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This article discusses the underlying principles for chi‐squared tests and provides an overview for the scope of their applications. Specific topics are as follows: (i) the test for specified variance of univariate normal random variables; (ii) approximate randomization tests for hypotheses of no association of response variables with a set of groups; (iii) approximate tests for hypotheses pertaining to parameters in assumed likelihood functions. For these topics, the chi‐squared test statistics are quadratic forms, and they address departures from randomness for a vector of differences between observed values of estimates of interest and their expected values under corresponding null hypotheses, relative to the inverse of the applicable covariance matrix. It is of note that, sufficient sample size is necessary for (ii) and (iii) to support an approximately chi‐squared distribution for the test statistic with degrees of freedom equal to the dimension of the vector that is addressed.