Suppose a psychologist proposes a
theory that there are two kinds of
"verbal intelligence" and "mathematical intelligence". Note that
these are inherently unobservable.
the theory is sought in the examination scores, from each of 10
different academic fields, of 1000 students. If each student is
chosen randomly from a large
then each student's 10 scores are random variables.
The psychologist's theory may
say that, for each of the 10 academic fields, the score averaged
over the group of all students who share some common pair of values
for verbal and mathematical "intelligences" is some
times their level of verbal intelligence plus another constant times
their level of mathematical intelligence, i.e., it is a
of those two "factors". The numbers, for this particular subject, by
which the two kinds of intelligence are multiplied to obtain the
expected score, are posited by the theory to be the same for all
intelligence level pairs, and are called "factor loadings"
for this subject.