📋 Sample Size Calculator
Find how many survey responses you need for a given confidence level and margin of error – with an optional correction for small populations.
Quick answer: At 95% confidence with a ±5% margin of error and p = 50%, you need n = 1.96² × 0.5 × 0.5 ÷ 0.05² = 384.15, rounded up to 385 responses.
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Sample Size Calculator inputs
Result
Required sample size
385
responses for ±5% at 95% confidence
- z-score
- 1.96
- Without population correction
- 385
Step by step
- z-score for 95% confidence: z = 1.96
- n₀ = z² × p(1 − p) ÷ e² = 1.96² × 0.5 × 0.5 ÷ 0.05² = 384.15
- Round up to a whole number of responses: n = 385
If you expect only part of the people you contact to respond, divide by the response rate – e.g. 385 ÷ 0.3 = 1,284 invitations at a 30% response rate.
Sample sizes by margin of error and confidence
| Margin of error | 90% confidence | 95% confidence | 99% confidence |
|---|---|---|---|
| ±1% | 6,764 | 9,604 | 16,588 |
| ±2% | 1,691 | 2,401 | 4,147 |
| ±3% | 752 | 1,068 | 1,844 |
| ±4% | 423 | 601 | 1,037 |
| ±5% | 271 | 385 | 664 |
| ±10% | 68 | 97 | 166 |
The sample size formula
For estimating a proportion – the share of people who agree, buy, vote and so on – the required sample size is:
n₀ = z² × p(1 − p) ÷ e² · with a finite population N: n = n₀ ÷ (1 + (n₀ − 1) ÷ N)
z is the critical value for your confidence level, p the expected proportion (as a decimal) and e the margin of error (as a decimal). Always round up.
Worked example
You want ±5% precision at 95% confidence and have no idea of the true proportion, so p = 0.5.
- z for 95% confidence = 1.96.
- n₀ = 1.96² × 0.5 × 0.5 ÷ 0.05² = 3.8416 × 0.25 ÷ 0.0025 = 384.15.
- Round up: 385 responses.
- If the population is only 2,000 people: n = 384.15 ÷ (1 + 383.15 ÷ 2,000) = 322.4, so 323 responses.
Quick reference (p = 50%, large population)
| Margin of error | 90% | 95% | 99% |
|---|---|---|---|
| ±10% | 68 | 97 | 166 |
| ±5% | 271 | 385 | 664 |
| ±3% | 752 | 1,068 | 1,844 |
| ±2% | 1,691 | 2,401 | 4,147 |
| ±1% | 6,764 | 9,604 | 16,588 |
How to use the calculator
Pick a confidence level, the margin of error you can accept and the proportion you expect. Add the population size only if it is small (a few thousand or less); otherwise leave it blank. The table below the result shows how the sample changes with other choices.
Tips
- Halving the margin of error needs four times the sample.
- If you will report results for subgroups (e.g. by region), each subgroup needs its own adequate sample.
- The formula assumes a simple random sample; a biased sample is not fixed by making it bigger.
Frequently asked questions
Why use 50% as the expected proportion?
p(1 − p) is largest when p = 50%, so it gives the biggest – most conservative – sample size. If earlier research suggests the true proportion is near 10% or 90%, using that value lets you survey fewer people for the same precision.
Does population size matter?
Surprisingly little, once the population is large. 385 responses give ±5% at 95% confidence whether the population is 100,000 or 300 million. It matters only when the sample would be a sizeable fraction of the population – then the finite population correction reduces the number needed.
What margin of error should I choose?
±5% at 95% confidence is the common default for opinion surveys. Use ±3% (about 1,068 responses) for decisions that hinge on small differences, and ±10% (97 responses) for rough, exploratory research.
What about people who don’t respond?
The result is the number of completed responses you need. Divide by your expected response rate to find how many people to invite: at a 20% response rate, 385 responses means inviting about 1,925 people.