# How many survey responses do you need? Know the number before you buy.

- Your sample size from the margin of error and confidence level
- The margin of error of a sample you already have
- How many people each group needs when you compare two

Set your margin of error, confidence level and population, and the calculator works out how many completed responses you need. It works the other way round too, and for comparing two groups.

For a large population you need 385 completed responses for a margin of error of 5% at a 95% confidence level, and 1,068 for 3%. A smaller population needs fewer: 278 out of 1,000. This calculator works out the number for your survey, the margin of error a sample gives you, and how many responses each group needs when you compare two.

## How it works

The calculator uses the standard formulas found in any statistics textbook.

- **Sample size.** Cochran's formula: n = z² × p × (1 − p) ÷ e². z is 1.645, 1.96 or 2.576 for 90%, 95% or 99% confidence, p is the split you expect (50% when you don't know, which gives the largest sample) and e is the margin of error.
- **A smaller population.** When you know how many there are in total (N), the sample shrinks to n ÷ (1 + (n − 1) ÷ N). It only matters when your sample is a real share of everyone, like the staff of one company.
- **Margin of error.** The other way round: e = z × √(p × (1 − p) ÷ n), times √((N − n) ÷ (N − 1)) when you know the population.
- **Comparing two averages.** Responses per group = 2 × ((zα + zβ) ÷ d)², where d is the difference you want to find in standard deviations (Cohen's d: 0.2 is small, 0.5 medium, 0.8 large), zα comes from the significance level and zβ from the power.
- **Comparing two percentages.** Responses per group = (zα × √(2 × p̄ × (1 − p̄)) + zβ × √(p₁(1 − p₁) + p₂(1 − p₂)))² ÷ (p₁ − p₂)², where p̄ is the average of the two.
- **Invitations.** Not everyone you invite answers. Divide the responses you need by the share you expect to answer: 385 at a 20% response rate means about 1,925 invitations.

Sources: William G. Cochran, Sampling Techniques, 3rd edition (1977); Jacob Cohen, Statistical Power Analysis for the Behavioral Sciences, 2nd edition (1988).

## Questions people ask

### How many responses do I need for a survey?

For a large population, 385 completed responses give you a margin of error of 5% either way at a 95% confidence level. For 3% you need 1,068. Put your own numbers into the calculator to get yours.

### What is a margin of error?

How far your result could be from the true answer. With a 5% margin, a result of 60% means the real figure is most likely between 55% and 65%.

### Which confidence level should I pick?

95% is the usual choice. 90% needs fewer responses and is fine for a quick read. 99% needs about 70% more and suits decisions that are hard to undo.

### Does the size of my population matter?

Only when it's small. Above about 20,000 the sample barely changes: 385 for a whole country, 377 out of 20,000 and 278 out of 1,000.

### What should I put in for the expected split?

The share you expect to answer one way, like 30% saying yes. If you don't know, leave it at 50%: that gives the largest, safest sample.

### How many responses do I need to compare two groups?

It depends on how big a difference you want to find. For a medium difference (a Cohen's d of 0.5) with 80% power at a 5% significance level you need 63 per group, and for a small one (0.2) 393 per group.

### What is statistical power?

The chance your survey finds a difference that's really there. 80% is the usual choice, which leaves a 1 in 5 chance of missing it. 90% needs about a third more responses.

### How many people should I invite?

Divide the responses you need by the share you expect to answer. If 1 in 5 answer, 385 responses means about 1,925 invitations. With Recruit you order completed responses, so you don't have to guess. https://samplelabs.io/recruit

### Does a bigger sample fix bad data?

No. Bots, AI-written and rushed answers don't cancel out, they move your result, and a bigger sample only buys more of them. Count the responses you'd keep. Protect checks every response while people answer. https://samplelabs.io/protect