Standard Errors and Confidence Intervals of Norm Statistics for Educational and Psychological Tests

Norm statistics allow for the interpretation of scores on psychological and educational tests, by relating the test score of an individual test taker to the test scores of individuals belonging to the same gender, age, or education groups, et cetera. Given the uncertainty due to sampling error, one would expect researchers to report standard errors for norm statistics. In practice, standard errors are seldom reported; they are either unavailable or derived under strong distributional assumptions that may not be realistic for test scores. We derived standard errors for four norm statistics (standard deviation, percentile ranks, stanine boundaries and Z-scores) under the mild assumption that the test scores are multinomially distributed. A simulation study showed that the standard errors were unbiased and that corresponding Wald-based confidence intervals had good coverage. Finally, we discuss the possibilities for applying the standard errors in practical test use in education and psychology. The procedure is provided via the R function check.norms, which is available in the mokken package.