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For an independent‐samples t‐test, required sample size can be estimated through R using pwr.t.test :

> pwr.t.test (n =, d =, sig.level =, power =, type = c(“two.sample”, “one.sample”, “paired”))

where, n = sample size per group, d = estimate of standardized statistical distance between means (Cohen's d), sig.level = desired significance level of the test, power = desired power level, and type = designation of the kind of t‐test you are performing (for our example, we are performing a two‐sample test).

Table 2.6 Power Estimates as a Function of Sample Size and Estimated Magnitude Under Alternative Hypothesis

Exact – Correlation: Bivariate Normal Model Tail(s) = Two, Correlation ρ H0 = 0, α err prob = 0.05
		Correlation ρ H1 = 0.1	Correlation ρ H1 = 0.2	Correlation p HI = 0.3	Correlation ρ HI = 0.4	Correlation ρ HI = 0.5
#	Power(1‐β err prob)	Total Sample Size	Total Sample Size	Total Sample Size	Total Sample Size	Total Sample Size
1	0.600000	488.500	121.500	53.5000	29.5000	18.5000
2	0.610000	500.500	124.500	54.5000	30.5000	18.5000
3	0.620000	511.500	126.500	55.5000	30.5000	19.5000
4	0.630000	523.500	129.500	56.5000	31.5000	19.5000
5	0.640000	535.500	132.500	58.5000	32.5000	19.5000
6	0.650000	548.500	135.500	59.5000	32.5000	20.5000
7	0.660000	561.500	138.500	60.5000	33.5000	20.5000
8	0.670000	574.500	142.500	62.5000	34.5000	21.5000
9	0.680000	587.500	145.500	63.5000	34.5000	21.5000
10	0.690000	601.500	148500	64.5000	35.5000	22.5000

It would be helpful at this point to translate Cohen's d values into R² values to learn how much variance is explained by differing d values. To convert the two, we apply the following transformation:

Table 2.7 contains conversions for r increments of 0.10, 0.20, 0.30, etc.

To get a better feel for the relationship between Cohen's d and r², we obtain a plot of their values (Figure 2.15).

As can be gleamed from Figure 2.15, the relationship between the two effect size measures is not exactly linear and increases rather sharply for rather large values (the curve is somewhat exponential).

Suppose a researcher would like to estimate required sample size for a two‐sample t‐test, for a relatively small effect size, d = 0.41 (equal to r of 0.20), at a significance level of 0.05, with a desired power level of 0.90. We compute:

> pwr.t.test (n =, d =0.41, sig.level =.05, power =.90, type = c(“two.sample”)) Two-sample t test power calculation n = 125.9821 d = 0.41 sig.level = 0.05 power = 0.9 alternative = two.sided NOTE: n is number in *each* group

Thus, the researcher would require a sample size of approximately 126. As R emphasizes, this sample size is per group, so the total sample size required is 126(2) = 252.

Table 2.7 Conversions for r → r2→ d.¹¹

r	r 2	d
0.10	0.01	0.20
0.20	0.04	0.41
0.30	0.09	0.63
0.40	0.16	0.87
0.50	0.25	1.15
0.60	0.36	1.50
0.70	0.49	1.96
0.80	0.64	2.67
0.90	0.81	4.13
0.99	0.98	14.04

Figure 2.15 Relationship between Cohen's d and R‐squared.