Standard Error of Measurement (SEM) & 95% Minimal Detectable Change (MDC95)

Why two noisy measurements require SQRT(2) scaling in MDC95

In psychometric testing, physical-therapy outcomes, and industrial gauge R&R, an individual’s single observed score X carries random within-subject measurement error with Standard Error of Measurement SEM = SD_baseline * SQRT(1 - r_xx), where r_xx is the test-retest Intraclass Correlation Coefficient (ICC).

When evaluating longitudinal change Delta_X = X_post - X_pre on the same subject, independent measurement errors from both timepoints add in variance: Var(X_post - X_pre) = SEM^2 + SEM^2 = 2 * SEM^2, so SD_diff = SQRT(2) * SEM. Multiplying by z_0.975 = 1.96 yields the 95% Minimal Detectable Change MDC_95 = 1.96 * SQRT(2) * SEM (and 1.645 * SQRT(2) * SEM for MDC_90): any individual score shift smaller than MDC_95 cannot be distinguished from random retest noise.

Standard Error of Measurement (SEM) Formula

Multiply baseline population/pooled SD by SQRT(1 - ICC) (clamping 1 - ICC >= 0).

=ROUND(B2 * SQRT(MAX(0, 1 - B3)), 4)

95% Minimal Detectable Change (MDC_95) Formula

Scale SEM by 1.96 * SQRT(2) (~2.7719 * SEM) to establish the minimum statistically real individual change threshold.

=ROUND(1.96 * SQRT(2) * E2, 2)

Further reading

• Weir, J. P. (2005) — Quantifying Test-Retest Reliability Using the Intraclass Correlation Coefficient and the SEM