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.


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.


Mannan, R. William and E. Charles. Meslow. “Bird populations and vegetation characteristics in managed and old-growth forests, northeastern Oregon.” Journal of Wildlife Management 48, 1219-1238, DOI:10.2307/3801783, 1984.


In [4], bird foraging behavior was investigated in an old-growth forest of Oregon. In the forest, 44% of the canopy volume was Douglas fir, 24% was ponderosa pine, 29% was grand fir, and 3% was western larch. The authors observed the behavior of several species of birds, one of which was the red-breasted nuthatch. They made 189 observations of this species foraging, recording 43 (“23%”) of observations in Douglas fir, 52 (“28%”) in ponderosa pine, 54 (“29%”) in grand fir, and 40 (“21%”) in western larch.

Using a chi-square test, we can test the null hypothesis that the proportions of foraging events are equal to the proportions of canopy volume. The authors of the paper considered a p-value less than 1% to be significant.

Using the above proportions of canopy volume and observed events, we can infer expected frequencies.

>>> import numpy as np
>>> f_exp = np.array([44, 24, 29, 3]) / 100 * 189

The observed frequencies of foraging were:

>>> f_obs = np.array([43, 52, 54, 40])

We can now compare the observed frequencies with the expected frequencies.

>>> from scipy.stats import chisquare
>>> chisquare(f_obs=f_obs, f_exp=f_exp)
Power_divergenceResult(statistic=228.23515947653874, pvalue=3.3295585338846486e-49)

The p-value is well below the chosen significance level. Hence, the authors considered the difference to be significant and concluded that the relative proportions of foraging events were not the same as the relative proportions of tree canopy volume.

Following are other generic examples to demonstrate how the other parameters can be used.

When just f_obs is given, it is assumed that the expected frequencies are uniform and given by the mean of the observed frequencies.

>>> chisquare([16, 18, 16, 14, 12, 12])
Power_divergenceResult(statistic=2.0, pvalue=0.84914503608460956)

With f_exp the expected frequencies can be given.

>>> 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]))