We consider the effect of aptitudes, knowledge, convictions, discernment of the context, experience with the content, modes of expression, and so on. We look at correlations among measures to see persistent ways of acting, knowing that errors of measurement and sample selection affect the correlations. Factor analysis provides a way to summarize the information latent in correlations. We decide what issues are common to a group of tests. The next factor has as much as possible in common with all the remaining variance.
Analysts use both geometry and mathematics.
Then they find a new vector, which is the least-squares best fit to all the other vectors. The next factor is the best fit to the residuals. We could obtain ninety-nine factors from a hundred tests, but most of these would seem essentially random. Ordinarily, a very small number of factors will account for all of the reliable variance. That number depends upon the reliability and diversity of the tests.