Survey Sample Size Calculator for a Finite Population (with Margin of Error)
Find how many completed survey responses you need for a target margin of error and confidence level, corrected for a finite population, or find the margin of error a given sample gives you. Includes response rate, design effect and sample size tables.
How it works
n₀ = z² × p × (1 − p) × deff ÷ e² · n = n₀ ÷ (1 + (n₀ − 1) ÷ N), rounded up (same as n = N z² p q ÷ [e² (N − 1) + z² p q]) · margin of error e = z × √(p (1 − p) deff ÷ n) × √((N − n) ÷ (N − 1)) · invitations = n ÷ response rate
This is Cochran’s sample size formula for estimating a proportion (such as the share of people who answer “yes”), with the finite population correction. The uncorrected n₀ assumes an effectively infinite population; when your sample is a noticeable share of a small group, each response tells you more and fewer are needed. The correction n₀ ÷ (1 + (n₀ − 1) ÷ N) is the exact form that follows from the (N − n) ÷ (N − 1) variance factor, and it is algebraically the same as the textbook formula n = N z² p q ÷ [e² (N − 1) + z² p q]. Some calculators use the simpler n₀ ÷ (1 + n₀ ÷ N), which gives a result one or two lower. Use 50% for the expected proportion when you have no prior estimate, because p(1 − p) is largest there and the sample is big enough for any true value. The z value comes from the confidence level: 1.645 for 90%, 1.96 for 95% and 2.576 for 99%. The design effect inflates the sample for cluster or stratified designs that are less efficient than simple random sampling. Switch to “Margin of error” to see how precise a survey with a given number of responses is. The result is completed responses; divide by your expected response rate to know how many people to invite. Sampling error is only part of survey error: a biased list or low response can matter more than the sample size.
Worked example
For a population of 10,000 at 95% confidence (z = 1.96), ±5% margin of error and p = 50%: n₀ = 1.96² × 0.5 × 0.5 ÷ 0.05² = 384.15. With the correction, n = 384.15 ÷ (1 + 383.15 ÷ 10,000) = 369.97, rounded up to 370 completed responses (385 without the correction). That is 3.7% of the population, and those 370 responses give a margin of error of ±5.00%. At a 25% response rate you would invite 1,480 people.
Sources: Sample size determination (Cochran’s formula, finite population correction), Wikipedia · Calculator.net: Sample size calculator (finite population correction factor) · Omni Calculator: Sample size calculator
Frequently asked questions
When does the population correction actually matter?
When your sample is a noticeable share of the population, roughly above 5%. For a national survey of millions it changes almost nothing, but for a 500-person company the 95%, ±5% sample drops from 385 to 218.
Why is 50% the default expected proportion?
The variance term p(1 − p) is largest at p = 0.5, so 50% guarantees a sample big enough for any true proportion. If you know the answer is near 10% or 90%, entering it gives a smaller sample that is still valid.
Is this the number of invitations to send?
No, it is the number of completed responses. Divide by your expected response rate: with 370 needed and a 25% response rate, invite about 1,480 people. Enter the rate under More options to have it worked out.
What z value goes with each confidence level?
1.645 for 90%, 1.96 for 95%, 2.326 for 98% and 2.576 for 99%. For a population of 10,000 at ±5%, those give 264, 370, 514 and 623 responses.
Why does my answer differ by one from another calculator?
Calculators differ in rounding z (1.96 versus 1.959964), in whether they use n₀ ÷ (1 + (n₀ − 1) ÷ N) or the simpler n₀ ÷ (1 + n₀ ÷ N), and in whether they round up. This one always rounds up, so the margin of error is met.
What is the design effect?
The ratio of your design’s variance to a simple random sample of the same size. Cluster samples (for example surveying whole classrooms) are less efficient, often with a design effect of 1.5 to 2, so they need proportionally more responses.
Related calculators
For planning purposes. The margin of error covers random sampling error only, not bias from non-response or an unrepresentative sample.