Compute the expected frequencies from a contingency table.

Given an n-dimensional contingency table of observed frequencies, compute the expected frequencies for the table based on the marginal sums under the assumption that the groups associated with each dimension are independent.

Parameters :

observed : array_like

The table of observed frequencies. (While this function can handle a 1-D array, that case is trivial. Generally observed is at least 2-D.)

Returns :

expected : ndarray of type numpy.float64, same shape as observed.

The expected frequencies, based on the marginal sums of the table.


>>> observed = np.array([[10, 10, 20],[20, 20, 20]])
>>> expected_freq(observed)
array([[ 12.,  12.,  16.],
       [ 18.,  18.,  24.]])

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