How many respondents does your survey need, and how will you defend that number when an examiner asks? “As many as I could get” is the answer examiners hear most and like least. A sample size is defensible when it follows from one of three things: a margin of error you want for an estimate, an effect size you want to be able to detect in a test, or a published table that applies your assumptions for you.
This article is part of our guide How to Write a Master’s Thesis by the Mastermind PhD team and relates to our Statistical Analysis service. Sources last checked: November 2026.
This guide works through all three with the same population of 2,000 employees, explains the finite population correction, shows four a priori power analyses in G*Power with their exact outputs, adds Green’s rule for regression, and ends with a methodology paragraph you can adapt.
What is sample size calculation? Sample size calculation is the process of determining, before data collection, how many units must be observed for the study’s estimates to reach a chosen precision or for its hypothesis tests to reach a chosen statistical power. For descriptive surveys it is driven by the confidence level, the margin of error and the expected proportion; for hypothesis tests it is driven by the significance level, the desired power (usually .80) and the smallest effect size worth detecting.
Why do the sample size calculation before you collect?
Two different questions hide behind “how many”. A descriptive question (“what proportion of employees intend to leave within a year?”) needs enough cases for the estimate to be precise: that is a margin-of-error calculation. An inferential question (“do intentions differ between departments?”) needs enough cases for the test to detect a real difference: that is a power calculation. Many theses need both, and the larger of the two numbers wins.
A sample size calculation done afterwards does not repair a small sample. A “post hoc power” figure computed from a non-significant result is a function of the p-value and adds no information, so examiners discount it. Calculate first, record the inputs, and keep the output screenshot for your appendix.
How does Cochran’s formula work?
Cochran (1977) gives the sample size for estimating a proportion in a large population:
n0 = Z² × p × (1 − p) / e²
where Z is the standard normal value for the confidence level, p is the expected proportion and e is the margin of error. When p is unknown, 0.5 is used because it maximizes p(1 − p) and therefore gives the most conservative sample.
Worked example at 95% confidence, e = 0.05, p = 0.5: n0 = 1.96² × 0.5 × 0.5 / 0.05² = 3.8416 × 0.25 / 0.0025 = 384.16, rounded up to 385.
| Confidence level | Z | Margin of error 3% | Margin 5% | Margin 7% |
|---|---|---|---|---|
| 90% | 1.645 | 752 | 271 | 139 |
| 95% | 1.960 | 1,068 | 385 | 196 |
| 99% | 2.576 | 1,844 | 664 | 339 |
Halving the margin of error quadruples the sample, which is why most master’s surveys settle on 5%.
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What does the finite population correction change?
Cochran’s n0 assumes an effectively infinite population. When the population N is known and not much larger than n0, the correction
n = n0 / (1 + (n0 − 1) / N)
reduces the required sample. For N = 2,000: n = 384.16 / (1 + 383.16 / 2,000) = 384.16 / 1.1916 = 322.4, rounded up to 323.
Yamane (1967) offers the shortcut n = N / (1 + N × e²), which is the same calculation with Z set to 2 instead of 1.96, so it runs slightly higher. For N = 2,000 and e = 0.05: n = 2,000 / (1 + 2,000 × 0.0025) = 2,000 / 6 = 333.3, rounded up to 334.
| Population N | Cochran with correction | Yamane | Krejcie and Morgan table |
|---|---|---|---|
| 200 | 132 | 134 | 132 |
| 500 | 218 | 223 | 217 |
| 1,000 | 278 | 286 | 278 |
| 2,000 | 323 | 334 | 322 |
| 5,000 | 357 | 371 | 357 |
| 10,000 | 370 | 385 | 370 |
| 100,000 | 383 | 399 | 383 |
All three columns assume 95% confidence, a 5% margin and p = 0.5. The differences are a few cases; what matters is naming the method you used and its assumptions.
Where does the Krejcie and Morgan table come from?
Krejcie and Morgan (1970) published the table that many universities still reproduce in their methodology courses. It is not a separate theory: it applies the formula
s = χ² N P (1 − P) / [d² (N − 1) + χ² P (1 − P)]
with χ² = 3.841 (the chi-square value for one degree of freedom at the .05 level, which equals 1.96²), P = 0.5 and d = 0.05. For N = 2,000 the formula gives 322.4, reported in the table as 322. The table tops out at 384 for populations of a million or more, which is why “384” appears in so many theses without explanation. If you cite the table, also state the three assumptions behind it; if your margin or confidence level differs, use the formula instead.
How do you run an a priori power analysis in G*Power?
G*Power (Faul et al., 2007) is a free program developed at Heinrich Heine University Düsseldorf that calculates the sample size needed to detect a given effect with a given power. The a priori procedure has four inputs: the test, the effect size, α (almost always .05, two-tailed) and the desired power (conventionally .80). Effect size conventions come from Cohen (1988): d of 0.2, 0.5 and 0.8 for small, medium and large mean differences; r of 0.1, 0.3 and 0.5 for correlations; f of 0.10, 0.25 and 0.40 for ANOVA; and f² of 0.02, 0.15 and 0.35 for regression.
| Planned analysis | G*Power test family and test | Effect size input | Other inputs | Required sample |
|---|---|---|---|---|
| Compare two independent groups | t tests, means: difference between two independent means | d = 0.5 (medium) | α = .05 two-tailed, power = .80, allocation 1:1 | 64 per group, 128 in total |
| Relationship between two continuous variables | Exact, correlation: bivariate normal model | r = 0.3 (medium) | α = .05 two-tailed, power = .80 | 84 |
| Compare three groups | F tests, ANOVA: fixed effects, omnibus, one-way | f = 0.25 (medium) | α = .05, power = .80, 3 groups | 159 in total (53 per group) |
| Predict an outcome from four predictors | F tests, linear multiple regression: fixed model, R² deviation from zero | f² = 0.15 (medium) | α = .05, power = .80, 4 predictors | 85 |
Choose the effect size from previous studies on your topic whenever possible; a “medium” effect is a convention, not a finding. If the only published study reported d = 0.35, planning for d = 0.5 will leave you underpowered. Entering d = 0.35 in the first row raises the requirement to about 130 per group.
Reporting template: “An a priori power analysis in G*Power 3.1 (Faul et al., 2007) for an independent-samples t test with a medium effect (d = 0.5), α = .05 (two-tailed) and power = .80 indicated a minimum of 64 participants per group. To allow for a 20% non-response rate, 160 questionnaires were distributed.”
Which rule applies to regression?
Green (1991) tested the common rules of thumb for multiple regression and recommended N ≥ 50 + 8m to test the overall R² and N ≥ 104 + m to test individual predictors, where m is the number of predictors. With four predictors that means 82 and 108 respectively; the G*Power figure of 85 for a medium effect sits between them. Report whichever rule you followed and why, and remember that Green’s rules assume a medium effect; smaller expected effects need the full power analysis.
For exploratory factor analysis and structural equation modeling, the sample size calculation depends on the number of items and factors rather than on an effect size, and the rules are different; our article on exploratory factor analysis covers them.
How do you adjust for non-response and attrition?
The calculated number is the achieved sample you need, not the number of invitations. Divide by the expected response rate: a target of 323 with an expected 40% response means about 808 invitations (323 / 0.40). For a two-wave study with 20% expected dropout, recruit 323 / 0.80 = 404 at wave one. State the assumed rate and its source (a previous study in the same setting, or your pilot) in the methodology chapter.
Which mistakes do examiners flag?
- Using 384 or 385 for a population of 600 without the finite population correction, or using Yamane for a nationwide population of millions without saying so.
- Calculating the size for the whole sample, then running comparisons on subgroups of 20.
- Reporting “a power analysis was conducted” without the effect size, α, power and software version.
- Confusing the 5% margin of error with the .05 significance level; they belong to different calculations.
- Computing post hoc power from a non-significant result and presenting it as justification.
- Treating 30 as a universal minimum. It is a rule of thumb for the central limit theorem, not a sample size for a thesis.
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Frequently asked questions
Is a sample of 30 enough for a master’s thesis?
Only for a pilot or a very small, fully enumerated population. Thirty cases split into two groups of 15 give power of only about .75 even for a very large effect (d = 1.0), and a margin of error near 18% for a proportion. Use a formula or G*Power and let the question decide.
Which online calculator is acceptable?
Any calculator that shows its inputs (confidence level, margin, population size, expected proportion) is acceptable if you report those inputs and the formula it implements. For hypothesis tests, G*Power is the standard because an examiner can reproduce its output.
What if I cannot reach the calculated sample size?
Report the target, the achieved number and the consequence: wider confidence intervals or reduced power to detect small effects. Do not quietly change the margin of error after collection to make the achieved sample look sufficient.
Do I need a power analysis for a purely descriptive survey?
No. A descriptive survey needs the margin-of-error calculation with the finite population correction. Power analysis applies when you plan tests of differences or relationships, which most theses do in at least one research question.
Does Likert scale data change the sample size calculation?
The formulas above treat summed or averaged scale scores as continuous, which is the usual practice. If you plan non-parametric tests on single items, G*Power can compute the sample size for the Wilcoxon-Mann-Whitney test directly (under t tests, means), and the requirement is somewhat higher than for the t test.
Sources
- Cochran, W. G. (1977). Sampling techniques (3rd ed.). Wiley.
- Yamane, T. (1967). Statistics: An introductory analysis (2nd ed.). Harper and Row.
- Krejcie, R. V., & Morgan, D. W. (1970). Determining sample size for research activities. Educational and Psychological Measurement, 30(3), 607–610. https://doi.org/10.1177/001316447003000308
- Faul, F., Erdfelder, E., Lang, A.-G., & Buchner, A. (2007). GPower 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior Research Methods, 39*(2), 175–191. https://doi.org/10.3758/BF03193146
- Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum.
- Green, S. B. (1991). How many subjects does it take to do a regression analysis? Multivariate Behavioral Research, 26(3), 499–510. https://doi.org/10.1207/s15327906mbr2603_7
- Heinrich Heine University Düsseldorf. GPower: Statistical power analyses for Mac and Windows*. https://www.psychologie.hhu.de/arbeitsgruppen/allgemeine-psychologie-und-arbeitspsychologie/gpower (accessed October 2026)
Sources last checked: November 2026. This article is general educational guidance and does not replace your university regulations or your supervisor’s instructions. How we prepare our content.