Both reliability and validity may be assessed mathematically. Internal consistency may be assessed by correlating performance on two halves of a test (split-half reliability); the value of the Pearson product-moment correlation coefficient is adjusted with the Spearman-Brown prediction formula to correspond to the correlation between two full-length tests. Other approaches include the intra-class correlation (the ratio of variance of measurements of a given target to the variance of all targets). A commonly used measure is Cronbach's a, which is equivalent to the mean of all possible split-half coefficients. Stability over repeated measures is assessed with the Pearson coefficient, as is the equivalence of different versions of the same measure (different forms of an intelligence test, for example). Other measures are also used.
Validity may be assessed by correlating measures with a criterion measure known to be valid. When the criterion measure is collected at the same time as the measure being validated the goal is to establish concurrent validity; when the criterion is collected later the goal is to establish predictive validity. A measure has construct validity if it is related to other variables as required by theory. Content validity, or face validity, is simply a demonstration that the items of a test are drawn from the domain being measured; it does not guarantee that the test actually measures phenomena in that domain.