Sample Size Determination

What Is Sample Size Determination?

Sample size determination is the process of calculating how many participants you need to recruit for your study before you begin collecting data. It ensures your study is large enough to detect a meaningful effect (if one exists) while avoiding unnecessary cost and effort from over-recruiting.

Getting the sample size right is one of the most important decisions in research design. Too small, and your study lacks the statistical power to find real effects. Too large, and you waste time, money, and participant goodwill.

"How many participants do I need?" is the single most common statistics question researchers ask. The answer is never a guess — it's a calculation.

When to Use It

Sample size determination should happen before you collect any data, during the study design phase. You need it when:

• Writing a grant proposal (funders require justification for sample size)
• Submitting to an IRB/ethics board (to ensure you're not exposing unnecessary participants to risk)
• Planning a dissertation or thesis study
• Designing a clinical trial (regulatory agencies require formal power calculations)
• Planning any study where you want to make a credible statistical inference

The Inputs You Need

To calculate sample size, you need four pieces of information. Three are chosen by the researcher; the fourth is what you solve for:

Formula

For an Independent Samples t-Test (Two Groups, Equal Size)

$$n = \frac{2\left(z_{1-\alpha/2} + z_{1-\beta}\right)^2}{d^2}$$

Where:

• $n$ = required sample size per group
• $z_{1-\alpha/2}$ = critical $z$-value for your significance level (1.96 for $\alpha = .05$, two-tailed)
• $z_{1-\beta}$ = critical $z$-value for your desired power (0.84 for 80% power; 1.28 for 90% power)
• $d$ = expected Cohen's d effect size

For a Paired Samples t-Test

$$n = \frac{\left(z_{1-\alpha/2} + z_{1-\beta}\right)^2}{d_z^2}$$

Where $d_z$ is the effect size for the difference scores (mean difference divided by the standard deviation of differences). Note that you need only one group, so $n$ is the total sample.

For One-Way ANOVA ($k$ Groups)

$$N = \frac{k\left(z_{1-\alpha/2} + z_{1-\beta}\right)^2}{f^2}$$

Where $f$ is Cohen's $f$ effect size and $N$ is the total sample size across all $k$ groups.

For Chi-Square Test of Independence

$$N = \frac{\left(z_{1-\alpha/2} + z_{1-\beta}\right)^2}{w^2}$$

Where $w$ is Cohen's $w$ effect size for chi-square tests.

Quick Reference Table

For $\alpha = .05$ (two-tailed) and power = .80, the per-group sample size for an independent t-test is:

Assumptions

1. You have a reasonable effect size estimate. This is the hardest part. Options include:
   • Results from a pilot study
   • Meta-analyses in your field
   • The smallest effect size of interest (SESOI) — the smallest effect that would be practically meaningful
2. You know which statistical test you will use. Different tests have different formulas.
3. Your design matches the formula. Equal vs. unequal group sizes, number of groups, number of predictors, etc.
4. You plan for attrition. The formula gives you the minimum needed for analysis; recruit extra to compensate for dropouts.

Worked Example

Scenario: An educational researcher wants to compare reading comprehension scores across three teaching methods (phonics-based, whole-language, and blended). A recent meta-analysis suggests a medium effect ($f = 0.25$). She wants 80% power at $\alpha = .05$.

Step 1: Identify parameters.

• $k = 3$ groups
• $f = 0.25$ (medium effect, Cohen's $f$)
• $\alpha = .05$, so $z_{0.975} = 1.96$
• Power = .80, so $z_{0.80} = 0.84$

Step 2: Apply the formula.

$$N = \frac{k(z_{1-\alpha/2} + z_{1-\beta})^2}{f^2} = \frac{3(1.96 + 0.84)^2}{0.25^2}$$

$$N = \frac{3(2.80)^2}{0.0625} = \frac{3 \times 7.84}{0.0625} = \frac{23.52}{0.0625} = 376.32$$

Step 3: Round up and distribute across groups.

$$N = 378 \text{ total} \quad (126 \text{ per group})$$

Step 4: Adjust for attrition. Expecting 10% dropout:

$$N_{adjusted} = \frac{378}{1 - 0.10} = \frac{378}{0.90} = 420 \text{ total} \quad (140 \text{ per group})$$

Interpretation: The researcher needs approximately 140 participants per group (420 total) to have 80% power for detecting a medium effect across three groups, accounting for 10% attrition.

Note: This approximation is useful for planning. For precise calculations, use software such as G*Power, which accounts for the exact non-central $F$-distribution. G*Power yields $n = 159$ per group for this scenario (the $z$-approximation is less precise for ANOVA).

Interpretation

A sample size calculation tells you the minimum number of participants needed under your stated assumptions. Key points for interpretation:

• The result is only as good as your effect size estimate. If the true effect is smaller than assumed, your study will be underpowered.
• Always round up to the next whole number — you cannot have 62.7 participants.
• For designs with groups, ensure each group meets the minimum, not just the total.
• Consider running a sensitivity analysis: calculate required $n$ for a range of plausible effect sizes (e.g., $d = 0.30$, $0.50$, $0.70$) to see how sensitive your design is.

Common Mistakes

1. Using "medium" as a default without justification. Reviewers will ask why you expect a medium effect. Cite prior research or explain your reasoning.
2. Calculating sample size after data collection. This is backwards. Post hoc sample size calculations are meaningless — the data are already collected.
3. Forgetting the design details. A repeated-measures design needs fewer participants than a between-subjects design. A 2x3 factorial ANOVA needs a different calculation than a one-way ANOVA.
4. Ignoring attrition, missing data, and exclusions. Always inflate your target $n$ to account for real-world data loss.
5. Confusing per-group and total sample size. If the formula gives $n = 64$ per group and you have two groups, you need 128 total — not 64 total.
6. Not specifying directionality. One-tailed tests require smaller samples than two-tailed tests, but they are harder to justify. Be clear about which you are using.

How to Report in APA Format

In the Method section, under Participants or Data Analysis Plan:

A priori sample size estimation was performed using G*Power 3.1 (Faul et al., 2009). For a one-way ANOVA with three groups, a medium effect size ($f$ = 0.25), $\alpha$ = .05, and power = .80, the minimum required sample was 159 per group (477 total). Anticipating approximately 10% attrition, we targeted 175 per group (525 total).

For a two-group design:

Based on prior research indicating a medium effect ($d$ = 0.50; Smith & Jones, 2024), a power analysis for an independent samples t-test (two-tailed, $\alpha$ = .05, power = .80) indicated a minimum of 64 participants per group. We recruited 75 per group to account for potential attrition.

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