Property 1 (of Cronbach’s Alpha): Let xj = tj + ej where each ej is independent of tj and all the ej are independent of each other. Also let x0 = and t0 = . Then the reliability of x0 ≥ α where α is Cronbach’s alpha.
Here we view the xj as the measured values, the tj as the true values and the ej as the measurement error values.
Proof: By Property 5 of Basic Concepts of Correlation, var(ti–tj) = var(ti) + var(tj) – 2 cov(ti, tj) , and since var(ti–tj) ≥ 0, it follows that var(ti) + var(tj) ≥ 2 cov(ti, tj). Since for each i there are k – 1 j for which j ≠ i, it follows that
It now follows by Property 5 of Basic Concepts of Correlation