Sample Size Calculator
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Results
You need a sample of 385 for ±5% at 95% confidence
| Margin of error | Sample size |
|---|---|
| ± 1% | 9,604 |
| ± 2% | 2,401 |
| ± 3% | 1,068 |
| ± 4% | 601 |
| ± 5% | 385 |
| ± 6% | 267 |
| ± 7% | 196 |
| ± 8% | 151 |
| ± 10% | 97 |
This is the number of completed responses you need. If you expect, say, a 25% response rate, invite about four times as many people.
How to use the sample size calculator
- Choose whether you want the sample size or the margin of error.
- Set the confidence level (95% is standard) and the margin of error you can accept, such as ±5 percentage points.
- Enter the proportion you expect. If you have no idea, leave it at 50%, which gives the largest and safest sample.
- If the whole population is small and known, such as 800 employees, enter it to apply the finite population correction.
Cochran's sample size formula
With a population of N: n = n₀ ÷ (1 + (n₀ − 1) ÷ N)
Margin of error for a sample of n: e = z × √(p(1 − p) ÷ n) × √((N − n) ÷ (N − 1))
Here z is the critical value for the confidence level (1.96 for 95%), p the expected proportion as a decimal, e the margin of error as a decimal and N the population size. The result is rounded up to the next whole person. These are the standard formulas in William G. Cochran, Sampling Techniques (3rd edition, Wiley, 1977), chapter 4, for estimating a proportion with simple random sampling. The last factor in the margin of error is the finite population correction; leave the population blank and it is 1.
Worked example
For 95% confidence and a ±5% margin with p = 50%: n₀ = 1.96² × 0.5 × 0.5 ÷ 0.05² = 384.1, which rounds up to 385 responses. If the population is only 1,000 people, n = 384.1 ÷ (1 + 383.1 ÷ 1,000) = 277.7, so 278 responses are enough. Running it in reverse, a sample of 1,000 from a large population gives a margin of error of ±3.1%.
Sample size by margin of error
Large population, expected proportion 50%:
| 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 |
| ±7% | 139 | 196 | 339 |
| ±10% | 68 | 97 | 166 |
Halving the margin of error roughly quadruples the sample: precision gets expensive quickly.
Does population size matter?
Less than most people expect. At 95% confidence and p = 50%:
| Population | Sample for ±5% | Sample for ±3% |
|---|---|---|
| 100 | 80 | 92 |
| 250 | 152 | 203 |
| 500 | 218 | 341 |
| 1,000 | 278 | 517 |
| 2,500 | 334 | 749 |
| 5,000 | 357 | 880 |
| 10,000 | 370 | 965 |
| 100,000 | 383 | 1,056 |
| 1,000,000 | 384 | 1,066 |
| Very large | 385 | 1,068 |
Once the population is more than a few thousand, the required sample barely changes. A well-drawn random sample of about 1,000 can describe a city or a whole country with similar precision.
Tips for planning a survey
- Plan for non-response. The result is completed responses. Divide by the response rate you expect: at a 20% response rate, invite 5 times as many people.
- Size for your subgroups. If you will report results for each region or age group separately, each group needs its own adequate sample.
- Random beats big. The formula assumes a random sample. A huge self-selected online poll can be far less accurate than a small random one.
- Margin of error is about sampling only. It does not cover biased wording, people who refuse to take part or untruthful answers.
- Estimating a mean? The analogous formula is n = (z × σ ÷ e)², which needs a guess at the standard deviation σ.
Once the data is in, the confidence interval calculator gives the interval for the proportion you found, including the Wilson interval for small samples.
Frequently asked questions
What sample size do I need for a 95% confidence level and 5% margin of error?
About 385 completed responses for a large population. Cochran’s formula gives n₀ = 1.96² × 0.5 × 0.5 ÷ 0.05² ≈ 384.1, which rounds up to 385.
How does population size affect the sample size?
For small populations the finite population correction lowers the number needed: n = n₀ ÷ (1 + (n₀ − 1) ÷ N). At 95% confidence and ±5%, a population of 1,000 needs 278, 10,000 needs 370 and 1,000,000 needs 384. Beyond a few thousand, size barely matters.
Why use 50% as the expected proportion?
p(1 − p) is largest at p = 50%, so it gives the most conservative (largest) sample when you do not know the answer in advance. If you are confident the proportion is near 20%, the requirement at 95% and ±5% falls from 385 to 246.
What is the margin of error for a sample of 1,000?
At 95% confidence with p = 50%, it is 1.96 × √(0.25 ÷ 1,000) ≈ ±3.1 percentage points, which is why many national polls interview about 1,000 people.
Does the sample size account for people who do not respond?
No. The result is the number of completed responses you need. Divide it by your expected response rate to find how many to invite: for 385 responses at a 25% response rate, invite about 1,540 people. Non-response can also bias results, which no sample size fixes.