Open any ten survey-based theses and you will meet the number 384, or the Yamane formula, several times over. Ask the author what the number is for and the usual answer is “so the sample is representative”. In fact 384 answers a very narrow question: how many respondents you need to estimate a single proportion within ±5 percentage points at 95% confidence. If your study compares two groups or fits a regression with ten predictors, that figure may be far too large or far too small.
Sample size is not a box to tick in the methods chapter. It depends on your question, on the size of effect you want to detect, and on how the data are gathered. This guide walks through each situation, with the inputs that you have to decide for yourself, because no software can decide them for you.
Where 384 actually comes from
The formula for estimating a proportion is n = z² × p × (1 − p) / e². With z = 1.96 for 95% confidence, p = 0.5 as the most conservative assumption, and a margin of error e = 0.05, you get 3.8416 × 0.25 / 0.0025 = 384.16, rounded up to 385.
The Yamane formula, n = N / (1 + N × e²), is essentially a simplified version of the same idea, assuming p = 0.5 and z close to 2, with a finite population correction built in. It is reasonable when your goal is to estimate a proportion. It has no justification when your goal is hypothesis testing or modelling, because it contains nothing about effect size or number of variables.
Three numbers you must choose
- Alpha: usually 0.05. If you are testing many hypotheses at once, consider a stricter threshold.
- Power: the probability of detecting an effect if it truly exists. The customary floor is 0.80; many biomedical funders expect 0.90.
- Effect size: the hardest input. Take it from prior studies, a pilot, or the smallest difference that would matter in practice — not from a default “medium” label because the software suggested it.
Effect size drives everything. For two independent groups, two-tailed alpha of 0.05 and power of 0.80: d = 0.8 needs about 26 per group, d = 0.5 needs 64 per group, and d = 0.3 needs about 176 per group. Halve the expected effect and the required sample roughly quadruples.
Quick reference by analysis type
| Goal | Assumptions | Approximate n |
|---|---|---|
| Estimate a proportion | p = 0.5, ±5%, 95% | 385 |
| Estimate a proportion | p = 0.5, ±3%, 95% | 1,068 |
| Compare two means | d = 0.5, alpha 0.05, power 0.80 | 64 per group |
| Correlation | r = 0.3, alpha 0.05, power 0.80 | about 84 |
| Correlation | r = 0.2, alpha 0.05, power 0.80 | about 194 |
You can reproduce all of these in G*Power, which is free, or with the pwr package in R. When reporting, name the software, the test and all four inputs so a reader can check your arithmetic.
Regression and factor analysis: rules of thumb and their limits
For multiple linear regression, the “50 + 8m” rule (m being the number of predictors) is often quoted for testing the overall model; testing individual coefficients needs more. For exploratory factor analysis, multivariate textbooks commonly recommend at least five cases per item, preferably ten. A 30-item scale therefore needs 150–300 usable responses.
Treat these as floors, not targets. They know nothing about whether your effect is small or large. If time allows, run a power analysis for the most important coefficient in the model (f² = 0.02 counts as a small effect) and take whichever figure is larger.
A fully worked survey example
Question: what proportion of first-year students at one university show signs of academic stress? Population N = 2,000. Sampling is by class, each with about 25 students.
- Base size: 384.16 (p = 0.5, ±5%, 95%).
- Finite population correction: n' = 384.16 / (1 + 383.16 / 2,000) = 384.16 / 1.1916 ≈ 323.
- Design effect for cluster sampling: DEFF = 1 + (m − 1) × ICC. With m = 25 and an assumed ICC of 0.05, DEFF = 1 + 24 × 0.05 = 2.2. Required: 323 × 2.2 ≈ 711.
- Non-response: expecting 80% usable returns, 711 / 0.8 ≈ 889. In practice, about 36 classes.
Step three is the one most theses skip. Students in the same class resemble one another more than students across the university, so 25 people from one class carry less information than 25 scattered across campus. Ignore the design effect and your reported confidence intervals will be narrower than reality.
When you cannot reach the calculated size
This is common: the population is 80 intensive-care nurses, or you can only reach 120 firms. Do not tweak inputs until the formula spits out the number you already have; reviewers spot this immediately. Instead:
- Report your actual sample and compute the minimum detectable effect for that size (a sensitivity analysis in G*Power).
- Emphasise estimates with confidence intervals rather than p-values alone.
- With a small population, consider a census rather than a sample.
- Never read a non-significant result from an underpowered study as evidence of “no difference”.
What you do not need
You do not need post hoc power computed from your observed effect; it is a re-expression of the p-value and adds nothing. You do not need a sample far beyond the calculation “just to be safe” when every response is costly; very large samples make trivial differences statistically significant. And you do not need to cite three different formulas in your methods chapter; one that matches your question is enough.
A model paragraph for your methods section
“Sample size was determined in G*Power 3.1 for a two-tailed independent-samples t-test with alpha = 0.05 and power = 0.80, assuming an effect size of d = 0.5 based on a prior pilot study. The minimum required was 64 participants per group. Allowing for 20% attrition, we recruited 80 per group.”
Everything a reader needs to verify your decision is in that paragraph. Next step: identify your primary analysis, find the closest published study that reports a comparable effect size, and run G*Power at three effect sizes — small, medium and your expected value — so you can see how sensitive your target is before you lock in a data-collection plan.
Câu hỏi thường gặp
Why is 384 the standard sample size?
It is the size needed to estimate a single proportion within ±5% at 95% confidence when the true proportion is unknown. It is not a universal standard and often does not fit comparative or regression studies.
Can I use the Yamane formula for regression?
It is not appropriate. Yamane is designed for estimating proportions and ignores effect size and number of predictors. Use a power analysis, or at least a predictor-based rule, for regression.
How do I calculate sample size in G*Power?
Choose the test family and statistical test, select a priori analysis, then enter alpha, desired power and the expected effect size. G*Power returns the minimum sample, which you then inflate for expected non-response.
What if my sample is smaller than the power analysis suggests?
Report it honestly and run a sensitivity analysis to show the smallest effect you could detect. Present confidence intervals and avoid interpreting non-significant results as proof of no effect.
How many participants do I need for factor analysis?
A common guideline is at least five responses per item, preferably ten, and rarely fewer than 100–150 overall. Strong, well-defined factors can work with less; weak ones need more.
