Statistical & Math
Cronbach’s Alpha (alpha) Internal Consistency & Spearman-Brown Prophecy
Calculate psychometric test reliability Cronbach’s alpha = (K / (K - 1)) * (1 - Sum_Item_Var / Total_Score_Var) and Spearman-Brown predicted reliability for scaled test length.
Audit survey & exam reliability before trusting composite scale scores!
Psychometricians, UX researchers, and clinical scale authors use Lee Cronbach’s (1951) coefficient alpha = (K / (K - 1)) * (1 - SUM(sigma_i^2) / sigma_total^2) to measure what fraction of composite test-score variance comes from shared true-score covariance across K items rather than uncorrelated item noise! Notice why total score variance sigma_total^2 must vastly exceed the sum of individual item variances SUM(sigma_i^2): when items correlate positively, pairwise covariances inflate sigma_total^2!
For a K = 10-item Likert scale with sum of item variances SUM(sigma_i^2) = 8.40 and composite total-score variance sigma_total^2 = 32.00, alpha = (10 / 9) * (1 - 8.40 / 32.00) = 0.8194 ("Good" reliability >= 0.80). Doubling the exam length (m = 2.0) lifts predicted reliability via the Spearman-Brown Prophecy Formula alpha* = (m * alpha) / (1 + (m - 1) * alpha) to 0.9008 ("Excellent" high-stakes clinical threshold >= 0.90)!
| Psychometric Scale Scenario (K = 10 Items, Total Var = 32.0) | Sum of Item Variances | Cronbach’s Alpha | Psychometric Interpretation |
|---|---|---|---|
| Cohesive Survey Scale | 8.40 | 0.8194 | Good Internal Consistency (>= 0.80) |
| 2x Length Exam (Spearman-Brown m = 2) | 8.40 (Scaled) | 0.9008 | High-Stakes Clinical / Certification (>= 0.90) |
| Noisy Items (Higher Unshared Variance) | 14.40 | 0.6111 | Questionable Consistency (< 0.70) |
Cronbach’s Coefficient Alpha = (K / (K - 1)) * (1 - Sum_Item_Var / Total_Score_Var)
Guards against single-item scales (K < 2) or non-positive variance inputs.
=IF(OR(B2 < 2, B3 <= 0, B4 <= 0), "Invalid Scale Inputs", ROUND((B2 / (B2 - 1)) * (1 - (B3 / B4)), 4))Spearman-Brown Prophecy Reliability for Length Multiplier m (B5)
Projects reliability alpha_new = (m * alpha) / (1 + (m - 1) * alpha) when scaling a K-item instrument by factor m > 0.
=IF(OR(B2 < 2, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Scale Inputs", ROUND((B5 * ((B2 / (B2 - 1)) * (1 - (B3 / B4)))) / (1 + (B5 - 1) * ((B2 / (B2 - 1)) * (1 - (B3 / B4)))), 4))Try the example
Use the original values or change them to test your understanding.