"How many participants do I need?" is one of the first questions in any research proposal, and one ethics committees and supervisors will ask you to justify. The answer depends on what you want to do with the data:
- If you want to estimate something, such as the percentage of customers who are satisfied, you need a sample size based on the margin of error.
- If you want to test a hypothesis, such as whether two groups differ, you need a sample size based on statistical power.
This guide covers both, with formulas, worked examples and quick-reference tables.
In this guide
Part 1: Sample size for surveys (margin of error)
The formula
To estimate a proportion with a given margin of error:
n₀ = z² × p × (1 − p) ÷ e²
- z is the z-value for your confidence level: 1.96 for 95%, 2.576 for 99%.
- p is the expected proportion. If you don't know it, use 0.5, which gives the largest (safest) sample size.
- e is the margin of error as a decimal (0.05 for ±5%).
Example: 95% confidence, ±5% margin, p = 0.5:
n₀ = 1.96² × 0.5 × 0.5 ÷ 0.05² = 384.2 → 385 respondents.
Finite population correction
If your population is small (for example, 2,000 employees in one company), you need fewer respondents:
n = n₀ ÷ (1 + (n₀ − 1) ÷ N)
For N = 2,000: n = 384.2 ÷ (1 + 383.2 ÷ 2,000) = 323 respondents.
Adjust for response rate
If you expect only 30% of invited people to respond, invite 323 ÷ 0.30 ≈ 1,077 people.
Quick-reference table (95% confidence, p = 0.5)
| Population size | ±3% | ±5% | ±10% |
|---|---|---|---|
| 500 | 341 | 218 | 81 |
| 1,000 | 517 | 278 | 88 |
| 5,000 | 880 | 357 | 95 |
| 10,000 | 965 | 370 | 96 |
| 100,000+ | 1,056 | 383 | 96 |
Beyond about 100,000, population size hardly matters: a national survey and a city survey need roughly the same sample for the same precision. The free survey sample size calculator does these calculations, including the response-rate adjustment.
The margin-of-error formula assumes a random sample. A convenience sample of 2,000 people recruited on social media is not more representative than a random sample of 400; it is just more precise about the wrong population. Describe your sampling method honestly.
Part 2: Sample size for hypothesis tests (power analysis)
The four ingredients
A power analysis links four quantities. Fix any three and you can solve for the fourth:
- Effect size: how big a difference or relationship you want to be able to detect (Cohen's d, f, r, f² or w).
- Significance level (α): usually .05.
- Power (1 − β): the probability of detecting the effect if it really exists. 80% is the conventional minimum; 90% is better for confirmatory studies.
- Sample size (n).
An a priori power analysis, done before data collection, solves for n. This is what proposals and ethics applications need.
Choosing the effect size
This is the hardest part, and the most important. In order of preference:
- The smallest effect that would matter in practice, for example a 5-point improvement on a clinical scale.
- Effect sizes from previous studies or a meta-analysis on the same topic. Published effects are often inflated, so consider using a somewhat smaller value.
- Cohen's conventions (small, medium, large) as a last resort.
| Test | Effect size | Small | Medium | Large |
|---|---|---|---|---|
| t-test | d | 0.20 | 0.50 | 0.80 |
| ANOVA | f | 0.10 | 0.25 | 0.40 |
| Correlation | r | 0.10 | 0.30 | 0.50 |
| Multiple regression | f² | 0.02 | 0.15 | 0.35 |
| Chi-square | w | 0.10 | 0.30 | 0.50 |
Quick-reference table (α = .05 two-tailed, power = .80, medium effect)
| Analysis | Effect size | Total sample needed |
|---|---|---|
| Independent t-test | d = 0.50 | 128 (64 per group) |
| Paired t-test | dz = 0.50 | 34 |
| One-way ANOVA, 3 groups | f = 0.25 | 159 (53 per group) |
| Pearson correlation | r = 0.30 | about 85 |
| Multiple regression, 3 predictors | f² = 0.15 | 77 |
| Multiple regression, 5 predictors | f² = 0.15 | 92 |
| Chi-square, df = 1 | w = 0.30 | 88 |
For a small effect the numbers grow fast: an independent t-test with d = 0.20 needs about 394 per group. For a large effect (d = 0.80), about 26 per group is enough.
Worked example
You plan to compare anxiety scores between a mindfulness group and a control group. A meta-analysis suggests d ≈ 0.45. With α = .05, two-tailed, and 90% power, the power analysis calculator gives 105 participants per group. Allowing for 15% dropout, you recruit 105 ÷ 0.85 ≈ 124 per group.
Rules of thumb for complex models
Power analysis for complex models is harder, so researchers often fall back on rules of thumb. Use them cautiously, and say that you did:
- Multiple regression: at least 10–15 cases per predictor, or N ≥ 50 + 8k for testing the overall model (Green, 1991).
- Logistic regression: at least 10 events (cases in the smaller outcome group) per predictor.
- Exploratory factor analysis: at least 5–10 respondents per item, and 200+ overall when communalities are modest.
- CB-SEM: about 200 as a common minimum for moderately complex models with good loadings; more for complex models or non-normal data. Monte Carlo simulation gives a model-specific answer.
- PLS-SEM: the inverse square root method (Kock & Hadaya, 2018) or a power analysis for the most complex regression in the model.
How to report your sample size justification
An a priori power analysis indicated that 105 participants per group would provide 90% power to detect an effect of d = 0.45 (based on a recent meta-analysis) in an independent-samples t-test with α = .05 (two-tailed). Allowing for 15% attrition, we aimed to recruit 124 participants per group.
Frequently asked questions
What sample size do I need for a survey?
For a large population, 95% confidence and a ±5% margin of error, you need about 385 respondents. Smaller populations need fewer once you apply the finite population correction, and you should invite more people to allow for non-response.
What is statistical power?
Power is the probability that a study will detect an effect of a given size if it really exists. 80% is the usual minimum, which means a 20% chance of missing a real effect.
Why is 80% power the standard?
It is a convention proposed by Jacob Cohen, balancing the cost of a larger sample against the risk of missing a real effect. Many funders now prefer 90% power for confirmatory studies.
Should I do a post hoc power analysis?
Generally no. Power calculated from the observed effect after the study adds no information beyond the p-value. Report confidence intervals for your effects instead.
Is there a free alternative to G*Power?
Yes. The free power analysis calculator on FreeStatisticalTools.com runs a priori power analyses for t-tests, ANOVA, correlation, proportions, chi-square and regression in your browser, with a power curve.