Editing Factor analysis (section)

===Example===
Suppose a psychologist has the hypothesis that there are two kinds of [[intelligence (trait)|intelligence]], "verbal intelligence" and "mathematical intelligence", neither of which is directly observed.{{Explanatory footnote|In this example, "verbal intelligence" and "mathematical intelligence" are latent variables.  The fact that they're not directly observed is what makes them latent.|name=latent variables|group=note}} [[Evidence]] for the hypothesis 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 [[population (statistics)|population]], then each student's 10 scores are random variables. The psychologist's hypothesis 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 [[Constant (mathematics)|constant]] times their level of verbal intelligence plus another constant times their level of mathematical intelligence, i.e., it is a linear combination of those two "factors". The numbers for a particular subject, by which the two kinds of intelligence are multiplied to obtain the expected score, are posited by the hypothesis to be the same for all intelligence level pairs, and are called '''"factor loading"''' for this subject. {{Clarify|date=July 2019}}  For example, the hypothesis may hold that the predicted average student's aptitude in the field of [[astronomy]] is

:{10 × the student's verbal intelligence} + {6 × the student's mathematical intelligence}.

The numbers 10 and 6 are the factor loadings associated with astronomy. Other academic subjects may have different factor loadings.

Two students assumed to have identical degrees of verbal and mathematical intelligence may have different measured aptitudes in astronomy because individual aptitudes differ from average aptitudes (predicted above) and because of measurement error itself. Such differences make up what is collectively called the "error" — a statistical term that means the amount by which an individual, as measured, differs from what is average for or predicted by his or her levels of intelligence (see [[errors and residuals in statistics]]).

The observable data that go into factor analysis would be 10 scores of each of the 1000 students, a total of 10,000 numbers. The factor loadings and levels of the two kinds of intelligence of each student must be inferred from the data.