How many responses do you need
Before launching a survey, you need to know how many completed responses will give results precise enough to act on. The answer depends on the margin of error you can accept, the confidence level you want and, to a lesser degree, the size of the population. The calculator gives the number of responses and, from your expected response rate, how many people to invite.
The formulas
- Sample size for a large population: n₀ = z² × p × (1 − p) ÷ e²
- Finite population correction: n = n₀ ÷ (1 + (n₀ − 1) ÷ N)
- Invitations = n ÷ response rate
Here z is the critical value (1.96 at 95%), p the expected proportion and e the margin of error, both as decimals. Use p = 50% when you have no prior estimate: it maximizes p × (1 − p) and gives the safest sample size.
Worked example
A company wants to measure customer satisfaction with a ±5% margin of error at 95% confidence, and expects 20% of invited customers to respond.
- n₀ = 1.96² × 0.5 × 0.5 ÷ 0.05² = 384.2, so 385 responses.
- At a 20% response rate, it must invite 385 ÷ 0.2 = 1,925 customers.
- If the company only has 10,000 customers, the correction lowers the target to 370 responses.
- For a tighter ±3%, it would need 1,068 responses from a large population.
Sample size by margin of error
At 95% confidence, p = 50%, large population:
| Margin of error | Responses | Invitations at 20% |
|---|---|---|
| ±1% | 9,604 | 48,020 |
| ±2% | 2,401 | 12,005 |
| ±3% | 1,068 | 5,340 |
| ±4% | 601 | 3,005 |
| ±5% | 385 | 1,925 |
| ±10% | 97 | 485 |
Why population size matters so little
Beyond a few thousand people, the population size barely changes the result: 385 responses give the same ±5% precision for a town of 50,000 as for the whole United States. What matters is the absolute number of responses, not the share of the population surveyed. The correction only helps when you survey a large part of a small group, such as the employees of one company.
Beyond the formula
- Subgroups: if you want to compare regions or age groups, each subgroup needs its own sample size.
- Bias beats precision: a large sample from the wrong people is worse than a small representative one. Watch for non-response bias, especially at low response rates.
- Experiments are different: for A/B tests and clinical trials, sample size depends on the effect you want to detect and the statistical power, not just the margin of error.
Use the margin of error calculator to check the precision of a survey you have already run.