Sample Size Calculator

Four choices set the number of respondents a survey needs: how precise the answer must be (the margin of error), how sure you want to be (the confidence level), how variable the answers are (the expected proportion, or the standard deviation of a measurement) and, only for a small population, how many units the population holds. Precision is expensive: because the margin shrinks with the square root of n, halving it takes four times the respondents.

The result is a minimum number of completed responses from a simple random sample. Non-response, clusters and unequal weights all raise the number you have to collect, as the sections below show.

Here e is the margin of error as a proportion (0.05 for ±5%), E the margin in the units of the measurement, p the expected proportion, and z the critical value for the confidence level: 1.6449 for 90%, 1.9600 for 95% and 2.5758 for 99%. Always round the result up, because the size is a minimum. Without a prior estimate of p use 50%, which maximises p(1 − p) and so guarantees at least the chosen precision whatever the true value.

This calculator uses the exact z value (1.959964 at 95%) rather than the rounded 1.96, so its unrounded n₀ for the default survey is 384.1459 where a textbook that rounds z gets 384.16. Both round up to 385.

1. Pick the calculation mode: a proportion (yes/no or percentage answers), a mean with a known σ, a mean whose σ is estimated (uses t), or the margin of error you get from a fixed sample size.
2. Choose the confidence level and the margin of error. For a proportion type the margin in percentage points (5 for ±5%); for a mean use the units of the measurement.
3. Enter the expected proportion (leave 50 if unsure) or the standard deviation of a mean from a pilot study or earlier research.
4. Optionally add the population size, if the population is small and known, and the expected response rate to see how many people to invite.
5. Press Calculate Sample Size. Use Copy link to this calculation to share the inputs.

A 95% confidence level, a margin of ±5 percentage points and no prior estimate of the proportion (p = 50%).

Those 385 respondents deliver a margin of ±4.99 points, just under the 5 that was asked for. For ±3 points the same formula gives n₀ = 1,067.07, so 1,068 responses.

The same survey in a company of 2,000 people. The population is small enough that the sample would be a noticeable share of it, so the finite population correction applies:

The correction saves 62 interviews. In a group of only 100 people, 80 responses are enough for the same precision. The smaller the population, the more the correction matters; for a large population it changes nothing.

You need 385 completed responses but expect only 30% of the people you contact to answer. Invite 385 ÷ 0.30 = 1,283.33, that is 1,284 people. Enter the response rate in the optional field to see this in the results. More invitations do not cure non-response bias: if the people who answer differ from those who do not, the estimate is off by more than the margin of error suggests.

Suppose the standard deviation of a measurement is about σ = 10 and you want a 95% margin of ±2 units.

The t-based size is 99, two more than the z-based 97, because t is larger than z for a finite sample. The multiplier itself depends on n, so the calculator searches upward from the z-based size for the smallest n that satisfies the inequality, rather than iterating on rounded values. The search is exact up to sample sizes of 100,000; beyond that the t multiplier and z differ by so little that the known-σ mode should be used.

Simple random sample, large population, expected proportion 50%:

The numbers grow with 1/e²: going from ±5% to ±1% multiplies the sample size by 25. About 1,070 respondents give ±3% at 95% confidence, the precision of a typical national poll, and ±2% needs 2,401.

95% confidence, ±5% margin, expected proportion 50%:

Once the population is above a few tens of thousands its size no longer matters: 383 respondents for 100,000 people and 384 for a million. The sample size depends on precision, not on how large the population is.

95% confidence, ±5% margin. The size depends on p(1 − p), so p and 100% − p need the same sample:

If earlier research tells you the answer is near 10% or 90%, you can survey 139 people instead of 385. If you assume 10% and the true value is 40%, the sample is too small for the margin you wanted, whereas assuming 50% can only make the survey larger than necessary, so use it when in doubt. For rare outcomes a margin of ±5 points is very wide compared with p itself, so choose the margin relative to p and check that both n × p and n × (1 − p) are at least 10, or use the Wilson interval of the proportion confidence interval calculator afterwards.

• Clusters and weights. Cluster sampling, stratification with unequal weights and other complex designs need a larger sample. Multiply n₀ by the design effect and round up: with a design effect of 1.5, the default 384.1459 becomes 576.2, so 577 responses.
• Subgroups. The margin applies to the group you measure. If you will report on four regions separately and want ±5% in each, you need 385 in each region, not 385 overall.
• Non-response. Divide by the response rate, as in the example above, and try to learn how responders differ from non-responders.
• Hypothesis tests. This page sizes a study for the precision of an estimate. When the goal is to detect a difference with a chosen power, use the A/B test sample size calculator or the statistical power calculator.
• Check the outcome. The margin of error calculator turns the responses you actually obtained into the interval you can report, and the confidence interval calculator builds it around a mean.

These use the same exact z values as the calculator. The t-based mean size has no one-line spreadsheet formula because t depends on n; the calculator finds it by search. The article sample size explained walks through the reasoning behind each formula.