Chi-Square Test

The chi-square test refers to a family of statistical tests that have been utilized to determine [Page 269]whether the observed (sampling) distribution or outcome differs significantly from an a priori or theoretically anticipated outcome or distribution. More simply stated, the test is formulated to determine whether the difference observed was due to a chance occurrence. This entry further describes the chi-square test and looks at its basic principles, applications, and limitations.

Although the most common chi-square test statistic is Pearson’s chi-square test, there are other test statistics that exist with the same theoretical foundation including Yates’s chi-square test, Tukey’s test of additivity, Cochran–Mantel–Haenszel test, and likelihood ratio tests. Although the chi-square test has been applied to a plethora of statistical applications, the fundamental utilization has been as a goodness-of-fit statistic and difference test by comparing a hypothesized distribution to an observed distribution.

In 1900, Karl Pearson developed the chi-square test and published his work entitled “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,” where he outlined the limitations of related measures and the functional utility of the test he developed. While the idea of determining whether standard distributions gave acceptable fits to data sets was well established early in Pearson’s career, detailed in his 1900 paper, he was determined to derive a test procedure that further advanced the problem of goodness of fit. As a result, the formulation of the chi-square statistic stands as one of the greatest statistical achievements of the 20th century.