Sample Size Calculator
Work out the minimum number of responses your survey or study needs. Choose between estimating a proportion (yes/no question) or a mean (average value), apply the finite population correction for small known groups, or reverse-solve: enter a sample size you already have and find the margin of error it delivers.
Formula
Worked example
95% confidence (z = 1.96), margin e = 0.05, p = 0.5: n = 1.96^2 * 0.5 * 0.5 / 0.05^2 = 0.9604 / 0.0025 = 384.16, rounded up to 385.
How the sample size formula works
For a proportion (yes/no question) the core formula is n = z^2 * p*(1-p) / e^2. The z-score comes from your confidence level: 1.96 for 95%, 2.58 for 99%. The term p*(1-p) is the variance of a proportion and peaks at p = 0.5, which is why an unknown split is the safe assumption. The margin of error e is squared in the denominator, so demanding twice the precision multiplies the required sample by four. For mean estimation the formula is n = (z * sigma / e)^2, where sigma is the estimated population standard deviation and e is the maximum acceptable error in the measurement unit. If you do not have a prior estimate of sigma, a quick pilot study of 30 observations or the range-divided-by-4 rule gives a reasonable starting point.
Finite population correction and reverse solving
The base formulas assume an effectively infinite population. When you are sampling from a small, known group, say the 800 employees of a company, the finite population correction n_adj = n / (1 + (n-1)/N) shrinks the requirement because each response carries proportionally more information. The effect is negligible once the population runs into the hundreds of thousands, which is why national polls quote a sample of roughly 1,000 regardless of country size. The reverse-solve mode flips the question: if you already have a fixed sample size (perhaps budget-constrained), it calculates the margin of error that sample delivers. This is useful for honest reporting, so your audience understands the precision of your results.
Accounting for non-response and planning your fieldwork
This calculator gives you the number of completed responses you need, not the number of people to contact. Because not everyone opens an email, answers a phone, or finishes a form, you must over-recruit. Toggle "Adjust for non-response" and enter your expected completion rate, for example, 30% for a cold email survey or 70% for a warm in-house panel. The calculator divides the required sample by that rate to tell you how many invitations to send. The confidence interval and margin of error only apply to completed responses, so partial responses or screened-out participants do not count toward the target.
Choosing your confidence level and margin of error
These two settings answer different questions. The confidence level is how often the procedure would capture the true value across many repeated studies. At 95%, 19 out of 20 intervals would contain the real figure. The margin of error is the half-width of that interval in percentage points (for proportions) or measurement units (for means). A 95% confidence level and a +/-5% margin of error is the accepted benchmark for most published survey research. Academic and clinical work often uses 99% for higher certainty. Internal or quick-turnaround research can tolerate 90% at +/-10% if the cost of more data collection is prohibitive.
Sample size reference at 95% confidence (p = 0.5, large population)
| Margin of error | Sample (95%) | Sample (99%) |
|---|---|---|
| +-1% | 9,604 | 16,588 |
| +-2% | 2,401 | 4,148 |
| +-3% | 1,068 | 1,844 |
| +-5% | 385 | 664 |
| +-10% | 97 | 166 |
Rounded up to the next whole response. Tightening the margin by half roughly quadruples the sample.
Frequently asked questions
Why is 50% the default expected proportion?
The variance term p*(1-p) is largest at p = 0.5, so assuming a 50/50 split produces the biggest sample requirement. Using it guarantees you are never under-powered, no matter how the responses actually break down. If you have a reliable prior estimate, say a feature is used by about 20% of users, entering it lowers the required sample.
What is the difference between the proportion and mean formulas?
Use the proportion formula when your question has a yes/no or pick-one answer and you report the result as a percentage. Use the mean formula when you are averaging a continuous measurement, like customer spend, test scores, or blood pressure. The mean formula needs an estimate of the standard deviation, which controls how spread out responses will be.
When do I need the finite population correction?
Only when your population is small and known. Leave it at 0 for large or unknown populations and the standard formulas apply. For groups under a few thousand the correction can meaningfully reduce how many responses you need. For example, surveying a team of 200 people at 95% confidence and +/-5% margin of error requires only 132 responses instead of 385.
Should I invite more people than this number?
Yes. This figure is the number of completed responses you need. Because not everyone replies, divide it by your expected response rate. For example, a target of 385 with a 30% response rate means inviting about 1,284 people. Toggle "Adjust for non-response" to have the calculator do this for you.
What does the reverse-solve mode tell me?
Switch to "Find margin of error" and enter the sample size you actually have (or can afford). The calculator tells you the margin of error your sample delivers at your chosen confidence level. This is useful when you are writing up results and need to honestly report precision, or when you want to check whether a pre-existing dataset is precise enough for your purpose.
How do I estimate the standard deviation for a mean calculation?
If you have run a small pilot study of 30 or more people, use the observed standard deviation. If not, use the range-divided-by-4 rule: estimate the plausible highest and lowest values, subtract them, and divide by 4. For example, if scores range from 0 to 100, sigma is roughly 25. It is better to over-estimate than under-estimate, as a higher sigma increases the required sample and keeps you from being under-powered.