Sample size is the part of the synopsis residents worry about most, and the part ethics committees check first. You do not need advanced statistics for most theses. Four formulas cover the large majority of MD/MS designs. This guide shows each one with a worked example.
The values every calculation needs
- Confidence level and Z value: 95% confidence gives Zα = 1.96 (two-sided).
- Power and Zβ: 80% power gives Zβ = 0.84; 90% power gives Zβ = 1.28.
- An expected value from earlier research: a prevalence, a mean and standard deviation, or a proportion in each group. Always cite the study you took it from. This is the step reviewers question most.
- Allowable error or expected difference: how precise you want to be, or the smallest difference worth detecting.
1. Estimating a prevalence or proportion (cross-sectional studies)
Formula: n = Z² × p × (1 − p) / d²
Where p is the expected proportion and d is the absolute allowable error.
Example: an earlier study reported a prevalence of 30%. With 95% confidence (Z = 1.96) and an absolute error of 5%:
- n = 1.96² × 0.30 × 0.70 / 0.05²
- n = 3.8416 × 0.21 / 0.0025
- n = 322.7, rounded up to 323
Adding 10% for non-response: 323 / 0.9 = 358.9, so 359 participants.
Tip: some studies use a relative error instead (for example, 20% of p). With p = 30%, that means d = 6%, which gives a smaller sample. Say clearly which one you used.
2. Comparing two means (two groups)
Formula, per group: n = 2 × (Zα + Zβ)² × σ² / Δ²
Where σ is the standard deviation and Δ is the difference in means you want to detect.
Example: comparing a pain score between two drugs. Earlier work suggests SD = 10, and a difference of 5 points is clinically important. With 95% confidence and 80% power:
- n = 2 × (1.96 + 0.84)² × 10² / 5²
- n = 2 × 7.84 × 100 / 25
- n = 62.7, so 63 per group, or 126 in total
3. Comparing two proportions (two groups)
Formula, per group: n = (Zα + Zβ)² × [p1(1 − p1) + p2(1 − p2)] / (p1 − p2)²
Example: a complication occurs in 40% of patients with the standard technique, and you expect 20% with the new one. With 95% confidence and 80% power:
- n = 7.84 × [0.40 × 0.60 + 0.20 × 0.80] / (0.20)²
- n = 7.84 × 0.40 / 0.04
- n = 78.4, so 79 per group
4. Correlation between two measurements
Formula: n = [(Zα + Zβ) / C]² + 3, where C = 0.5 × ln[(1 + r) / (1 − r)]
Where r is the expected correlation coefficient from earlier studies.
Example: you expect r = 0.3 between two measurements. With 95% confidence and 80% power:
- C = 0.5 × ln(1.3 / 0.7) = 0.5 × 0.619 = 0.3095
- n = (2.8 / 0.3095)² + 3 = 81.8 + 3
- n = 84.8, so 85 participants
How to write it in the synopsis
Write the formula, define every symbol, state each value and its source, show the result, and add the allowance for drop-outs. For example:
"Based on the prevalence of 30% reported by [author, year], with 95% confidence and an absolute precision of 5%, the minimum sample size was calculated as 323 using the formula n = Z²pq/d². Allowing for 10% non-response, 359 participants will be enrolled."
Common mistakes
- No source for the expected value.
- Using the prevalence formula for a comparative study.
- Forgetting that formulas 2 and 3 give the number per group.
- Rounding down instead of up.
- A sample size you cannot reach in your data-collection period. If the number is too large, revisit the design, the outcome or the precision with your guide, rather than quietly enrolling fewer patients.
Software you can use
Free tools such as OpenEpi, G*Power and the sample size modules in many statistics packages give the same results as these formulas, and reviewers accept them if you name the tool and the values entered.
Need it done for your title?
The Full Plan for any title on this site includes the sample size worked out with the formula, values and justification, along with the full synopsis. See a complete sample synopsis, read our guide to the synopsis format, or browse titles on the all specialities page.