Mauchly’s test for sphericity can be run in the majority of statistical software, where it tends to be the default test for sphericity. Mauchly’s test is ideal for mid-size samples. It may fail to detect sphericity in small samples and it may over-detect in large samples.
If the test returns a small p-value (p ≤.05), this is an indication that your data has violated the assumption. The following picture of SPSS output for ANOVA shows that the significance “sig” attached to Mauchly’s is .274. This means that the assumption has not been violated for this set of data.
Kempthorne uses the randomization-distribution and the assumption of unit treatment additivity to produce a derived linear model , very similar to the textbook model discussed previously.  The test statistics of this derived linear model are closely approximated by the test statistics of an appropriate normal linear model, according to approximation theorems and simulation studies.  However, there are differences. For example, the randomization-based analysis results in a small but (strictly) negative correlation between the observations.   In the randomization-based analysis, there is no assumption of a normal distribution and certainly no assumption of independence . On the contrary, the observations are dependent !