*Classical test theory* may be regarded as roughly synonymous with *true score theory*. The term "classical" refers not only to the chronology of these models but also contrasts with the more recent psychometric theories, generally referred to collectively as item response theory, which sometimes bear the appelation "modern" as in "modern latent trait theory".

The central model of classical test theory is that observed test scores (x) are composed of a true score (T) and an error score (E):

(Eq. 1)

where T and E are independent. Using a variety of justifications, E is often assumed to be a random variable having a normal distribution:

(Eq. 2)

(Eq. 3)

(Eq. 4)

Additionally, parallel tests are defined such that:

(Eq. 5)

From these assumptions, certain results follow:

(Eq. 6)

(Eq. 7)

(Eq. 8)

(Eq. 9)

(Eq. 10)

[insert an explanation of these equations]

It is worth noting the implications of classical test theory for test-takers. Tests are fallible, imprecise tools. The score acheived by an individual is rarely the individual's true score. This *observed score* is almost always the true score occluded by some degree of error. This error may push the observed score higher or lower.

It is also worth noting that nothing about these models refutes human development or improvement. A person may learn skills, knowledge or even so called "test-taking skills" which may translate to a higher true-score.

See also psychometrics, standardized test.