Direct, vec and other squares, and sample variance-covariance of dioptric power

Matrices can be multiplied in several ways. As a result one can define a number of distinct squares of the dioptric power matrix. Additional squares can be defined for matrices vectorized by means of the vec and vech operators. These various squares can form the basis for the definition of variance and covariance of samples of dioptric powers. The complete form of the variance of dioptric powers is a 4 x 4-matrix (a variance-covariance matrix) with 10 distinct elements; four of them represent the variances of the four elements of the dioptric power matrix and the other six are the covariances between those four elements. For thin systems there are only six distinct elements of the variance-covariance matrix, three of which are variances and three covariances. Examples are included that show the calculation of the different types of squares and the variance-covariance matrix for a sample of equivalent powers of thick bitoric lenses and for a sample of powers of thin systems, including conventional refractive errors. Variances calculated in the past, including those of nearest equivalent spheres and Gartner's and Churm's variances turn out to be components (or combinations of components) of the generalized variance defined here. They are valid as far as they go but they do not completely represent the dispersion of a sample of dioptric powers. The complete variance-covariance matrix does represent the dispersion fully and thus opens the way for the formal statistical analysis of measurements of dioptric power.