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First Approach to Internal Consistency
  • 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.