• Consider a test of 100 items randomly divided into Form A and Form B.
• Consider the test-retest form of reliability:
  • Form A is administered at time t
  • Form B is administered shortly thereafter at time t+1.
  • The correlation between performance on Form A and Form B is a measure of the test's reliability.
• Assume now that respondents take the tests in both forms, A and B, at the same sitting.
  • The correlation between performance on Form A and Form B is still a measure of its reliability.
  • Assume instead that the 100 items are not divided into Form A and Form B, but the random items are interspersed in the same test and numbered as odd and even.
  • The correlation between the odd and even items is still a measure of the test's reliability.
• This is called the "split-halves" technique in assessing internal consistency, and it has several forms:
  • Performance on the first 50 items could be correlated with performance on the last 50.
  • Performance on the first 25 and last 25 could be correlated with performance on the middle 50.
  • And so on.
• G.F. Kuder and M.W. Richardson (1937) developed various reliability formulas to apply to such "parallel tests."
  • Kuder-Richardson Formula 20 was shown by L.J. Cronbach (1951) to produce the mean correlation of all possible ways of splitting a test into two halves.
  • Since then, the formula has been known as Cronbach's alpha.
    

Where:

k = number of indicators

r = item intercorrelations in one off-diagonal, so it is
the sum of the intercorrelations.

As the average intercorrelations among the items (r) increases, and as the number of items (k) increases, the value of alpha increases--so the more reliable the scale.

Programs that calculate scale reliability commonly employ Cronbach's alpha.