Information Loss in Grade Point Conversion

Some information gets lost when numerical scores evaluating performances are converted into letter grades. We propose to measure this information loss through the proportion of variance lost due to grouping. We study various properties of this measure, including its invariance in location and scale equivariant families. The information loss typically decreases with an increase in the number of levels of letter grades. However, it is not appropriate to have too many levels. The optimum number of levels may be determined, either by visual inspection when the information loss becomes marginal/stable, or by minimizing the sum of the information loss and a penalty term, the latter being taken as linear in the number of levels. We also address the problem of determining the groups, or equivalently, the boundaries so that the information loss is minimized, given a fixed number of groups. Finding these optimal boundaries is a computationally intensive exercise even for moderate size data, unless the number of groups is very small. We recommend an alternative way by fitting an appropriate probability distribution. When the probabilistic nature of the data is known, the boundary points turn out to be the solutions to a system of equations; however these solutions may not necessarily have any closed form. We derive the exact or approximate solutions of these equations when the composite scores follow a probability distribution belonging to the Uniform, Triangular, and Gaussian family.