Measurements in mental tests through person space

If a test is administered to n- persons, the test score can be viewed as a point in n – dimensional person space where each person is considered in an axis and the variables are represents as points or vectors. The orthogonality of the axes is better indication since individuals are assumed to be independent. Such presentation helps to make useful inferences about the subjects and various parameters of the test, item statistics along with geometrical interpretation of such computations. For a test consisting of one-right-rest wrong type items, the paper gives theoretical formulation of mental testing through person space, primarily in terms of length of observed score vector and angle separating the vector with the vector representing the maximum possible score in the test. In addition, method is described to obtain (i) test reliability, as per the theoretical definition, via a single administration of the test, which is extended to find reliability of a battery of tests (ii) difficulty, discriminating values of items and test along with their relationships. Empirical verification of the proposed methods are undertaken.