Scaling and combining examination marks

Scaling and combining examination marks

by J.D.Biggins and K.K. Yue

Brit. J. Math. Statist. Psych. (1993) 46, 153-179.

Abstract

Biggins, Loynes and Walker (1986) considered the problem of scaling and combining examination marks from several papers to obtain transformed marks and an overall measure of each candidate's performance in the examination. Their approach is to obtain the transformations and the overall marks by the minimisation of a suitably chosen loss function subject to a single constraint. In the main, following Broyden (1983), they consider the case where the allowed transformation is multiplication by a constant (which varies from paper to paper). This paper discusses the same problem but with a richer class of possible transformations, the main example being regression splines with end-point restrictions. These end-point restrictions will mean that the curve can be forced to preserve the mark range, by passing through (0,0) and (100,100), for example. If grades rather than marks are returned the problem becomes the well-studied one of scaling categorical attributes. Our formulation applies in this case also, allowing us to connect the proposals with existing scaling literature. The general issue of how to incorporate the expectation that transformation curves should not be too far from the $45^\circ$ line is also addressed, with the device of using fictitious candidates, introduced by Biggins {\it et al.}\ (1986), being extended to this context.

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