A note on the inadequacy of percentage grading scales for almost all ability distributions.

Abstract

AbstractThis study investigated distributions of letter grades (A,B,C,D,F) assigned to test scores according to the percentages adopted by Canadian universities, assuming realistic score distributions of ten different shapes with various means and standard deviations. The grade distributions corresponding to 83 out of 90 score distributions were highly anomalous, and the remaining 7 were far from ideal. Therefore, in the majority of practical testing situations, the percentage grading method is inadequate because of purely statistical properties of a scale based on fixed percentages.It is commonplace for instructors in high schools and universities to assign letter grades to test scores according to percentage scales. The following scale has been adopted by most Canadian universities and is published in many catalogs and on transcripts of credits: A -- 80% to 100%, B -- 70% to 79%, C -- 60% to 69%, D -- 50% to 59%, and F -- below 50%.Scales of this type have been used for about a century and have become quite familiar in educational institutions. However, they are difficult to reconcile with objective testing methods that have been developed by psychometricians and test theorists during the same historical period. Cureton (1971, p. 4) noted that, as actually used in practice, the percentages have the status of ranks. And recently Frary, Cross & Weber (1992, p. 2) remarked that ...it is widely perceived in measurement circles that teacher practices and beliefs concerning testing and grading are more in keeping with the 1890s than the 1990s.Distributions of Ability and Distributions of GradesThe present note examines how specific distributions of test scores determine corresponding distributions of grades based on the above scale. Assuming realistic probability distributions of test scores, we inquire as to whether or not the resulting distributions of letter grades turn out to be reasonable. For a normal distribution, this information can be obtained analytically. The same is true for various other well - known probability densities, including the uniform density.One suspects, however, that test scores frequently have unusual distribution shapes which are not represented by familiar probability densities. In these cases, calculations are not feasible. Furthermore, most distributions of test scores are discrete rather than continuous. The present study employed computer simulation to obtain the desired information in these more realistic cases.The simulations were based on ten distribution shapes, including several of the non - normal distributions characteristic of data obtained in psychological research. For each shape, the study investigated all combinations of three selected means and three selected standard deviations which are typical of test scores in practice. Assigning the grades, A, B, C, D, and F according to the percentages mentioned above, the program determined the percent of examinees receiving each grade.Simulated Test Score DistributionsThe following distributions of scores were examined:1. Normal. Normal deviates with mean 0 and standard deviation 1 were generated by the method of Box and Muller (1958), using the transformation Equation where U and V are uniformly distributed pseudorandom numbers on the interval (0,1). This variable was further transformed by multiplying by a constant and adding a constant in order to produce a desired mean and standard deviation. The program also imposed a discrete structure by computing the greatest integer less than X, in order to represent test scores which are a sum of scores on items answered correctly. 2. Uniform. A uniformly distributed variable with mean 0 and standard deviation 1 was generated from X = 12(U - .5), where U has mean 0 and standard deviation 1, and further transformed to a discrete variable with a predetermined mean and standard deviation, as described above. For further details concerning simulation of distributions, see Devroye (1986). …