How to use this calculator
- Enter or choose confidence level.
- Enter or choose margin of error (±).
- Enter or choose expected proportion.
- Enter or choose population size (optional).
- The responses needed updates as you type. Use Copy result or Share to keep it.
Method
n₀ = z² × p(1 − p) ÷ E²; with population N: n = n₀ ÷ (1 + (n₀ − 1) ÷ N); round up
- z
- standard normal value for the two-sided confidence level (1.96 for 95%)
- p
- expected proportion
- E
- margin of error as a decimal
- N
- population size
Worked example
- For 95% confidence, z = 1.959964.
- n₀ = 1.959964² × 0.5 × 0.5 ÷ 0.05² = 3.841459 × 0.25 ÷ 0.0025 ≈ 384.15.
- Round up: 385 responses.
To estimate a percentage within ±5 points with 95% confidence, you need 385 completed responses.
What the result means
This is the number of completed responses, not invitations. If you expect a 20% response rate, invite about five times as many people.
Assumptions and limits
- A simple random sample, estimating a single proportion, using the normal approximation to the binomial distribution.
- Clustered or stratified designs need a design effect; comparing two groups or estimating a mean needs a different formula.
- A random sample only fixes random error, not bias from who chooses to respond.
Questions
Why does a population of millions need only about 385 people?
Precision depends mostly on the sample size, not the population. The population matters only when the sample is a sizeable share of it, which the finite population correction handles.
Why use 50% for the expected proportion?
p(1 − p) is largest at 50%, so it gives the most conservative (largest) sample. If you know the answer is near 10% or 90%, a smaller sample suffices.