Sample size calculation for research

Which sample size method fits your study

The right method depends on what the study is trying to do. A survey that wants to estimate how many teachers use a smart classroom is a different problem from a study that wants to detect a difference between two teaching methods.

If your thesis has several objectives, calculate the sample size for the main analysis and check that it is adequate for the others. The largest requirement usually wins.

Cochran’s formula for surveys

When the aim is to estimate a proportion in a population, such as the percentage of MSMEs in a district that use digital payments, Cochran’s formula gives the sample needed for a chosen precision:

n₀ = Z² × p × (1 − p) / e²

• Z is the value for your confidence level: 1.96 for 95%, 2.576 for 99%.
• p is the proportion you expect. If you have no earlier estimate, use 0.5, which gives the largest (safest) sample.
• e is the margin of error you can accept, written as a decimal: 0.05 for ±5 percentage points.

Worked example

With 95% confidence, p = 0.5 and e = 0.05: n₀ = 1.96² × 0.5 × 0.5 / 0.05² = 384.16. Round up to 385. This is where the familiar figure of 384 or 385 in so many theses comes from. It applies to a large population and to the assumptions above, not to every study.

Finite population correction

If the population is small and known, the sample can be reduced:

n = n₀ / (1 + (n₀ − 1) / N)

For a population of 2,000 registered MSMEs: n = 384.16 / (1 + 383.16 / 2000) = 322.4, so 323. For a population of 500 nurses in a district’s government hospitals, it falls to about 218. The published table by Krejcie and Morgan (1970), which many Indian theses cite, is built on the same logic and gives very similar numbers.

Allow for non-response

The formula gives the number of usable responses you need, not the number of questionnaires to send out. If you expect 70% of people to respond usably, divide by 0.7: 323 / 0.7 = 462 questionnaires. Say so in the methodology chapter.

A note on Slovin’s formula

Many theses use Slovin’s formula, n = N / (1 + Ne²). It is simple, but it does not state a confidence level or an expected proportion, and methodologists have criticised it for that reason. Most examiners will accept Cochran’s formula more readily, because every assumption is visible.

Power analysis in G*Power

When the aim is to test a hypothesis, the question changes. You need enough participants to have a good chance of detecting an effect of the size you care about, if it exists. That is power analysis. G*Power is free, widely used and accepted by examiners, and it is developed at Heinrich Heine University Düsseldorf.

Power analysis links four quantities: significance level, power, effect size and sample size. Fix three and G*Power gives you the fourth. For a sample size calculation you choose:

1. The test you will run, from the statistical test guide.
2. Significance level (α), usually 0.05.
3. Power (1 − β), usually 0.80, meaning an 80% chance of detecting a real effect of the assumed size.
4. Effect size, the size of the difference or relationship you expect.

Where the effect size comes from

This is the input that matters most and is most often guessed. In order of preference: use effect sizes reported in earlier studies close to yours; use a pilot study; or use the smallest effect that would matter in practice. Cohen’s conventional small, medium and large values (for example d = 0.2, 0.5, 0.8) are a last resort, and you should say that is what you used. Assuming a large effect makes the sample small, which is why examiners look hard at this number.

Running it in G*Power

Choose the test family and statistical test, set “Type of power analysis” to “A priori”, enter the effect size, α and power, and click Calculate. Save or screenshot the input and output window; it makes a good appendix item and answers the examiner’s question before it is asked.

Sample size for SEM and regression

Structural equation modelling is common in management, commerce and social science theses, and so is uncertainty about sample size. You will meet several rules of thumb:

• A minimum of 200 is often cited as a reasonable size for covariance-based SEM (AMOS, lavaan), with more needed for complex models.
• Ratios of cases to parameters, such as 10 or 20 cases per estimated parameter, are also cited.
• The “10-times rule” for PLS-SEM (ten times the largest number of arrows pointing at any construct) is widely used but has been criticised for giving samples that are too small.

Rules of thumb are easy to cite and hard to defend when an examiner asks why. Two better options:

• 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 / p_min)², where p_min is the smallest path coefficient you expect to be significant. For p_min = 0.2 that is 155; for 0.15 it is 275.
• For any SEM, a power analysis for the model, using tools built for SEM, such as the RMSEA-based approach of MacCallum and colleagues or a Monte Carlo simulation. These need more set-up; ask a statistician if the model is complex.

For multiple regression, the G*Power calculation in the table above is more defensible than older rules such as “50 + 8 × number of predictors”, though the two often give similar numbers for small models.

Sample size for qualitative studies and small populations

Qualitative research

Interviews and focus groups are not sized with a formula. The usual justification is saturation: you continue until new interviews add little that is new to your themes. State in advance roughly how many interviews you expect and why, keep a record of when new codes stopped appearing, and report it. Numbers from published studies with similar aims can support your estimate.

When the calculated sample is not available

Sometimes the population is simply small, such as the heads of all engineering colleges in one district, or access is limited. Then say so plainly. Study the whole population if you can (a census), report the power you actually have for your main analysis, and describe your results with confidence intervals. An honest statement of a limitation is far better than a sample size worked backwards to match the responses you managed to collect.

Mixed methods

Size each strand on its own terms: a formula or power analysis for the quantitative strand, and a saturation-based justification for the qualitative one.

What to write in your synopsis and methodology chapter

A sample size section needs only a paragraph or two, but it must let a reader repeat your calculation. Include:

• The target population, and its size if known, with the source of that figure (census, department records, association membership).
• The sampling method: simple random, stratified, cluster, systematic or purposive, and why (see our sampling techniques guide).
• The method used to calculate the sample size, with the formula or software named (e.g. G*Power 3.1).
• Every input value, and where each came from: confidence level, margin of error, expected proportion or effect size, power.
• The adjustment for expected non-response, with the rate you assumed.
• The number of responses actually received and usable, and how that compares with the target.

Here is an example of the level of detail examiners like: “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 is an illustration.)

Your synopsis should state the planned calculation; your methodology chapter should state the plan and what actually happened. If you are designing the study and want the calculation checked before data collection, our research methodology consulting covers that.