Sample size calculation for medical research and MD/MS theses: formulas with worked examples

An undersized study cannot answer its own question: a real difference may go undetected, and a non-significant result tells the reader little. Ethics committees and thesis protocol reviewers therefore ask how the sample size was chosen, and reporting guidelines such as CONSORT and STROBE require authors to explain it. Yet "all patients during the study period" remains one of the most common justifications in Indian theses, and one of the most commonly queried.

This guide covers the formulas for the designs most often used in postgraduate research, with worked examples you can follow on a calculator.

The four things every calculation needs

1. The primary outcome, and whether it is a proportion (for example, the percentage with anaemia) or a mean (for example, systolic blood pressure).
2. An expected value from a previous study, pilot data or routine records: the expected prevalence, or the expected mean and standard deviation.
3. The precision or difference that matters: how precisely you want to estimate a prevalence, or the smallest difference between groups that is clinically important.
4. The error rates you accept: the significance level (α, usually 5%) and, for comparisons, the power (1 − β, usually 80% or 90%).

1. Estimating a prevalence or proportion

For a cross-sectional study whose main aim is to estimate a proportion:

n = Z² × p × (1 − p) / d²
where p is the expected proportion and d is the absolute precision (margin of error).

Worked example. A previous study found that 30% of antenatal women in a similar district were anaemic. You want to estimate the prevalence within ±5 percentage points with 95% confidence.

n = 1.96² × 0.30 × 0.70 / 0.05² = 3.8416 × 0.21 / 0.0025 = 322.7, so 323 women.

Precision matters a great deal: with d = 0.10 the same calculation gives 81, and with d = 0.03 it gives 897. Choose d deliberately and justify it; a common rule is that d should be much smaller than p itself when p is small.

2. Estimating a mean

n = Z² × σ² / d²
where σ is the expected standard deviation and d the precision in the same units.

Worked example. To estimate mean systolic blood pressure within ±2 mmHg, with an expected SD of 12 mmHg: n = 3.8416 × 144 / 4 = 138.3, so 139 participants.

3. Comparing two means

For a trial or comparative study with two independent groups of equal size and a continuous outcome:

n per group = 2 × (Z_α/2 + Z_β)² × σ² / Δ²
where Δ is the smallest clinically important difference between means.

Worked example. You expect pain scores to have an SD of 10 points and consider a 5-point difference clinically important; α = 0.05, power 80%.

(1.96 + 0.84)² = 7.84, so n = 2 × 7.84 × 100 / 25 = 62.7, giving 63 per group (126 in total).

4. Comparing two proportions

n per group = (Z_α/2 + Z_β)² × [p₁(1 − p₁) + p₂(1 − p₂)] / (p₁ − p₂)²

Worked example. The complication rate with standard care is 40% and you expect 25% with the new technique; α = 0.05, power 80%.

n = 7.85 × (0.24 + 0.1875) / 0.15² = 7.85 × 0.4275 / 0.0225 = 149.1, giving 150 per group.

This is the simple unpooled formula. Software using a pooled-variance or continuity-corrected formula may give a slightly larger number; report which method you used.

Adjusting the calculated number

For non-response or loss to follow-up

Divide by the expected retention: n_final = n / (1 − dropout rate). With 323 participants and 10% expected non-response, 323 / 0.9 = 358.9, so recruit 359.

For a small, known population

When sampling from a finite population of size N, apply the finite population correction: n_adj = n / [1 + (n − 1) / N]. For n = 323 drawn from a population of 1,000, n_adj = 323 / 1.322 = 244.3, so 245.

For cluster sampling

If you sample villages, wards or schools rather than individuals, multiply by the design effect (often 1.5 to 2 in community surveys, ideally estimated from earlier studies).

Designs that need other methods

Paired designs, more than two groups, regression models, survival outcomes, diagnostic accuracy studies, non-inferiority trials and agreement studies each have their own formulas. Use dedicated software (G*Power, OpenEpi or R) or a statistician; the principle of specifying the outcome, expected values, important difference, α and power is the same.

How to report it

A reviewer should be able to reproduce your number from the methods section. A complete statement looks like this:

"Assuming a prevalence of anaemia of 30% based on a previous study in the district [ref], a sample of 323 was required to estimate prevalence with an absolute precision of 5% and 95% confidence. Allowing for 10% non-response, we aimed to recruit 359 women."

Sources and further reading

• OpenEpi, free sample-size calculators
• EQUATOR Network: CONSORT and STROBE items on sample size

Our statistical analysis service calculates and documents sample size for theses, protocols and ethics submissions, and analyses the data once it is collected.