Sample Size and Power Calculator

Work out how many participants you need to estimate a percentage or a mean, or to compare two groups, or the power or precision of the sample you have. Each answer comes with a paragraph for your methods section or ethics form.

Calculator

Plan your sample

Your study

EstimateCompare

Outcome

PercentageMean

One group, a yes-or-no answer: a prevalence survey, or the share of students who do something.

Find

Sample sizeMargin

Confidence level

90%95%99%

Expected percentage

From an earlier study or a pilot. If you only know it is between 10% and 90%, use 50.

Margin of error

± points

In percentage points: ±5 around 30% is 25% to 35%.

Population sizeOptional

Only for a small, known population, such as one college's students.

Design effectOptional

For cluster sampling. A simple random sample is 1.

Non-response or dropoutOptional

The share you expect to lose.

Enter the expected percentage and margin of error to see the sample size.

The formula

Estimate a percentage

n = z² × p(1 − p) ÷ d²

Estimate a mean

n = (z × σ ÷ d)²

Compare two percentages

n = (zα√(2p̄q̄) + zβ√(p₁q₁ + p₂q₂))² ÷ (p₁ − p₂)²

Compare two means

n = 2(zα + zβ)² × σ² ÷ Δ²

n: sample size; for each group when comparing
z: 1.96 for 95% confidence (1.645 for 90%, 2.576 for 99%)
p: expected proportion, as a decimal: 30% is 0.30
d: margin of error, in the units of the estimate
σ: standard deviation of the measurement
zα: 1.96 for a two-sided test at the 0.05 level
zβ: 0.84 for 80% power, 1.28 for 90%
p₁, p₂: expected proportions in the two groups; p̄ is their average, and q = 1 − p
Δ: the difference in means worth detecting

Good to know

Round up, and allow for losses by dividing: with 10% expected to drop out, 323 ÷ 0.9 = 359. Adding 10% instead gives 356, which falls short.
When comparing groups, the answer is for each group: two groups of 64 make 128.
For a percentage under 10%, ±5 points is too wide a margin. Naing and colleagues (2006) suggest half the percentage: ±2 points for 4%. Above 90%, use half of what is left: ±1 point for 98%.
For two means, the answer comes from the t-test itself, as G*Power's does. The z formula above runs low for small groups: for an effect size of 2 it gives 4 per group where the t-test needs 6.
Sampling whole clusters (classes, wards, villages) needs more people. Enter a design effect from an earlier survey of the same kind; immunisation coverage surveys have found about 2.

Worked example

A prevalence survey, with anaemia expected in about 30%

01p = 0.30, d = 0.05, z = 1.96 for 95% confidence
02n = 1.96² × 0.30 × 0.70 ÷ 0.05² = 322.7, rounded up to 323
03with 10% non-response: 323 ÷ 0.90 = 358.9, rounded up to 359

Invite 359 students, so that about 323 take part.

How it works

The reasoning behind it

To estimate a percentage or a mean, the question is precision. A sample's estimate lands near the true value but rarely on it, and the margin of error says how near: z standard errors either side. The standard error shrinks with the square root of the sample, so halving the margin takes four times as many people.

To compare two groups, two mistakes have to be kept rare: finding a difference that isn't there (the significance level, α, usually 0.05) and missing one that is (β; power is 1 − β, usually 80%). The sample grows with the square of zα + zβ and shrinks with the square of the effect size, the difference worth detecting divided by the standard deviation. Halving the difference quadruples the sample.

The answer is the number you need to finish with. Round it up, allow for the people who won't respond or will drop out, and say in your methods where the expected percentage or the standard deviation came from: an earlier study, a pilot, or an estimate you can defend.

Questions

Frequently asked

How do I calculate the sample size for a prevalence study?

Use Cochran's formula, n = z²p(1 − p) ÷ d², with p the prevalence you expect and d the margin of error, both as decimals. For an expected 30%, ±5 points and 95% confidence: 1.96² × 0.3 × 0.7 ÷ 0.05² = 322.7, so 323.

What expected percentage should I use if I don't know it?

Take it from an earlier study of a similar population, or run a small pilot. If you can only say it lies between 10% and 90%, use 50%: it gives the largest sample, 385 at ±5 points and 95% confidence. Below 10% or above 90%, 50% is not safe, because the margin there should be smaller.

Where do I get the standard deviation?

From a published study that measured the same thing in similar people, or from a pilot. If you only know the usual range of values, a quarter of it is a rough guide. Say in your methods where the figure came from.

What is statistical power, and why 80%?

Power is the chance that your study detects a difference that really exists. At 80%, one real difference in five would be missed. 80% is the usual minimum; 90% is common in clinical trials and needs about a third more participants.

Why does G*Power give a different answer?

For two means it shouldn't: both use the t-test itself, so an effect size of 0.5 at 80% power and the 0.05 level needs 64 per group in each. The textbook z formula gives 63. For two percentages, G*Power offers several tests. Fisher's exact test needs more than Fleiss's formula gives, so if you will analyse with it, use the continuity-corrected figure shown here.

How do I allow for dropout or non-response?

Divide by the share you expect to keep. For 323 with 10% non-response, 323 ÷ 0.9 = 358.9, so invite 359. Adding 10% instead gives 356, which is too few once 10% of them are lost.

Lab Tools