‘How did you decide on your sample size?’ is one of the questions thesis examiners ask most often. A good answer names a recognised method, shows the numbers that went into it, and explains how you handled practical limits such as expected non-response. This guide explains the three methods students use most: Cochran’s formula for surveys, Yamane’s simplified formula, and power analysis for experiments and group comparisons. It includes worked examples you can check with the free sample size calculator and power analysis calculator.
First, decide what your sample size is for
The right method depends on your research question:
- Descriptive surveys that estimate a proportion or percentage in a population (for example, the percentage of teachers who use technology daily) are usually sized with Cochran’s or Yamane’s formula. These methods focus on the precision of an estimate.
- Studies that test a hypothesis, such as comparing groups, testing a correlation or running a regression, should be sized with a power analysis. This asks how many participants are needed to have a good chance of detecting an effect of a given size.
Many survey-based theses also test hypotheses. In that case it is good practice to calculate both and use the larger number, or at least to show that your sample gives adequate power for your main analysis.
Cochran’s formula
Cochran (1977) gave the classic formula for estimating a proportion with a specified margin of error:
n₀ = Z² × p × (1 − p) / e²
- Z is the critical value for your confidence level: 1.96 for 95% (the usual choice), 2.576 for 99%.
- p is the expected proportion. If you do not know it, use 0.5, which gives the largest (most conservative) sample.
- e is the margin of error, for example 0.05 for ±5 percentage points.
With 95% confidence, p = 0.5 and e = 0.05: n₀ = 1.96² × 0.5 × 0.5 / 0.05² = 384.16, which rounds up to 385. This is the familiar ‘385’ seen in many theses, and it applies to large populations.
Finite population correction
When the population is small or known, Cochran’s finite population correction reduces the required sample:
n = n₀ / [1 + (n₀ − 1) / N]
For a population of N = 1,000 university staff: n = 384.16 / (1 + 383.16 / 1000) = 277.8, so you need 278 respondents. This matches the value of 278 for a population of 1,000 in the widely used table by Krejcie and Morgan (1970), which is based on the same principle.
Yamane’s formula
Yamane (1967) presented a simplified formula that assumes 95% confidence and p = 0.5:
n = N / (1 + N × e²)
For N = 1,000 and e = 0.05: n = 1000 / (1 + 1000 × 0.0025) = 285.7, rounded up to 286. Yamane’s result is slightly larger than Cochran’s corrected value, because the formula is an approximation. It is popular in social science theses because it is easy to apply, but it can only be used when the population size is known and it does not let you change the confidence level. If your supervisor has no preference, Cochran’s formula with the finite population correction is the more general choice.
Power analysis for hypothesis tests
Power analysis links four quantities: the effect size, the significance level (α), the power (1 − β) and the sample size. Fix any three and you can calculate the fourth. For an a priori sample size calculation you choose:
- α: usually .05, two-tailed.
- Power: usually .80, meaning an 80% chance of detecting the effect if it exists (Cohen, 1988). Some fields use .90.
- The expected effect size: ideally taken from previous studies or a pilot study. If there is no better information, Cohen’s (1988) conventions are often used: for t tests, d = 0.20 (small), 0.50 (medium) and 0.80 (large).
Typical results for an independent samples t test with α = .05 (two-tailed) and power = .80 are shown below; you can reproduce them with our power analysis calculator or with G*Power (Faul et al., 2007).
| Expected effect (Cohen’s d) | Participants per group | Total N |
|---|---|---|
| 0.20 (small) | 394 | 788 |
| 0.50 (medium) | 64 | 128 |
| 0.80 (large) | 26 | 52 |
The table shows why the expected effect size matters so much: halving the effect roughly quadruples the sample needed. For other analyses, G*Power can calculate sample sizes for correlations, ANOVA and regression. For example, detecting a correlation of r = .30 with α = .05 and power = .80 requires about 84 participants, and a one-way ANOVA with three groups and a medium effect (f = 0.25) requires about 159 participants in total (53 per group).
Rules of thumb (use with caution)
You may come across rules such as ‘at least 30 per group’ or, for multiple regression, N ≥ 50 + 8m, where m is the number of predictors (Tabachnick & Fidell, 2019). These can be useful sanity checks, but they are not a substitute for a power analysis when you test hypotheses, and examiners generally prefer a calculated justification.
Adjusting for non-response
The formulas give the number of completed, usable responses you need. If you expect that only some invited people will respond, inflate the number you invite:
Number to invite = required n / expected response rate
If you need 278 completed questionnaires and expect a 70% response rate, invite at least 278 / 0.70 = 397 people. State the response rate you assumed and, in the results chapter, report the response rate you actually achieved.
How to write your sample size justification
A good justification in the methodology chapter is short and specific. Examples:
The target population consisted of 1,000 secondary school teachers in the district. Using Cochran’s (1977) formula with a 95% confidence level, a 5% margin of error and an assumed proportion of .50, with the finite population correction, the minimum required sample was 278. Allowing for an expected response rate of 70%, questionnaires were distributed to 400 teachers.
An a priori power analysis was conducted in G*Power 3.1 (Faul et al., 2007) for an independent samples t test. Assuming a medium effect size (d = 0.50) based on previous studies, α = .05 (two-tailed) and power of .80, the required sample was 64 participants per group (N = 128).
Common mistakes
- Using 384/385 for a small, known population without applying the finite population correction.
- Using Cochran’s or Yamane’s formula alone to justify an experimental study; these formulas do not consider statistical power.
- Choosing a ‘large’ effect size simply to get a smaller sample, without evidence that a large effect is realistic.
- Confusing the number invited with the number of usable responses.
- Calculating the sample size after the data have been collected and presenting it as if it had been planned. If you do a power calculation afterwards, describe it honestly as a sensitivity analysis.
- Ignoring subgroup analyses: if you plan to compare subgroups, each subgroup needs enough participants.
Quick decision guide
- Estimating a percentage in a large population → Cochran (385 at 95% ± 5%).
- Known, smaller population → Cochran with finite population correction (or Yamane, if your department prefers it).
- Comparing groups, testing correlations or regression → power analysis with a justified effect size.
- Always → adjust for expected non-response and report the achieved response rate.
Once your data are in, our guide on choosing a statistical test will help you plan the analysis.
Roman Urdu mein khulasa
Thesis ka sample size do tareeqon se nikala jata hai. Agar survey se kisi population ka percentage estimate karna hai to Cochran ka formula istemal karein: 95% confidence aur ±5% error par bari population ke liye 385, aur 1,000 ki population ke liye correction ke baad 278. Yamane ka formula (N = 1,000 par 286) asaan hai magar sirf maloom population ke liye hai. Agar groups compare karne hain ya hypothesis test karni hai to power analysis karein (d = 0.50, α = .05, power = .80 par har group mein 64). Response rate ka andaza laga kar zyada logon ko questionnaire bhejein.
Frequently asked questions
Why do so many theses use 384 or 385?
It is Cochran’s sample size for a large population with 95% confidence, ±5% margin of error and p = 0.5. For a smaller known population, apply the finite population correction.
Should I use Cochran or Yamane?
Both give similar results for known populations. Cochran’s formula is more general because you can change the confidence level and the expected proportion; Yamane’s formula is a simplified version.
What sample size do I need for a t test?
With α = .05, power = .80 and a medium effect (d = 0.50), you need 64 participants per group. Smaller expected effects need much larger samples.
Where do I get the effect size for a power analysis?
Preferably from previous studies or a pilot study on a similar population. If none exist, use Cohen’s conventions and say so.
What if I cannot reach the required sample size?
Report the achieved sample honestly, discuss the reduced precision or power as a limitation, and consider reporting a sensitivity analysis showing the smallest effect your sample could detect.
Related calculators
- Sample Size Calculator for Thesis
- Power Analysis Calculator
- Cohen’s d Effect Size Calculator
- Test chooser: which statistical test should I use?
References
- Cochran, W. G. (1977). Sampling techniques (3rd ed.). Wiley.
- Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
- Faul, F., Erdfelder, E., Lang, A.-G., & Buchner, A. (2007). G*Power 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
- 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
- Tabachnick, B. G., & Fidell, L. S. (2019). Using multivariate statistics (7th ed.). Pearson.
- Yamane, T. (1967). Statistics: An introductory analysis (2nd ed.). Harper & Row.
Written and maintained by the ScholarCalc team. Last reviewed: November 2026.