KR-20 / KR-21 Test Reliability Calculator

How it works

KR-21 = k/(k−1) × [1 − M(k − M) ÷ (k σ²)]; KR-20 = k/(k−1) × [1 − Σ p(1−p) ÷ σ²]; SEM = σ √(1 − r); Spearman–Brown length factor n = r_target(1 − r) ÷ [r(1 − r_target)]

Reliability asks how consistently a test would rank the same people if they took it again, and for tests scored right or wrong the Kuder–Richardson formulas estimate it from a single administration by comparing the variance of total scores with the variance the items contribute individually: when items hang together, total-score variance is large relative to the sum of item variances and reliability approaches one. KR-20 uses each item's proportion correct; KR-21 needs only the mean and standard deviation but assumes equal item difficulty and therefore runs low. Reliability sets the standard error of measurement — how far an observed score may sit from the person's true score — and the Spearman–Brown formula shows how many more similar items would be needed to reach a target, which is why high-stakes tests are long.

Worked example

A 40-item test with mean 28 and SD 6.5 has KR-21 = 40/39 × (1 − 28 × 12 ÷ (40 × 42.25)) = 0.822; using the item p-values, KR-20 is about 0.85 — good. The standard error of measurement is about 2.6 points, so a student scoring 30 is probably (68%) between 27 and 33; reaching 0.90 would take about 66 items.

Frequently asked questions