Statistical & Psychometrics
Kuder-Richardson Formula 20 (KR-20) Dichotomous Test Reliability
Calculate internal-consistency reliability KR-20 = (K / (K - 1)) * (1 - SUM(p_i * q_i) / σ_X^2) for binary right/wrong certification exams and diagnostic checklists.
Exact internal consistency for binary 0/1 multiple-choice exams without item-by-examinee covariance matrices!
When test items are scored dichotomously (1 = correct / symptom present, 0 = incorrect / absent), individual item variance simplifies to the Bernoulli product p_i * q_i = p_i * (1 - p_i), where p_i is the proportion of examinees answering item i correctly! G. Frederic Kuder and Marion W. Richardson (Psychometrika, 1937) proved that Cronbach’s alpha reduces identically to KR-20 = (K / (K - 1)) * (1 - SUM(p_i * (1 - p_i)) / s_total^2) across K binary items and total-score variance s_total^2!
Audit a 5-item clinical competency mini-quiz with item pass rates p = [0.80, 0.70, 0.60, 0.50, 0.40]: Bernoulli item variances p*(1-p) = [0.16, 0.21, 0.24, 0.25, 0.24] sum to 1.10. Against observed total-score variance s_total^2 = 2.75, KR-20 = (5 / 4) * (1 - 1.10 / 2.75) = 0.7500 (acceptable internal consistency >= 0.70)!
| Total Test Variance s_total² (K = 5, Σpq = 1.10) | Variance Ratio Σpq / s_total² | KR-20 Reliability | Psychometric Benchmark |
|---|---|---|---|
| s_total² = 2.75 (Baseline Cohort) | 0.4000 | 0.7500 | Acceptable Research / Classroom Scale (>= 0.70) |
| s_total² = 3.50 (Higher True-Score Spread) | 0.3143 | 0.8571 | High-Stakes Certification Grade (>= 0.80) |
| Invalid s_total² <= 0 or K < 2 | Undefined | "Invalid Test Variance" | Variance Guard |
Kuder-Richardson KR-20 Reliability Coefficient
Computes (K / (K - 1)) * (1 - SUM(C2:C6) / E2) from item pass proportions B2:B6, Bernoulli item variances C2:C6 = p*(1-p), and total-score variance E2.
=IF(OR(COUNT(B2:B6) < 2, E2 <= 0), "Invalid Test Variance", ROUND((COUNT(B2:B6) / (COUNT(B2:B6) - 1)) * (1 - (SUM(C2:C6) / E2)), 4))Sum of Bernoulli Dichotomous Item Variances Σ p_i (1 - p_i)
Totals individual binary item variances across C2:C6 (maximized at 0.25 per item when item difficulty p_i = 0.50).
=ROUND(SUM(C2:C6), 4)Try the example
Use the original values or change them to test your understanding.