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Cohen's sample sizes for power 0.80 at α = 0.05: t tests, χ² and one-way ANOVA
| Tool | Hypothesis Tests |
| Source | Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences, 2nd ed. Lawrence Erlbaum Associates, Hillsdale, NJ. Chapter 2 (t tests, medium and large d), Table 7.4 (χ², w = 0.3, df 1), Table 8.4.4 (ANOVA, f = 0.25, k = 4); confirmed by G*Power 3. |
| Level | Graduate |
Problem
How many observations are needed to detect a medium effect (Cohen's d = 0.5) with power 0.80 in a two-sided test at α = 0.05, (a) with two independent groups and (b) with one sample or paired differences; how many for a large effect d = 0.8 with two groups; and what total N for a χ² test with w = 0.3 on 1 df and what n per group for a four-group ANOVA with f = 0.25 at the same power?
Inputs to Enter
- Sample-Size Planning group: target effect 0.5 (d, w or f), target power 0.80, α = 0.05, two-sided
- Select the test whose sample size is wanted: Two-sample t, Paired t (or One-sample t), Chi-square goodness of fit (df 1), One-way ANOVA (k = 4)
- Repeat with target effect 0.8 for the large-effect two-sample case
Answers
- Two-sample t, d = 0.5: n = 64 per group by the exact noncentral t (power at n = 64: 0.8015, G*Power); normal approximation 2(z_α + z_β)²/d² = 62.8 → 63
- One-sample or paired t, d = 0.5: n = 34 (power 0.8077 at 34; 33 falls below 0.80)
- Two-sample t, d = 0.8: n = 26 per group
- χ² with w = 0.3, df = 1: N = 88 (Cohen Table 7.4, G*Power 88)
- One-way ANOVA, f = 0.25, k = 4: n = 45 per group (Cohen Table 8.4.4)
Notes
The planning inputs default to effect 0.5 and power 0.80, so the default page already shows n = 34 for the default paired test; the other values need the test select changed. The exact noncentral t is the tool's own AS 243 implementation (test_hypothesis_tests.php).