Sample Size Calculator
Find how many people you need to survey. Choose a confidence level and margin of error, leave the expected proportion at 50% if you are unsure, and add your population size to apply the finite population correction. The result shows Cochran's formula worked through with your numbers. Free, no sign-up.
Results
385
Required sample size
1.96
Z-score used
Infinite
Population
How this was calculated
n₀ = z²p(1−p) / e² = 1.96² × 0.5 × (1 − 0.5) / 0.05² = 0.9604 / 0.0025 = 384.16
Rounding up to a whole respondent gives a required sample size of 385. This assumes simple random sampling; add extra respondents to cover expected non-response.
Sample size at common margins of error (95% confidence)
| Margin of error | Required sample size |
|---|---|
| ±3% | 1,068 |
| ±5% | 385 |
| ±10% | 97 |
How to Use the Sample Size Calculator
- 1. Choose the confidence level. 90%, 95% or 99% (z = 1.645, 1.96 or 2.576). 95% is the usual default for surveys and research.
- 2. Set the margin of error. This is the plus-or-minus you can accept on a percentage result, such as 5%. A smaller margin needs a larger sample.
- 3. Set the expected proportion. Leave it at 50% unless you know roughly what share will answer "yes". 50% is the most cautious choice because it gives the largest sample.
- 4. Add the population size, if you know it. Leave it blank for a very large or unknown population. A figure applies the finite population correction, which lowers the answer when your population is small.
- 5. Read the sample size. The result is rounded up to a whole respondent. A table below shows the sample sizes at 3%, 5% and 10% margins for your settings.
Cochran's Formula
For a yes-or-no question in a large population, the required sample size is n₀ = z² × p × (1 − p) ÷ e², where z is the z-score for your confidence level, p is the expected proportion and e is the margin of error, both as decimals. For a known population N the correction is n = n₀ ÷ (1 + (n₀ − 1) ÷ N). The calculator applies it and caps the result at N, since you cannot sample more people than exist.
95% confidence, 5% margin, p = 50%
n₀ = 1.96² × 0.5 × 0.5 ÷ 0.05² = 0.9604 ÷ 0.0025 = 384.16, rounded up to 385
Population of 10,000: n = 384.16 ÷ (1 + 383.16 ÷ 10,000) = 369.98, so 370
Population of 1,000: n = 384.16 ÷ (1 + 383.16 ÷ 1,000) = 277.74, so 278
Population of 500: n = 217.49, so 218
The last three lines show why population size matters only when the sample is a sizeable fraction of it. Above about 100,000 the correction makes almost no difference, which is why national polls survey roughly a thousand people whatever the country's size.
What Changes the Answer
| Setting | Sample size (large population, p = 50%) |
|---|---|
| 90% confidence, 5% margin | 271 |
| 95% confidence, 5% margin | 385 |
| 99% confidence, 5% margin | 664 |
| 95% confidence, 3% margin | 1,068 |
| 95% confidence, 10% margin | 97 |
Halving the margin of error multiplies the sample by about four, because e is squared in the denominator. If the true proportion is not 50%, the requirement falls: with p = 20%, 95% confidence and a 5% margin you need 246.
Limits and Adjustments
- Simple random sampling is assumed. Cluster sampling, panels and convenience samples need a larger size or a design effect, and a margin of error means nothing for a self-selected sample.
- Non-response. The number is completed responses, not invitations. If you expect a 40% response rate, invite 385 ÷ 0.40, about 963 people.
- Subgroups. If you will report results for a subgroup, such as one age band, that subgroup needs its own adequate sample.
- Proportions only. This tool is for percentages. For estimating a mean, Cochran's n₀ = z²σ² ÷ e² needs a standard deviation; to compare two groups you need a power analysis.
- Rounded z-values. The calculator uses 1.645, 1.96 and 2.576, which round the exact quantiles slightly, so a hand calculation with more decimals may differ by one.
Which Tool Do You Need?
| Your question | Use |
|---|---|
| How many people to survey for a given margin of error | This Sample Size Calculator |
| What margin of error a given sample gives | Margin of Error Calculator |
| How many participants to detect an effect in a t-test or a comparison of two proportions | Statistical Power & Sample Size Calculator |
| Sample size split across strata | Stratified Sample Size Calculator |
| An interval around a result you already have | Confidence Interval Calculator |
Frequently Asked Questions
- How is the sample size calculated?
- Using Cochran's formula n₀ = z² p (1 − p) ÷ e², where z is the z-score for the confidence level, p the expected proportion and e the margin of error. If you give a population size N, it applies n = n₀ ÷ (1 + (n₀ − 1) ÷ N) and rounds up.
- What confidence level should I choose?
- 95% is the standard for most surveys and academic work. Choose 99% when errors are costly and you can afford a larger sample, and 90% for quick exploratory work where a smaller sample is acceptable.
- Why is the expected proportion set to 50%?
- p(1 − p) is largest at p = 0.5, so 50% gives the biggest, safest sample size. Use a different value only if earlier data or a pilot study suggests the true share is far from half.
- What if I do not know my population size?
- Leave it blank. The calculator then uses the large-population formula, which is slightly conservative for small populations and exact enough for any population above a few tens of thousands.
- Does a bigger population always need a bigger sample?
- No. Beyond a point, the required sample barely changes: at 95% confidence and a 5% margin you need 383 for a population of 100,000 and 385 for 10 million. Population size only matters when the sample would be a large share of it.
- How many survey responses do I need for 95% confidence and a 5% margin of error?
- 385 for a large population, assuming simple random sampling and p = 50%. For a population of 1,000 it is 278, and for 10,000 it is 370. Plan for more invitations than that to allow for people who do not respond.