Internal consistency reliability is the extent to which the measurements of a test remain consistent over repeated tests of the same subject under identical conditions. An experiment is reliable if it yields consistent results of the same measure, i.e. it doesn’t yield random error in measurement. It is unreliable if repeated measurements give different results.
Since there are inaccuracies when taking measurements, even when the same measurements are taken twice there can be differences. We can therefore partition an observed value of x into the true value of x and an error term. Thus we have x = t + e.
Definition 1: The reliability of x is a measure of internal consistency and is the correlation coefficient rxt of x and t.
Proof: Since the covariance is additive (i.e. Property 1 of Correlation – Advanced),
Since ti and ei are independent, cov(t, e) = 0, and so