Sample Size Formula Explained with Examples

Created:March 8, 2026

Last Updated:March 11, 2026

Understanding the formulas behind sample size calculations gives you the ability to critically evaluate study designs, communicate with statisticians, and make informed decisions when planning research. In this tutorial, we break down the sample size formula for six commonly used statistical tests, each with a step-by-step worked example and code in R.

If you're looking for a high-level process rather than formulas, see our How to Determine Sample Size for a Study guide. For a deeper dive into the concept of statistical power, see Power Analysis: A Complete Guide.

Skip the math? Use our Sample Size & Power Analysis Calculator to compute sample sizes instantly with interactive power curves.

The General Framework

All sample size formulas share the same underlying logic. You need four quantities, and knowing any three lets you solve for the fourth:

$\alpha$

Significance level

$1 - \beta$

Statistical power

Effect size

Sample size

The general pattern for most formulas is:

n \propto \frac{(z_{\alpha} + z_{\beta})^2}{\text{ES}^2}

This tells us two key relationships:

Sample size is inversely proportional to the square of the effect size. For example, if you double the effect size, the denominator becomes $(2d)^2 = 4d^2$ , which is $4\times$ larger — so the required $n$ shrinks to $\frac{1}{4}$ of its original value (a 75% reduction).
Higher power or stricter alpha increases the z-values in the numerator, increasing the required $n$ .

Note: The manual formulas below use $z$ -values (normal approximation), while R's pwr package uses the $t$ -distribution, which has heavier tails. As a result, software-computed sample sizes are often slightly larger than the manual formula results. The software values are more accurate in practice, especially for smaller sample sizes.

Sample Size Formulas by Test Type

1. Two-Sample t-Test (Independent Groups)

Compares the means of two independent groups. The effect size is Cohen's d, the standardized mean difference.

Formula

n_{\text{per group}} = \frac{2(z_{\alpha/2} + z_{\beta})^2}{d^2}

where $d = \frac{|\mu_1 - \mu_2|}{\sigma}$ is Cohen's d.

Worked Example

Detect a medium effect (d = 0.5) with 80% power at $\alpha = 0.05$ :

$z_{\alpha/2} = z_{0.025} = 1.96$
$z_{\beta} = z_{0.20} = 0.84$
$n = \frac{2(1.96 + 0.84)^2}{0.5^2} = \frac{2 \times 7.84}{0.25} = \frac{15.68}{0.25} = 62.72$
Round up: n = 63 per group (126 total)

library(pwr)

result <- pwr.t.test(
    d = 0.5, sig.level = 0.05,
    power = 0.8, type = "two.sample"
)
cat("n per group:", ceiling(result$n))
# n per group: 64

Unequal groups? With allocation ratio r = n₂/n₁: $n_1 = \frac{(1 + 1/r)(z_{\alpha/2} + z_{\beta})^2}{d^2}$ , $n_2 = r \times n_1$

2. Paired t-Test

Compares means from the same subjects measured twice (e.g., before/after). The correlation between measurements reduces the required sample size.

Formula

n_{\text{pairs}} = \frac{2(z_{\alpha/2} + z_{\beta})^2(1 - \rho)}{d^2}

where $\rho$ is the correlation between paired measurements.

Worked Example

Detect d = 0.5 with 80% power, $\alpha = 0.05$ , and correlation $\rho = 0.6$ :

$n = \frac{2(1.96 + 0.84)^2(1 - 0.6)}{0.5^2} = \frac{15.68 \times 0.4}{0.25} = \frac{6.272}{0.25} = 25.09$
Round up: n = 26 pairs

Compare to the independent t-test which needed 63 per group — the paired design needs only 26 pairs because the correlation reduces variance.

library(pwr)

# Paired t-test
result <- pwr.t.test(
    d = 0.5, sig.level = 0.05,
    power = 0.8, type = "paired"
)
cat("n pairs:", ceiling(result$n))
# n pairs: 34 (without rho adjustment)

# With rho = 0.6, effective d:
rho <- 0.6
d_eff <- 0.5 / sqrt(2 * (1 - rho))
result2 <- pwr.t.test(
    d = d_eff, sig.level = 0.05,
    power = 0.8, type = "one.sample"
)
cat("n pairs (rho=0.6):", ceiling(result2$n))
# n pairs (rho=0.6): 28

3. Two-Proportion Test

Compares two proportions (e.g., conversion rates in A/B testing). Uses the arc-sine transformation or the direct proportion formula.

Formula

n_{\text{per group}} = \frac{\left[z_{\alpha/2}\sqrt{2\bar{p}(1-\bar{p})} + z_{\beta}\sqrt{p_1(1-p_1) + p_2(1-p_2)}\right]^2}{(p_1 - p_2)^2}

where $\bar{p} = (p_1 + p_2)/2$ is the average proportion.

Worked Example

Detect a change from 10% to 15% conversion, 80% power, $\alpha = 0.05$ :

$p_1 = 0.10, p_2 = 0.15, \bar{p} = 0.125$
$n = \frac{[1.96\sqrt{2(0.125)(0.875)} + 0.84\sqrt{0.10(0.90) + 0.15(0.85)}]^2}{(0.05)^2} = 684.8 \approx 685$
n = 685 per group (1,370 total)

The code below uses Cohen's h (arcsine transformation) rather than the raw proportion difference, which is why it gives 681 instead of 685.

library(pwr)

# Cohen's h effect size
h <- ES.h(0.10, 0.15)
result <- pwr.2p.test(
    h = h, sig.level = 0.05,
    power = 0.8, alternative = "two.sided"
)
cat("n per group:", ceiling(result$n))
# n per group: 681

4. One-Way ANOVA

Compares means across three or more groups. The effect size is Cohen's f (related to $\eta^2$ ).

Formula

ANOVA sample size uses the non-central F-distribution. The approximation is:

n_{\text{per group}} \approx \frac{(z_{\alpha/2} + z_{\beta})^2}{f^2} + \frac{k - 1}{2k}

where k is the number of groups and $f = \sqrt{\eta^2 / (1 - \eta^2)}$ .

Worked Example

3 groups, medium effect (f = 0.25), 80% power, $\alpha = 0.05$ :

Using the pwr package: n = 53 per group (159 total).

library(pwr)

result <- pwr.anova.test(
    f = 0.25, k = 3,
    sig.level = 0.05, power = 0.8
)
cat("n per group:", ceiling(result$n))
# n per group: 53

5. Chi-Square Test

Tests associations between categorical variables. The effect size is Cohen's w.

Formula

n = \frac{\chi^2_{\alpha, df}}{w^2} \quad \text{(approximate)}

where $w = \sqrt{\sum \frac{(p_{o} - p_{e})^2}{p_{e}}}$ and df = (rows - 1)(columns - 1).

Worked Example

2×2 table (df = 1), medium effect (w = 0.3), 80% power, $\alpha = 0.05$ :

Using software: n = 88 total.

library(pwr)

result <- pwr.chisq.test(
    w = 0.3, df = 1,
    sig.level = 0.05, power = 0.8
)
cat("Total n:", ceiling(result$N))
# Total n: 88

6. Multiple Regression

Tests whether a set of predictors explains a significant portion of variance. The effect size is Cohen's $f^2$ .

Formula

n = \frac{(z_{\alpha/2} + z_{\beta})^2}{f^2} + p + 1

where p is the number of predictors and $f^2 = \frac{R^2}{1 - R^2}$ .

Cohen's conventions: $f^2 = 0.02$ (small), $f^2 = 0.15$ (medium), $f^2 = 0.35$ (large).

Worked Example

3 predictors, medium effect ( $f^2 = 0.15$ ), 80% power, $\alpha = 0.05$ :

Using software: n = 77 total.

library(pwr)

result <- pwr.f2.test(
    f2 = 0.15,       # Cohen's f²
    u = 3,            # numerator df (predictors)
    sig.level = 0.05,
    power = 0.8
)
# Total n = v + u + 1
total_n <- ceiling(result$v) + 3 + 1
cat("Total n:", total_n)
# Total n: 77

Complete Worked Example: Planning a Study from Scratch

Scenario

A psychologist wants to test whether cognitive behavioral therapy (CBT) reduces anxiety scores more than a waitlist control group. Previous studies report that CBT reduces anxiety by about 8 points on the GAD-7 scale, with a pooled standard deviation of 12 points. The study expects 10% dropout. Budget allows recruiting up to 100 participants.

Step 1: Calculate effect size

d = \frac{8}{12} = 0.667

This is a medium-to-large effect size.

Step 2: Set parameters

$\alpha = 0.05$ (two-sided)
Power = 0.80
Test: independent two-sample t-test

Step 3: Calculate sample size

n = \frac{2(1.96 + 0.84)^2}{0.667^2} = \frac{15.68}{0.4449} = 35.24

Round up: 36 per group (72 total).

Step 4: Adjust for dropout

n_{\text{adjusted}} = \frac{72}{1 - 0.10} = \frac{72}{0.90} = 80

Recruit 80 participants (40 per group). This is within the budget of 100.

Conclusion

The study needs 80 participants (40 per group) to detect a Cohen's d of 0.667 with 80% power at the 0.05 significance level, accounting for 10% dropout. This is feasible within the budget.

Try this example yourself with our Sample Size & Power Analysis Calculator. Select "Mean Difference (Independent t-test)", set d = 0.667, power = 0.80, and alpha = 0.05.

Test Your Understanding

Practice applying the sample size formulas with these problems. Work through each step before revealing the answer.

Problem 1Medium

A researcher wants to compare the mean reaction times of two groups. They expect a difference of 30 ms with a pooled standard deviation of 50 ms. Calculate the required sample size per group for 80% power at α = 0.05 (two-sided).

Problem 2Medium

Using the two-sample t-test formula

n = \frac{2(z_{\alpha/2} + z_\beta)^2}{d^2}

, what happens to the required sample size when you double the effect size from 0.3 to 0.6?

The sample size doubles

The sample size is cut in half

The sample size is reduced to one-quarter

The sample size stays the same

Problem 3Hard

An e-commerce company wants to detect a 2 percentage point increase in purchase rate from a baseline of 5% (i.e., from 5% to 7%). Calculate the sample size per group using the two-proportion formula at 80% power and α = 0.05 (two-sided).

Problem 4Easy

Why do paired t-tests generally require fewer participants than independent t-tests?

Paired tests use a different significance level

The correlation between paired measurements reduces the effective variance

Paired tests have higher Type I error rates

The effect size is always larger in paired designs

Problem 5Hard

A colleague tells you: "I used the t-test sample size formula and it says I need 393 participants per group to detect a small effect (d = 0.2). That's way too many. How can I reduce this?" What would you advise?

Think About It

Take a moment to consider this question and formulate your answer before viewing the solution.

Can the research design be changed to require fewer participants?
Is a small effect size truly what they expect, or could they justify a larger one?
What trade-offs are involved in each approach?
Are there any approaches that would NOT be appropriate?

Sample Size Formula Explained with Examples

Created:March 8, 2026

Last Updated:March 11, 2026

Skip the math? Use our Sample Size & Power Analysis Calculator to compute sample sizes instantly with interactive power curves.

The General Framework

All sample size formulas share the same underlying logic. You need four quantities, and knowing any three lets you solve for the fourth:

$\alpha$

Significance level

$1 - \beta$

Statistical power

Effect size

Sample size

The general pattern for most formulas is:

n \propto \frac{(z_{\alpha} + z_{\beta})^2}{\text{ES}^2}

This tells us two key relationships:

Sample size is inversely proportional to the square of the effect size. For example, if you double the effect size, the denominator becomes $(2d)^2 = 4d^2$ , which is $4\times$ larger — so the required $n$ shrinks to $\frac{1}{4}$ of its original value (a 75% reduction).
Higher power or stricter alpha increases the z-values in the numerator, increasing the required $n$ .

Sample Size Formulas by Test Type

1. Two-Sample t-Test (Independent Groups)

Compares the means of two independent groups. The effect size is Cohen's d, the standardized mean difference.

Formula

n_{\text{per group}} = \frac{2(z_{\alpha/2} + z_{\beta})^2}{d^2}

where $d = \frac{|\mu_1 - \mu_2|}{\sigma}$ is Cohen's d.

Worked Example

Detect a medium effect (d = 0.5) with 80% power at $\alpha = 0.05$ :

$z_{\alpha/2} = z_{0.025} = 1.96$
$z_{\beta} = z_{0.20} = 0.84$
$n = \frac{2(1.96 + 0.84)^2}{0.5^2} = \frac{2 \times 7.84}{0.25} = \frac{15.68}{0.25} = 62.72$
Round up: n = 63 per group (126 total)

library(pwr)

result <- pwr.t.test(
    d = 0.5, sig.level = 0.05,
    power = 0.8, type = "two.sample"
)
cat("n per group:", ceiling(result$n))
# n per group: 64

Unequal groups? With allocation ratio r = n₂/n₁: $n_1 = \frac{(1 + 1/r)(z_{\alpha/2} + z_{\beta})^2}{d^2}$ , $n_2 = r \times n_1$

2. Paired t-Test

Compares means from the same subjects measured twice (e.g., before/after). The correlation between measurements reduces the required sample size.

Formula

n_{\text{pairs}} = \frac{2(z_{\alpha/2} + z_{\beta})^2(1 - \rho)}{d^2}

where $\rho$ is the correlation between paired measurements.

Worked Example

Detect d = 0.5 with 80% power, $\alpha = 0.05$ , and correlation $\rho = 0.6$ :

$n = \frac{2(1.96 + 0.84)^2(1 - 0.6)}{0.5^2} = \frac{15.68 \times 0.4}{0.25} = \frac{6.272}{0.25} = 25.09$
Round up: n = 26 pairs

Compare to the independent t-test which needed 63 per group — the paired design needs only 26 pairs because the correlation reduces variance.

library(pwr)

# Paired t-test
result <- pwr.t.test(
    d = 0.5, sig.level = 0.05,
    power = 0.8, type = "paired"
)
cat("n pairs:", ceiling(result$n))
# n pairs: 34 (without rho adjustment)

# With rho = 0.6, effective d:
rho <- 0.6
d_eff <- 0.5 / sqrt(2 * (1 - rho))
result2 <- pwr.t.test(
    d = d_eff, sig.level = 0.05,
    power = 0.8, type = "one.sample"
)
cat("n pairs (rho=0.6):", ceiling(result2$n))
# n pairs (rho=0.6): 28

3. Two-Proportion Test

Compares two proportions (e.g., conversion rates in A/B testing). Uses the arc-sine transformation or the direct proportion formula.

Formula

n_{\text{per group}} = \frac{\left[z_{\alpha/2}\sqrt{2\bar{p}(1-\bar{p})} + z_{\beta}\sqrt{p_1(1-p_1) + p_2(1-p_2)}\right]^2}{(p_1 - p_2)^2}

where $\bar{p} = (p_1 + p_2)/2$ is the average proportion.

Worked Example

Detect a change from 10% to 15% conversion, 80% power, $\alpha = 0.05$ :

$p_1 = 0.10, p_2 = 0.15, \bar{p} = 0.125$
$n = \frac{[1.96\sqrt{2(0.125)(0.875)} + 0.84\sqrt{0.10(0.90) + 0.15(0.85)}]^2}{(0.05)^2} = 684.8 \approx 685$
n = 685 per group (1,370 total)

The code below uses Cohen's h (arcsine transformation) rather than the raw proportion difference, which is why it gives 681 instead of 685.

library(pwr)

# Cohen's h effect size
h <- ES.h(0.10, 0.15)
result <- pwr.2p.test(
    h = h, sig.level = 0.05,
    power = 0.8, alternative = "two.sided"
)
cat("n per group:", ceiling(result$n))
# n per group: 681

4. One-Way ANOVA

Compares means across three or more groups. The effect size is Cohen's f (related to $\eta^2$ ).

Formula

ANOVA sample size uses the non-central F-distribution. The approximation is:

n_{\text{per group}} \approx \frac{(z_{\alpha/2} + z_{\beta})^2}{f^2} + \frac{k - 1}{2k}

where k is the number of groups and $f = \sqrt{\eta^2 / (1 - \eta^2)}$ .

Worked Example

3 groups, medium effect (f = 0.25), 80% power, $\alpha = 0.05$ :

Using the pwr package: n = 53 per group (159 total).

library(pwr)

result <- pwr.anova.test(
    f = 0.25, k = 3,
    sig.level = 0.05, power = 0.8
)
cat("n per group:", ceiling(result$n))
# n per group: 53

5. Chi-Square Test

Tests associations between categorical variables. The effect size is Cohen's w.

Formula

n = \frac{\chi^2_{\alpha, df}}{w^2} \quad \text{(approximate)}

where $w = \sqrt{\sum \frac{(p_{o} - p_{e})^2}{p_{e}}}$ and df = (rows - 1)(columns - 1).

Worked Example

2×2 table (df = 1), medium effect (w = 0.3), 80% power, $\alpha = 0.05$ :

Using software: n = 88 total.

library(pwr)

result <- pwr.chisq.test(
    w = 0.3, df = 1,
    sig.level = 0.05, power = 0.8
)
cat("Total n:", ceiling(result$N))
# Total n: 88

6. Multiple Regression

Tests whether a set of predictors explains a significant portion of variance. The effect size is Cohen's $f^2$ .

Formula

n = \frac{(z_{\alpha/2} + z_{\beta})^2}{f^2} + p + 1

where p is the number of predictors and $f^2 = \frac{R^2}{1 - R^2}$ .

Cohen's conventions: $f^2 = 0.02$ (small), $f^2 = 0.15$ (medium), $f^2 = 0.35$ (large).

Worked Example

3 predictors, medium effect ( $f^2 = 0.15$ ), 80% power, $\alpha = 0.05$ :

Using software: n = 77 total.

library(pwr)

result <- pwr.f2.test(
    f2 = 0.15,       # Cohen's f²
    u = 3,            # numerator df (predictors)
    sig.level = 0.05,
    power = 0.8
)
# Total n = v + u + 1
total_n <- ceiling(result$v) + 3 + 1
cat("Total n:", total_n)
# Total n: 77

Complete Worked Example: Planning a Study from Scratch

Scenario

Step 1: Calculate effect size

d = \frac{8}{12} = 0.667

This is a medium-to-large effect size.

Step 2: Set parameters

$\alpha = 0.05$ (two-sided)
Power = 0.80
Test: independent two-sample t-test

Step 3: Calculate sample size

n = \frac{2(1.96 + 0.84)^2}{0.667^2} = \frac{15.68}{0.4449} = 35.24

Round up: 36 per group (72 total).

Step 4: Adjust for dropout

n_{\text{adjusted}} = \frac{72}{1 - 0.10} = \frac{72}{0.90} = 80

Recruit 80 participants (40 per group). This is within the budget of 100.

Conclusion

The study needs 80 participants (40 per group) to detect a Cohen's d of 0.667 with 80% power at the 0.05 significance level, accounting for 10% dropout. This is feasible within the budget.

Try this example yourself with our Sample Size & Power Analysis Calculator. Select "Mean Difference (Independent t-test)", set d = 0.667, power = 0.80, and alpha = 0.05.

Test Your Understanding

Practice applying the sample size formulas with these problems. Work through each step before revealing the answer.

Problem 1Medium

Problem 2Medium

Using the two-sample t-test formula

n = \frac{2(z_{\alpha/2} + z_\beta)^2}{d^2}

, what happens to the required sample size when you double the effect size from 0.3 to 0.6?

The sample size doubles

The sample size is cut in half

The sample size is reduced to one-quarter

The sample size stays the same

Problem 3Hard

Problem 4Easy

Why do paired t-tests generally require fewer participants than independent t-tests?

Paired tests use a different significance level

The correlation between paired measurements reduces the effective variance

Paired tests have higher Type I error rates

The effect size is always larger in paired designs

Problem 5Hard

Think About It

Take a moment to consider this question and formulate your answer before viewing the solution.

Can the research design be changed to require fewer participants?
Is a small effect size truly what they expect, or could they justify a larger one?
What trade-offs are involved in each approach?
Are there any approaches that would NOT be appropriate?

Sample Size Formula Explained with Examples

The General Framework

Sample Size Formulas by Test Type

1. Two-Sample t-Test (Independent Groups)

Formula

Worked Example

2. Paired t-Test

Formula

Worked Example

3. Two-Proportion Test

Formula

Worked Example

4. One-Way ANOVA

Formula

Worked Example

5. Chi-Square Test

Formula

Worked Example

6. Multiple Regression

Formula

Worked Example

Complete Worked Example: Planning a Study from Scratch

Scenario

Step 1: Calculate effect size

Step 2: Set parameters

Step 3: Calculate sample size

Step 4: Adjust for dropout

Conclusion

Test Your Understanding

View Step-by-Step Solution

View Step-by-Step Solution

Think About It

I've thought about it - Show me the solution

Sample Size Formula Explained with Examples

The General Framework

Sample Size Formulas by Test Type

1. Two-Sample t-Test (Independent Groups)

Formula

Worked Example

2. Paired t-Test

Formula

Worked Example

3. Two-Proportion Test

Formula

Worked Example

4. One-Way ANOVA

Formula

Worked Example

5. Chi-Square Test

Formula

Worked Example

6. Multiple Regression

Formula

Worked Example

Complete Worked Example: Planning a Study from Scratch

Scenario

Step 1: Calculate effect size

Step 2: Set parameters

Step 3: Calculate sample size

Step 4: Adjust for dropout

Conclusion

Test Your Understanding

View Step-by-Step Solution

View Step-by-Step Solution

Think About It

I've thought about it - Show me the solution

Sample Size Formula Explained with Examples

The General Framework

Sample Size Formulas by Test Type

1. Two-Sample t-Test (Independent Groups)

Formula

Worked Example

2. Paired t-Test

Formula

Worked Example

3. Two-Proportion Test

Formula

Worked Example

4. One-Way ANOVA

Formula