For a PhD, sample size is calculated with Cochran’s formula for surveys, a power analysis for comparisons or relationships, and saturation for qualitative interviews. But in synopsis presentations and DC meetings, many scholars are still giving a round number that somebody else used. Examiners have learnt this, and they ask how the number was worked out. If you can explain your sample size in two sentences, with the inputs named, you are safe in the pre-submission seminar and in the viva-voce also.
Mostly PhD research needs one of three methods. Cochran’s formula is for surveys that estimate a proportion or a mean. Power analysis in G*Power is for studies that compare groups or test relationships. Structural equation modelling has its own set of rules and methods. We go through each with actual numbers, and end with what to write in your methodology chapter.
Any calculator can do the arithmetic. The inputs are judgement calls about your own study, so sit with your guide and discuss about them. For clinical or experimental work, show them to a statistician also. Last reviewed September 2026.
Which sample size method fits your study
Which method is right for you depends on what your study wants to do. Take a survey that wants to estimate how many teachers are using a smart classroom. A study that wants to find a difference between two teaching methods has a different problem altogether.
| Your study | Method to use | What you need to decide |
|---|---|---|
| Survey estimating a proportion or mean in a population (e.g. awareness of a scheme among farmers) | Cochran’s formula, with finite population correction if the population is small | Confidence level, margin of error, expected proportion |
| Comparing groups or testing a relationship (t-test, ANOVA, correlation, regression) | Power analysis, e.g. in G*Power | Test, expected effect size, significance level, power |
| Structural equation modelling (AMOS, SmartPLS, lavaan) | Power analysis for the model, or the inverse square root method for PLS-SEM, checked against rules of thumb | Number of constructs and paths, smallest path you expect to detect |
| Experiment or clinical study | Power analysis based on the primary outcome | Clinically or practically meaningful effect, drop-out rate |
| Qualitative interviews or focus groups | No formula. Justify it by saturation and the purpose of the study | How you will decide that more interviews are adding little |
Many theses have more than one objective. In that case calculate the sample size for the main analysis first, and then check whether the same number is enough for the other analyses. Mostly the largest number wins.
Cochran’s formula for surveys
Suppose your aim is to estimate a proportion in a population, like what percentage of MSMEs in a district are using digital payments. Here Cochran’s formula gives the sample you need for the precision you have chosen.
n₀ = Z² × p × (1 − p) / e²
- Z is the value for your confidence level. It is 1.96 for 95% and 2.576 for 99%.
- p is the proportion you are expecting. When there is no earlier estimate, take 0.5. That gives the largest sample, so it is the safe choice.
- e is the margin of error you are ready to accept, written as a decimal. For ±5 percentage points it is 0.05.
Worked example
Take 95% confidence, p = 0.5 and e = 0.05. Then n₀ = 1.96² × 0.5 × 0.5 / 0.05² = 384.16, and you round it up to 385. The 384 or 385 you see in so many theses comes from here only. Remember that this number holds for a large population and for these same assumptions. For a study with other assumptions the number will be different.
Finite population correction
When the population is small and you know its size, the sample can be reduced.
n = n₀ / (1 + (n₀ − 1) / N)
Say there are 2,000 registered MSMEs. Then n = 384.16 / (1 + 383.16 / 2000) = 322.4, which becomes 323. If the population is 500 nurses in a district’s government hospitals, the sample comes down to about 218. Many Indian theses quote the published table of Krejcie and Morgan (1970). It works on the same logic and gives very close numbers.
Allow for non-response
The formula gives the number of usable responses. How many questionnaires you send out is a separate number. Suppose you expect 70% of people to return a usable response. Then divide by 0.7, so 323 / 0.7 = 462 questionnaires. Write this adjustment clearly in the methodology chapter.
A note on Slovin’s formula
Slovin’s formula, n = N / (1 + Ne²), is used in many theses because it is simple. The trouble is that it does not state any confidence level or expected proportion, and methodologists have criticised it on that point. Most examiners will accept Cochran’s formula more easily, as every assumption is visible in it.
Power analysis in G*Power
If your aim is to test a hypothesis, the question becomes different. You need enough participants to have a good chance of finding an effect of the size you care about, if that effect is really there. This is what power analysis does. G*Power is free and widely used, and examiners accept it. It is developed at Heinrich Heine University Düsseldorf.
A power analysis links four numbers, namely significance level, power, effect size and sample size. Give G*Power any three of them and it will calculate the remaining one for you. Here sample size is the unknown, so you decide these four.
- The test you are going to run, which you can pick with the help of the statistical test guide.
- Significance level (α), mostly taken as 0.05.
- Power (1 − β), usually 0.80. This means an 80% chance of detecting a real effect of the assumed size.
- Effect size, that is the size of difference or relationship you are expecting.
| Analysis in G*Power | Assumed effect | Sample needed (α = 0.05, power = 0.80) |
|---|---|---|
| Independent t-test, two-tailed | Medium, d = 0.5 | 64 per group, 128 in total |
| One-way ANOVA, 3 groups | Medium, f = 0.25 | About 53 per group, 159 in total |
| Correlation, two-tailed | r = 0.3 | About 85 |
| Multiple regression, 5 predictors (R² different from zero) | Medium, f² = 0.15 | 92 |
| Multiple regression, 10 predictors | Medium, f² = 0.15 | 118 |
Where the effect size comes from
Effect size is the input that matters most, and many times it is the one scholars simply guess. First look for effect sizes reported in earlier studies that are close to yours. If there are none, a pilot study is the next source. After that, take the smallest effect that would make a difference in practice. Cohen’s conventional small, medium and large values (for example d = 0.2, 0.5, 0.8) are the last option, and if you use them, say so in the thesis. If you assume a big effect, G*Power will give you a small sample, and examiners know it, so they check this number closely.
Running it in G*Power
Select the test family and the statistical test, and set “Type of power analysis” to “A priori”. Type the effect size, α and power in their boxes and then only click Calculate. Save it or take screenshot of the input and output window and keep it in the appendix, so when the examiner asks, the proof is already there.
Sample size for SEM and regression
Structural equation modelling is common in management, commerce and social science theses, and confusion about its sample size is equally common. You will come across several rules of thumb.
- A minimum of 200 is many times quoted as a fair size for covariance-based SEM (AMOS, lavaan). For complex models you need more than that.
- Ratios of cases to parameters are also quoted, like 10 or 20 cases for each estimated parameter.
- The “10-times rule” for PLS-SEM is used widely. It takes ten times the largest number of arrows pointing at any construct. It has been criticised for giving samples that are too small.
These rules of thumb is easy to quote, but if the examiner asks why 200 and not 150, most scholars have no answer. Two better options are there.
- For PLS-SEM, the inverse square root method of Kock and Hadaya (2018). At 5% significance and 80% power, the minimum sample is (2.486 / pmin)², where pmin is the smallest path coefficient you expect to come out significant. With pmin = 0.2 this gives 155, and with 0.15 it gives 275.
- For any SEM, a power analysis for the model, done with tools made for SEM. The RMSEA-based approach of MacCallum and colleagues is one example, and a Monte Carlo simulation is another. Setting these up takes more work than G*Power, and for a complex model it is wiser to get a statistician involved.
For multiple regression, the G*Power numbers in the table above are easier to defend than older rules like “50 + 8 × number of predictors”. With small models, though, both mostly give similar numbers.
Sample size for qualitative studies and small populations
Qualitative research
For interviews and focus groups there is no formula. Usually the number is justified by saturation. In simple words, you keep doing interviews till the new ones are giving hardly anything fresh for your themes. Before starting, write down roughly how many interviews you expect and why. Note at which point new codes stopped coming, and report that point. Published studies with similar aims help, as you can quote how many interviews they did.
When the calculated sample is not available
Sometimes the population itself is small, like the heads of all engineering colleges in one district. In other studies access is the problem. In both cases write it plainly in the thesis and don’t try to hide it. If possible, study the whole population (a census). Report the power you really have for the main analysis, and describe the results with confidence intervals. An honest limitation is much better than a sample size worked backwards to fit whatever responses you managed to collect.
Mixed methods
Each strand is sized as per its own logic. The quantitative strand gets a formula or a power analysis, and for the qualitative strand you give a justification based on saturation.
What to write in your synopsis and methodology chapter
One or two paragraphs are enough for the sample size section. What matters is that a reader can repeat your calculation from it, so put in the following points.
- The target population, and its size if you know it, with where that figure came from (census, department records, association membership).
- The sampling method, whether simple random, stratified, cluster, systematic or purposive, and the reason for choosing it (see our sampling techniques guide).
- Which method you used to calculate the sample size, naming the formula or the software (e.g. G*Power 3.1).
- Each input value along with its source. That means confidence level, margin of error, expected proportion or effect size, and power.
- How you adjusted for expected non-response, and what rate you assumed for it.
- How many responses you actually got and could use, and how that number compares with the target.
This is the kind of detail examiners like to see. “The sample size was calculated with G*Power 3.1 for a two-tailed independent samples t-test. Assuming an effect size of d = 0.45, based on Kumar and Rao (2021), α = 0.05 and power = 0.80, the required sample was 79 per group. Allowing for 20% attrition, 99 students per group were recruited.” (The study cited here is only an illustration.)
At synopsis stage the data is not collected yet, so you give only the planned calculation there. The methodology chapter gives the plan and also what actually happened in the field. If you are still designing the study and want the calculation checked before data collection, our research methodology consulting covers that.
Want a specialist to help with this in your own work? Statistics Training and Analysis Review (SPSS, R, AMOS, SmartPLS): a free first consultation, then a written scope and fixed quote.
Sources
- G*Power — Heinrich Heine University Düsseldorf (download and manual)
- Krejcie, R. V. and Morgan, D. W. (1970). Determining sample size for research activities. Educational and Psychological Measurement 30(3)
- Kock, N. and Hadaya, P. (2018). Minimum sample size estimation in PLS-SEM: the inverse square root and gamma-exponential methods. Information Systems Journal 28(1)
FAQ
Questions scholars ask
Is 384 always the right sample size for a survey?
No. 384 is what Cochran’s formula gives for a large population at 95% confidence, ±5% margin and p = 0.5. If your population is smaller, your margin is different, or your analysis compares groups, the number changes.
Can I use a sample size that my senior used for a similar thesis?
If their study is really similar, you can start from their inputs. Still, you have to do your own calculation and justify the figure. Many synopses write “previous studies used 300”, and that is not a calculation.
What if I get fewer responses than the calculated sample?
Report the number you actually got. Then work out the power or margin of error you really achieved, and discuss it under limitations. Some scholars redo the calculation to match what came back, which is wrong.
Is a bigger sample always better?
Only up to a point. With a very large sample, even tiny differences of no importance become statistically significant, so report effect sizes. And collecting a huge sample costs you time and money. That effort could go into a better instrument or sampling method.
Does sample size depend on the number of questionnaire items?
That depends on which analysis you are running. In factor analysis and SEM, the number of items and constructs does affect what you need. For a simple survey estimate it makes no difference. For factor analysis some people quote ratios like “10 respondents per item”. You can use such a ratio as a rough check, but it is not a justification by itself.
