Sample Size Calculator

How it works

A sample size calculation runs the confidence interval formula backwards. You decide how wide an interval you can live with (the margin of error E) and how sure you want to be that it contains the truth (the confidence level), and the formula tells you how many observations that takes. Halving the margin of error takes four times the sample, because precision improves with the square root of n.

For a proportion the required size depends on how far the true proportion is from 50%, which is where a proportion is hardest to pin down. If you have no prior estimate, 50% is the conservative choice and is what most survey sample-size tables assume: 385 people for ±5 points at 95% confidence, 1,068 for ±3 points.

For a mean you need an estimate of the spread of the measurement, σ, from a pilot study, earlier work or a rough rule such as range ÷ 4. The sample size is (z* × σ ÷ E)².

When the population itself is small, say a company of 500 employees, you do not need as large a sample, because each observation is a bigger share of the whole. The finite population correction shrinks the sample size accordingly; it matters once the sample would be more than about 5% of the population.

Formula

proportion:  n₀ = z*² × p(1 − p) ÷ E²        (p and E as fractions)
mean:        n₀ = (z* × σ ÷ E)²
finite population:  n = n₀ ÷ (1 + (n₀ − 1) ÷ N)
round n up to the next whole number
z* = 1.644854 (90%), 1.959964 (95%), 2.575829 (99%)

Example

To estimate a proportion to within ±5 percentage points at 95% confidence with no prior estimate (p = 50%): n₀ = 1.959964² × 0.5 × 0.5 ÷ 0.05² = 384.15, so 385 people are needed. For ±3 points it rises to 1,067.07, so 1,068.

If the population is only 1,000 people, the correction gives 384.15 ÷ (1 + 383.15 ÷ 1000) = 277.73, so 278 people are enough.

To estimate a mean IQ score to within ±3 points at 95% confidence with σ = 15: n = (1.959964 × 15 ÷ 3)² = 96.04, so 97 people.

Assumptions and limitations

• The sample will be a simple random sample. Clustered or stratified designs need a design-effect adjustment that is not included.
• The proportion formula uses the normal approximation to the binomial, which is adequate for the sample sizes it produces; for very small results (under about 30) treat the figure as a rough guide.
• The mean formula treats σ as known. If σ is a guess, a larger sample than shown protects against guessing low.
• A population size of 0 or blank means the population is treated as unlimited.
• The result is the number of completed, usable responses. Allow for non-response on top of it.
• z* is computed from Acklam's inverse-normal approximation refined with one Halley step.

Frequently asked questions