SPECTRAL ANALYSIS OF TEST SCORES

Abstract

Spectral analysis based on group representation theory has been shown to be applicable to various statistical problems by several authors. We discuss in this paper an application of such analysis to the data of test scores consisting of binary correct-incorrect answers for each test items. The group considered here is the symmetric group on p letters, which is denoted by Sp. The spectral analysis proceeds as follows.First, we identify the invariant subspaces under the operation of Sp, and find the projection matrices onto the spaces. Then, we project the data vector onto such spaces, and interpret the result by looking at the projections. Numerical examples are also shown to illustrate the present analysis.