Читать книгу Applied Univariate, Bivariate, and Multivariate Statistics - Daniel J. Denis - Страница 56

Having reviewed the concept of covariance, we need a way to account for the covariance of many variables. For this, we write the sample covariance in matrix form:

where s_jk are the covariances for variables j by k. The population covariance matrix ∑ can be analogously defined:

where along the main diagonal of the covariance matrix are variances σ₁₁, σ₂₂, etc., for variables 1, 2, etc., up to σ_pp, the variance of the p^th variable.

When we standardize the covariance matrix, dividing each of its elements by respective products of standard deviations, we obtain the correlation matrix:

where r₁₂ is the correlation between variables 1 and 2, etc., and r_1p is the correlation between variable 1 and the p^th variable.

An example of a correlation matrix (Heston, 1948) is that between different tests on the GRE (Graduate Record Examination):

Intercorrelations Among The G.R.E. Tests Of General Education Math P.S. B.S. Soc. Lit. Arts Exp. Voc. Mathematics .55 .44 .51 .36 .35 .52 .38 Physical Science .55 .49 .43 .20 .40 .32 .29 Biological Science .44 .49 .57 .42 .42 .46 .50 Social Studies .51 .43 .57 .54 .40 .61 .59 Literature .36 .20 .42 .54 .39 .53 .54 Arts .35 .40 .42 .40 .39 .42 .52 Effecive Expression .52 .32 .46 .61 .53 .42 .66 Vocabulary .38 .29 .50 .59 .54 .52 .66

From the matrix, we can see that most correlations are low to moderate, with the correlation between Effective Expression and Vocabulary relatively large at a value of 0.66. The correlation between Physical Science and Vocabulary is relatively small, equaling 0.29.