What about norm referenced tests, I hear you ask ...

In this article, Brennan (1972) adopts the orthodox view that norm referenced tests rely on the assumption of normally distributed test scores.

Brennan (1972) uses / describes / advocates for the standard classical test theory line about the importance of understanding the distributions behind correlation coefficients and the need for the normality (i.e. the normality assumption). A psychometrician must be mindful of the normality assumption when working in the CTT paradigm.

Here are some examples from the article:

*“These symmetric cut-off points are, in turn, basically a result of the preoccupation of test theory with the normal distribution, which is, of course, symmetric.*^{2} Unfortunately, however, not all reasonable distributions of test scores are normal.”

^{2} “Kelley (1939) notes that the upper and lower 27 per cent of the cases constitute optimal groups for determining discrimination indices only when the criterion test scores are normally distributed.” (page 291)

*“Some of the correlation type of discrimination indices are also affected by non-normal test score distributions. For example, lack of normality precludes the use of tetrachoric correlation coefficient. Also, unless one is willing to assume that student responses to dichotomous items are essentially continuous and normally distributed, the biserial coefficient should not be used. Neither the point biserial correlation coefficient, nor the phi coefficient necessitates normality assumptions;” (symbols for the types of correlation coefficients have been removed) (page 297)*

R.L. Brennan (1972) *A generalized upper-lower item discrimination index*. Educational and Psychological Measurement, 32, 289-303.

SOURCE: http://journals.sagepub.com/doi/abs/10.1177/001316447203200206?journalCode=epma