Cohen’s h Arcsine Two-Proportion Effect Size & Variance-Stabilized Z-Test

Why a 4% lift from 1% to 5% carries 3x the statistical signal of 51% to 55%

Binomial variance p * (1 - p) shrinks rapidly near 0% and 100%, so raw percentage-point differences (p_1 - p_2) are NOT equally detectable across the probability scale! Jacob Cohen (*Statistical Power Analysis for the Behavioral Sciences*, 1988) stabilized variance via the angular arcsine-square-root mapping phi(p) = 2 * ASIN(SQRT(p)) (in radians), defining standardized effect size h = phi(p_1) - phi(p_2) with conventional benchmarks |h| = 0.20 (Small), 0.50 (Medium), and 0.80 (Large).

Test two experiments with the exact same +0.04 raw difference on n_1 = n_2 = 200 subjects: lifting conversion from 1% to 5% yields h = 0.4510 - 0.2003 = 0.2507 and arcsine z = h * SQRT(n_1*n_2 / (n_1+n_2)) = 2.5069 (significant at 95%), whereas lifting 51% to 55% yields h = 0.0802 and z = 0.8017 (indistinguishable from sampling noise).

Cohen’s h Radian Arcsine Formula

Guard that both proportions lie in [0, 1] and evaluate 2 * ASIN(SQRT(p_1)) - 2 * ASIN(SQRT(p_2)) directly in radians (do NOT wrap in DEGREES!).

=IF(OR(B2 < 0, B2 > 1, B3 < 0, B3 > 1), "INVALID_PROB", ROUND(2 * ASIN(SQRT(B2)) - 2 * ASIN(SQRT(B3)), 4))

Variance-Stabilized Arcsine Z-Statistic & Benchmark Tier

Multiply unrounded h by harmonic-root sample factor SQRT((n_1 * n_2) / (n_1 + n_2)) to test H_0: p_1 = p_2.

=LET(h_raw, 2 * ASIN(SQRT(B2)) - 2 * ASIN(SQRT(B3)), ROUND(h_raw * SQRT((B4 * B5) / (B4 + B5)), 4))

Further reading

• Cohen, J. (1988) — Statistical Power Analysis for the Behavioral Sciences (Chapter 6: Differences Between Proportions)