<Dd> (number of rows − 1) (number of columns − 1) = (3 − 1) (4 − 1) = 6 . (\ displaystyle ((\ text (number of rows)) - 1) ((\ text (number of columns)) - 1) = (3 - 1) (4 - 1) = 6. \,) </Dd> <P> If the test statistic is improbably large according to that chi - squared distribution, then one rejects the null hypothesis of independence . </P> <P> A related issue is a test of homogeneity . Suppose that instead of giving every resident of each of the four neighborhoods an equal chance of inclusion in the sample, we decide in advance how many residents of each neighborhood to include . Then each resident has the same chance of being chosen as do all residents of the same neighborhood, but residents of different neighborhoods would have different probabilities of being chosen if the four sample sizes are not proportional to the populations of the four neighborhoods . In such a case, we would be testing "homogeneity" rather than "independence". The question is whether the proportions of blue - collar, white - collar, and no - collar workers in the four neighborhoods are the same . However, the test is done in the same way . </P> <P> In cryptanalysis, chi - squared test is used to compare the distribution of plaintext and (possibly) decrypted ciphertext . The lowest value of the test means that the decryption was successful with high probability . This method can be generalized for solving modern cryptographic problems . </P>

Chi square is a useful test for determining