How to choose the right sample size for your research

"How many respondents do I need?" is one of the first questions a researcher asks, and one of the first an examiner asks about the methodology chapter. The answer is not a fixed number like 100 or 300. It depends on what you want your results to say. This article explains the factors, the most common formula, and when that formula is not enough.

Why sample size matters {#why}

• Too small: the results are imprecise, and may miss a difference or relationship that really exists, so you wrongly conclude "no significant differences".
• Too large: time, effort and cost for little gain, and every tiny difference may become "statistically significant" without mattering in practice.

The goal is a sample that is adequate for your research question, with a written justification of how you chose it.

What determines the sample size {#factors}

Cochran's formula, worked through {#cochran}

The most common way to size a sample for estimating a proportion is Cochran's formula:

n₀ = Z² × p × (1 − p) ÷ e²

where:

• Z is the standard score for the confidence level: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%,
• p is the expected proportion, 0.5 when you don't know it,
• e is the margin of error as a decimal (5% = 0.05).

Example: 95% confidence, 5% margin of error, p = 0.5:

n₀ = (1.96)² × 0.5 × 0.5 ÷ (0.05)² = 3.8416 × 0.25 ÷ 0.0025 = 384.16, so about 384 respondents (many round up to 385 to be safe).

The finite population correction {#finite-population}

When your population is limited and of known size (N), such as 2,000 students in a faculty, the sample you need is smaller:

n = n₀ ÷ (1 + (n₀ − 1) ÷ N)

Example: n = 384.16 ÷ (1 + 383.16 ÷ 2,000) = 384.16 ÷ 1.1916 = about 322 respondents.

A ready table {#table}

Sample size for 95% confidence, a 5% margin of error and p = 0.5, after the finite population correction, rounded to the nearest whole number:

These are almost exactly the figures in the Krejcie & Morgan (1970) table that many theses cite, because it rests on the same assumptions. If you cite the table, state its assumptions with it.

And the effect of the margin of error and confidence level, for a large, unlimited population:

Don't forget the response rate {#response-rate}

If you need 322 answers and expect 60% of those you invite to respond, invite about 322 ÷ 0.60 = 537 people. Report how many questionnaires were distributed, returned and usable, because the examiner will ask about the gap between them.

When the formula is not enough {#beyond}

Cochran's formula answers one question: how many do I need to estimate a proportion or mean in the population with a given precision? When your research question is testing a difference or a relationship, the better tool is a power analysis, usually done in the free G*Power program. It needs: the test you will run, the effect size you expect, the significance level (usually 0.05) and the power you want (usually 0.80).

Well-known examples at a 0.05 significance level and 0.80 power:

• Independent-samples t-test with a medium effect (d = 0.5): about 64 per group.
• A correlation of r = 0.30: about 85 respondents (Cohen, 1992).
• Multiple regression: Green's (1991) rule of thumb for testing the overall model is 50 + 8 × the number of predictors. For five predictors: 50 + 40 = 90.

Factor analysis has its own rules too. See our article on questionnaire validity and reliability.

Small samples and their limits {#small-samples}

Some studies cannot reach a large sample: a rare population, a pilot study, or an experiment with a single group. That is acceptable when you:

• say so plainly in the study's limitations,
• choose suitable tests, such as non-parametric tests, or exact p-values for small groups,
• report effect sizes and confidence intervals, not p-values alone,
• don't generalise to a population the sample does not represent.

Size alone is not enough: how you sample {#sampling}

A sample of 1,000 people chosen because they were easiest to reach can be less representative than a random sample of 300. So describe how the sample was chosen (simple random, stratified, cluster or convenience) with the same care as its size. In a stratified sample, the size is split across strata in proportion to each one's share of the population.

In Pulseform {#pulseform}

In the Sample & Quotas tab of the survey settings:

• Calculate Sample Size Statistically from the population size, margin of error and confidence level, using the formula above with the finite population correction,
• Target Sample Size Quota, with progress towards it shown on the results page,
• Automatically close survey when target sample collection quota is reached.

If you have a field team, a collector quota sets how many answers each collector sends, to share the sample between them. The details are in Survey settings and Collector quotas.

References {#sources}

The original sources of the formulas, thresholds and rules of thumb in this article:

• Cochran, W. G. (1977). Sampling techniques (3rd ed.). John Wiley & Sons.
• Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
• Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155–159.
• 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.
• Green, S. B. (1991). How many subjects does it take to do a regression analysis? Multivariate Behavioral Research, 26(3), 499–510.
• Krejcie, R. V., & Morgan, D. W. (1970). Determining sample size for research activities. Educational and Psychological Measurement, 30(3), 607–610.
• MacCallum, R. C., Widaman, K. F., Zhang, S., & Hong, S. (1999). Sample size in factor analysis. Psychological Methods, 4(1), 84–99.