This widely used test of the sphericity assumption assesses the null hypothesis that the variances of the differences are equal. Put another way, it assesses the assumption that the differences scores of paired levels of a repeated measures factor have equal population variances. Provided the data are derived from a multivariate normal population, a significant Mauchly test value (W) indicates that the null hypothesis can be rejected and that the assumption of sphericity has been violated (i.e., the variances of the differences are not equal). As the sphericity assumption is always met for ANOVA designs with a repeated measures factor that has two levels, a Mauchly test is not needed in such a case. The test has been subjected to the criticism that it fails to detect violations of sphericity in small samples and to detect them too often when the sample is large. The test was devised in 1940 by John W. Mauchly (1907-1980), a physicist who co-designed the first general purpose electronic digital computer.
See Compound symmetry, Greenhouse-Geisser epsilon, Hunyh-Feldt epsilon, Repeated measures analysis of variance, Sphericity, Sphericity assumption