Sample size, and the four numbers it needs

Power, alpha, effect size and variability — and where each one comes from.

A sample size calculation answers: how many participants do I need to detect a difference this large, if it is really there? It needs four things.

Alpha (α) — the risk you will accept of finding a difference that is not real. Conventionally 0.05.

Power (1 − β) — the chance of finding a real difference. Conventionally 80%, sometimes 90%.

Effect size — the smallest difference that would actually matter clinically. This is a judgement, not a calculation.

Variability — the standard deviation, taken from a previous study or a pilot.

Why it matters

Too small and your study cannot find the effect it was designed to find — a non-significant result then means nothing, because you never had the power to detect anything. Too large and you have put people through a study unnecessarily. Ethics committees ask for this calculation, and examiners check that the number you recruited matches the number you justified.

An example

Comparing mean VAS pain between two groups. From a previous study the standard deviation is about 1.8 points. You decide a 1.2-point difference is the smallest that matters clinically. At α = 0.05 and 80% power, that needs roughly 36 participants per group — 72 in total. Allowing 15% dropout, recruit 84.

Common mistakes

• Choosing the effect size to make the number small enough to be achievable. That is working backwards, and it shows.
• Calculating for the primary objective and then reporting secondary outcomes as though they were powered.
• Forgetting dropout. A 20% loss to follow-up is normal in a twelve-week study.
• Quoting a sample size with no source for the standard deviation.

Read next

• What a p-value is, and what it is not — The single most misreported number in postgraduate research.
• Objectives, and why your results are organised by them — One objective, one analysis, one paragraph of results.