Reliabilities of Item Score, Person's Score, and Attribute Mastery Probability, and Their Relationship to the Classical Test Theory

In the classical test theory, the observed score X is defined as the sum of the true score T and the error of measurement E, where the covariance of T and E is 0, and the variances are not zero. The reliability of score X is defined by the ratio of the variances of T and X, ρ = σ²(T)/σ²(X). When X is the sum of n item scores, X = X₁ + X₂ + ⋯ + X_n, then the lower bound of the reliability, Cronbach’s α, is given by the following equation, which is expressed by the observed item score variances of X and item scores X_i: 9.1 https://www.w3.org/1998/Math/MathML"> α ≥ n n − 1 { 1 − ∑ i = 1 n σ 2 ( X i ) σ 2 ( X ) } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203883372/bec7ce46-4277-48ed-bc4c-ce2fd4cf1e63/content/eq9_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>