The bivariate distribution of test scores and scores on the common factor of test items is derived as a function of mathematics of the usual parameters. This is given for three cases. The first, where the test score is the number of items answered correctly; the second, the multivariate distribution of responses to the test items of the nature that it could have come from a normal multivariate distribution of abilities in the group tested; and the third, the matrix of the tetrachoric item intercorrelations has a rank of 1. An alternate condition is that the item characteristic curve is a normal ogive. It is possible to check empirically whether or not any given data set conforms to these conditions. Mathematical expressions are given for: 1) the univariate frequency distribution of the test score; 2) the standard error of measurement for testees at any given ability level; 3) the regression of test scores on ability; and 4) an index of discriminating power of the test for individuals at any given ability level. The regression of test score on ability is never linear. Rectangular and U-shaped distributions of test score may be obtained if items have tetrachoric intercorrelations of 0.50 or more. The standard measurement error is smallest for those to whom the test is least discriminating. If a test is to be used for selection or rejection of testees, all items should be equally hard. (SGK).