The data from k groups are arranged either (a) all in a single variable with another categorical variable indicating ``levels,'' or (b) in multiple columns each of whose variables represents a ``level.'' In either case, (i) the original variables 's are converted to new variables 's via the transformation if necessary; otherwise, leave it to ``no change.'' Then, (ii) they are moved to the column Variable in the table below, and (iii) the residual is calculated.
When the nonnormality cannot be eliminated by the use of transformation, the Kruskal-Wallis test is appropriate for the hypothesis testing. Here the null hypothesis is that k population distributions (not necessarily normal) are identical. It calculates the test statistic and the p-value . By rejecting we can find some evidence supporting that not all the distributions are the same.