The mean score statistic is one of the statistics used in the generalized Cochran-Mantel-Haenszel tests. It is applicable when the response levels (columns) are measured at an ordinal scale.
If the two variables are independent of each other in all strata, the asymptotic distribution of the mean score statistic is the chi-square distribution with (R-1) degrees of freedom, where R is the number of the treatment groups (rows).
The mean score statistic tends to take on higher values if the mean scores of the response vary across the treatment groups in at least one strata. This statistic is also called “nonparametric ANOVA statistic”.