Standard Error of Measurement & Score Confidence Band Calculator

How it works

SEM = SD × √(1 − r); true-score estimate = M + r(X − M); SE(true) = SD √(r(1 − r)); SE of a difference = SEM √2; z = (X − cut) ÷ SEM

No test score is exact: the standard error of measurement converts a test's reliability into score points, giving the spread of scores a person would obtain over many hypothetical retests. A 15-point IQ-type scale with reliability 0.90 has an SEM of about 4.7, so the familiar ±5 band around a score covers only 68% and a 95% band is ±9. Because scores above the mean contain more positive error on average, the best estimate of a true score is pulled toward the mean by the unreliability; and because comparing two scores doubles the error variance, differences that look meaningful — between subtests, siblings or before and after a programme — are often within noise. The testing Standards ask that scores be reported with these bands, particularly near cut scores where classification is uncertain.

Worked example

With reliability 0.90 and SD 15 the SEM is 4.74: an observed 112 has a 95% band of 103–121 and an estimated true score of 110.8. Compared with a score of 104 the 8-point gap is 1.19 standard errors of the difference (p = 0.23) — not reliably different — and there is only a 66% chance the true score is above a cut of 110.

Frequently asked questions