How to calculate sample size for your thesis (with worked examples)

"How many patients do I need?" is usually the first statistical question a resident asks — and the one ethics committees and university boards scrutinise most closely. A sample size isn't a number you pick; it's a number you justify, with a formula, assumptions from published literature, and an allowance for real-world losses.

What every sample size calculation needs

Whatever your design, you'll need to state:

• The primary outcome — one outcome drives the calculation, not all of them.
• An expected value from prior studies — a prevalence, a mean and standard deviation, or proportions in two groups. Cite the study you took it from.
• The confidence level — almost always 95% (Z = 1.96).
• Precision or power — how narrow your estimate should be (descriptive studies), or the probability of detecting a real difference (comparative studies, usually 80%).
• An allowance for non-response or drop-out — typically 10–20%.

1. Estimating a single proportion (prevalence studies)

For cross-sectional studies estimating how common something is:

n = Z² × p × (1 − p) / d²

where p is the expected prevalence and d the absolute precision.

Example. A previous study found anaemia in 30% of antenatal women. You want your estimate within ±5 percentage points:

n = 1.96² × 0.30 × 0.70 / 0.05² = 0.8067 / 0.0025 ≈ 323

Allowing for 10% non-response: 323 / 0.9 ≈ 359 participants.

Absolute vs relative precision. "5% precision" can mean ±5 percentage points (absolute) or ±5% of 30%, i.e. ±1.5 points (relative). The second gives a sample roughly eleven times larger. Always say which one you used.

2. Comparing two means

For comparing a continuous outcome (blood pressure, HbA1c, pain score) between two groups:

n per group = 2 × (Zα/2 + Zβ)² × σ² / Δ²

With 95% confidence and 80% power, (1.96 + 0.84)² = 7.84.

Example. Systolic BP has a standard deviation of 10 mmHg, and a 5 mmHg difference is clinically meaningful:

n = 2 × 7.84 × 10² / 5² = 1568 / 25 ≈ 63 per group

3. Comparing two proportions

For a binary outcome (complication yes/no) between two groups:

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

Example. Complications occur in 40% with standard care and you expect 20% with the new technique:

n = 7.84 × (0.24 + 0.16) / 0.20² = 3.136 / 0.04 ≈ 79 per group

Software that applies a continuity correction will give a somewhat larger number — that's expected, and more conservative.

Tools that help

You don't need to calculate by hand. G*Power, OpenEpi and nMaster (from CMC Vellore) all handle these designs and more. Save a screenshot of the inputs and output for your records — committees sometimes ask for it.

Mistakes that get synopses sent back

1. No source for the assumptions. An expected prevalence or SD must come from a cited study, ideally in a similar population.
2. Using a "convenient" number. "All patients admitted during the study period" is a sampling frame, not a sample size justification — though it can be acceptable if you show it exceeds the calculated minimum.
3. Ignoring the design. Cluster sampling needs a design effect (often 1.5–2); matched designs need different formulas.
4. Powering for every objective. Calculate for the primary objective and say so.
5. Forgetting feasibility. If your department sees 150 eligible patients a year and you need 359, revisit the design, duration or precision before submitting.

A template paragraph

"Based on a prevalence of 30% reported by [Author, Year], with 95% confidence and an absolute precision of 5%, the minimum sample size calculated using the formula n = Z²p(1−p)/d² was 323. Allowing for 10% non-response, 359 participants will be enrolled."

If your design doesn't fit these formulas — diagnostic accuracy, correlation, non-inferiority, survival — the principle is the same but the formula changes. That's where a statistician saves you a round of corrections.