResult(statistic=23.310344827586206, pvalue=0.01597569253412758)

ddof is the change to make to the default degrees of freedom.

>>> chisquare([16, 18, 16, 14, 12, 12], ddof=1)
Power_divergenceResult(statistic=2.0, pvalue=0.7357588823428847)

The calculation of the p-values is done by broadcasting the chi-squared statistic with ddof.

>>> chisquare([16, 18, 16, 14, 12, 12], ddof=[0, 1, 2])
Power_divergenceResult(statistic=2.0, pvalue=array([0.84914504, 0.73575888, 0.5724067 ]))

f_obs and f_exp are also broadcast. In the following, f_obs has shape (6,) and f_exp has shape (2, 6), so the result of broadcasting f_obs and f_exp has shape (2, 6). To compute the desired chi-squared statistics, we use axis=1:

>>> chisquare([16, 18, 16, 14, 12, 12],
...           f_exp=[[16, 16, 16, 16, 16, 8], [8, 20, 20, 16, 12, 12]],
...           axis=1)
Power_divergenceResult(statistic=array([3.5 , 9.25]), pvalue=array([0.62338763, 0.09949846]))

For a more detailed example, see Chi-square test.