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# Sample size

- Concept

How many people are needed to detect the effect that matters with the accepted risk of error. It is calculated before the study starts.

Updated Nov 5, 2026

## Key points

- Ingredients: α, power, smallest relevant effect, variability (SD or control proportion) and expected losses.
- Effect size weighs most: n depends on the inverse square of the difference; halving it multiplies n by about 4.
- A smaller α or higher power increases n.
- Greater variability increases n.
- Adjust for losses: final n = calculated n / (1 − proportion lost).

## What it is

A sample size calculation estimates how many participants a study needs to detect a clinically relevant difference at a given α and power. Its ingredients are α, power (1 − β), the effect size to detect, the variability (or the expected proportion in the control group) and the expected losses. A study that is too small cannot answer its question, and one that is too large exposes more people than needed.

## Key facts

- Two proportions (approximation): n per group = (zα/2 + zβ)² × [p1(1 − p1) + p2(1 − p2)] / (p1 − p2)².
- Two means (approximation): n per group = 2 × (zα/2 + zβ)² × SD² / difference².
- With two-sided α = 0.05 (z = 1.96) and 80% power (z ≈ 0.84), (1.96 + 0.84)² = 7.84.

## Worked example

- Data: a trial expects to cut reinfection from 20% (control) to 10% (treatment); two-sided α = 0.05 and 80% power.
- Variances: 0.20 × 0.80 = 0.16 and 0.10 × 0.90 = 0.09; sum = 0.25.
- Difference = 0.20 − 0.10 = 0.10; squared = 0.01.
- n per group = 7.84 × 0.25 / 0.01 = 196.
- If 10% losses are expected: 196 / 0.90 = 217.8, rounded up to 218 per group.
- If the expected effect were smaller (20% vs 15%): 7.84 × (0.16 + 0.1275) / 0.0025 ≈ 902 per group.
- Reading: halving the difference sought multiplies the sample size by more than 4 (approximation with a simple formula).

## Markers

- n
- α
- 1 − β
- SD

## Common pitfalls

- Picking the expected effect so the sample comes out small, instead of using the smallest clinically important effect.
- Forgetting to adjust for losses or non-adherence.
- Calculating the sample size after the study has finished.

## Memory aid

‘Small effect, big sample’: the difference sought enters the denominator squared.

## Test yourself

Think of the answer, then tap the question to see it.

## What inputs are needed to calculate a trial’s sample size?

α, the desired power, the smallest relevant effect, the variability (SD or control proportion) and the expected losses.

## Which input changes the sample size most?

The effect size: n varies with its inverse square, so halving it multiplies n by about 4.

## How does n change when power goes from 80% to 90%?

It increases: more people are needed to lower the chance of a false negative.

## If the calculation gives 300 per group and 25% losses are expected, how many should be enrolled?

300 / 0.75 = 400 per group.

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## Connections

## Related 4

- [Statistical power](https://durgical.com/en/epidemiology/statistical-power/)
- [Type I and type II errors](https://durgical.com/en/epidemiology/type-i-and-type-ii-errors/)
- [Standard error](https://durgical.com/en/epidemiology/standard-error/)
- [Randomised controlled trial](https://durgical.com/en/epidemiology/randomised-controlled-trial/)

[← All of epidemiology](https://durgical.com/en/epidemiology/)

Educational content on epidemiological and statistical methods; the examples use made-up numbers to teach the calculation. Not a clinical or legal guide: current Colombian rules are in the official sources.

Sources: [Sample sizes required](https://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm) (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012); [What are statistical tests?](https://www.itl.nist.gov/div898/handbook/prc/section1/prc13.htm) (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012).

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Representative model, for educational purposes only.

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License: CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Cite as: DurgiPedia (durgical.com), https://durgical.com/en/epidemiology/sample-size/