Sample Size and Sampling Methods for Journal Papers

“Why this many participants?” is one of the questions reviewers ask most often, and “because 384 is the standard number” is rarely a satisfying answer. The right sample size depends on the research question, the design, the analysis and the precision or effect you need to detect, while the sampling method decides whom the results can be generalised to. This guide explains both in plain language, with worked examples of Cochran’s and Yamane’s formulas, the basics of power analysis, sample size in qualitative studies, and a template for reporting your justification.

Why Sample Size and Sampling Matter to Reviewers

Two separate questions sit behind every sample, and reviewers judge them separately.

• Is the sample large enough? Too few participants and real effects go undetected, estimates have wide margins of error, and complex models such as regression or structural equation modelling become unstable. Too many can waste resources and, in clinical or animal research, expose more participants to risk than necessary.
• Is the sample the right one? A very large sample drawn the wrong way can still be badly biased. Ten thousand responses to an online poll shared on social media tell you about people who saw and chose to answer the poll, not about the whole population.

Sample size therefore controls precision and power, while the sampling method controls representativeness. A paper needs both arguments. Reporting guidelines reflect this: CONSORT asks trial authors to explain how the sample size was determined, and STROBE asks observational studies to explain how the study size was arrived at. Our guide to writing a reproducible methods section shows where this information fits in the manuscript.

Population, Sampling Frame and Sampling Unit

Before choosing a method, pin down three terms.

• Target population — everyone the conclusions are meant to cover, for example “all secondary-school mathematics teachers in Maharashtra”.
• Sampling frame — the actual list or mechanism used to reach them, such as a state teacher register or a list of schools. Gaps between population and frame (unregistered teachers, private schools missing from the list) create coverage bias.
• Sampling unit — what is selected: individuals, households, classrooms, firms, villages, hospital records.

When the unit selected differs from the unit analysed — selecting schools but analysing students, for instance — the design is clustered, and that affects both the analysis and the sample size, as discussed below.

State all three in your methods section. “A sample of 200 respondents was selected”, with no frame, procedure or number approached, leaves reviewers unable to judge bias.

Probability and Non-Probability Sampling Methods

In probability sampling, every member of the frame has a known, non-zero chance of selection, which allows sampling error to be estimated and results to be generalised statistically. In non-probability sampling, selection depends on access or judgement, so generalisation rests on argument rather than statistics.

Non-probability samples are acceptable in many published studies, but the paper must name the method honestly and limit its claims. Calling a Google Forms link shared in WhatsApp groups “random sampling” is a frequent and easily spotted error.

Power Analysis Basics

For studies that test hypotheses — comparing groups, testing correlations or regression coefficients — sample size is best set by a power analysis. It combines four quantities; fix any three and the fourth follows:

1. Significance level (α) — the accepted risk of a false positive, usually 0.05.
2. Power (1 − β) — the probability of detecting an effect that really exists, usually 0.80 or 0.90.
3. Effect size — the smallest difference or association worth detecting.
4. Sample size — what you are solving for.

Free software such as G*Power, or packages in R and other statistics programs, performs these calculations for t-tests, ANOVA, chi-square, correlation and regression. Record the software, test family and every input so reviewers can reproduce the number. Power analysis must be done before data collection; “post-hoc power” computed from the observed effect adds nothing beyond the p-value and is widely discouraged.

Choosing an Effect Size You Can Defend

The effect size is the input that most changes the answer, and the one most often chosen carelessly. Halving the effect size roughly quadruples the sample needed. Defensible sources, in order of preference:

• The smallest effect that matters in practice — for example, a 5-mark gain that a school would consider worth the cost of an intervention.
• Earlier studies or meta-analyses on the same outcome and population, bearing in mind that published effects are often inflated.
• A pilot study, used cautiously because small pilots give imprecise estimates.
• Conventional benchmarks, only when nothing better exists, and stated as such.

For multiple regression, a frequently quoted rule of thumb is at least 50 + 8m participants to test the overall model, where m is the number of predictors, but a power analysis based on f² is more defensible. Structural equation models and factor analyses generally need larger samples, often several hundred, depending on model complexity.

Cochran’s Formula With Worked Examples

When the aim is to estimate a proportion — the percentage of farmers using crop insurance, say — rather than test a difference, sample size is set by the precision you want. Cochran’s formula for a large population is:

n₀ = Z² × p × (1 − p) / e²

where Z is the value for the confidence level (1.96 for 95%), p is the expected proportion, and e is the margin of error expressed as a decimal.

This is the source of the familiar figure of 384. It applies only to estimating one proportion with simple random sampling at 95% confidence and ±5% precision; it is not a universal minimum and does not justify a sample for regression or group comparisons. For cluster samples, multiply by the design effect (often 1.5–2 or more) to account for similarity within clusters.

Yamane’s Formula and When Not to Use It

Yamane’s simplified formula is popular in management, education and social science theses because it needs only the population size and margin of error:

n = N / (1 + N × e²)

Worked example: for a population of N = 2,000 employees and e = 0.05, n = 2,000 / (1 + 2,000 × 0.0025) = 2,000 / 6 = 333.3, rounded up to 334.

Yamane’s formula is a shortcut that assumes 95% confidence and maximum variability (p = 0.5); for very large populations it approaches 1 / e², which is 400 for a ±5% margin. It gives figures close to Cochran’s formula with the finite population correction, slightly larger because it effectively rounds Z to 2.

Its limits are the same as Cochran’s, and reviewers increasingly point them out:

• It answers a precision question, not a power question, so it cannot justify a sample for hypothesis tests.
• It assumes random sampling; applied to a convenience sample, the calculated margin of error is not meaningful.
• It ignores non-response, clustering and planned subgroup analyses.

Prefer a power analysis whenever your main results are inferential tests.

Sample Size in Qualitative Research

Qualitative studies do not use power calculations. The usual justification is saturation — the point at which further interviews or observations add little new information relevant to the research question. Empirical studies of interview data have found that most themes often appear within the first 9–12 interviews in fairly homogeneous groups, while a richer understanding of those themes may need more, and varied or complex topics need more still.

Saturation is a judgement, not a number, so describe how you assessed it:

• Track new codes per interview and report when they levelled off.
• State a stopping rule in advance, such as “two consecutive interviews producing no new codes after a minimum of ten”.
• Explain sampling choices: purposive criteria, maximum variation, or theoretical sampling in grounded theory.

Some methodologists argue that saturation fits poorly with reflexive or interpretive approaches and suggest the concept of information power instead: the more focused the aim, the more specific the participants, the stronger the theory and the richer the dialogue, the fewer participants are needed. Either way, avoid quoting a number without reasoning; our overview of research methodology explains how qualitative designs differ from quantitative ones.

Reporting Your Sample Size Justification

A good justification lets a reviewer recalculate your number. Include the target, the method, every input with its source, the adjustment for non-response, and what was actually achieved.

Also report the response rate and how non-responders compare with responders where known; a flow diagram is expected in trials (CONSORT) and helpful in surveys. If the achieved sample fell short, say so and state what effect the final sample could detect. Do not recalculate the target after seeing the data to make the achieved number look adequate.

Finally, make the data behind the analysis findable: if you submit to JMVRA, a Data Availability Statement is expected before the reference list, and it is the natural place to say where anonymised survey or interview data can be requested.

Quick Checklist

1. Define population and frameWho, where, and the list used to reach them
2. Name the sampling methodProbability or not, stated honestly
3. Pick the right calculationPrecision formula or power analysis
4. Justify the inputsEffect size, p and margin from real sources
5. Adjust for lossesNon-response, attrition and design effect
6. Report target vs achievedApproached, consented, completed, analysed

Questions Researchers Often Ask

In Summary

Sample size and sampling method answer two different questions: how precise or powerful your study is, and whom its findings represent. Define the population and frame, choose a method you can describe honestly, and calculate the size from the analysis you actually plan — Cochran or Yamane for estimating proportions, power analysis for testing hypotheses, saturation or information power for qualitative work. Then report every input, the non-response adjustment and the achieved numbers. Check the rest of your manuscript against the pre-submission checklist and the publication guidelines before you submit.

References and Sources

• EQUATOR Network — Reporting guidelines (CONSORT, STROBE, COREQ and others)
• STROBE — Reporting observational studies
• APA Style — Journal Article Reporting Standards (JARS)