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Sample Size Calculator for Surveys

How Many Survey Responses You Need for a Chosen Margin of Error

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How the Sample Size Calculator works

This sample size calculator tells you how many completed responses a survey needs to hit a chosen margin of error. Set the margin, the confidence level and, if you know it, the size of the group you are surveying, and it returns the sample needed with the working shown.

Too few responses and the results are noise; too many and you waste time and budget. The right number depends on how precise you need to be, how confident you want to be, and, for small groups, how many people there are in total. It does not depend on the population once that is large.

How the sample size calculator works

The formula starts with n₀ = z² × p(1 − p) ÷ e², where z is the critical value for the confidence level, p the expected proportion and e the margin of error. For a population of N it then applies the correction n = n₀ ÷ (1 + (n₀ − 1) ÷ N) and rounds up.

First example. For ±5% at 95% with p = 50%, n₀ = 1.96² × 0.25 ÷ 0.05² ≈ 384.2, so 385 responses.

Second example. For a company of 10,000 staff the correction brings that down to 370.

How many people should I invite? Divide the sample by the response rate you expect. At a 20% response rate, 385 completes needs about 1,925 invitations.

Sample size at 95% confidence and a 50% proportion by population size

Population ±10% margin ±5% margin ±3% margin ±1% margin
100 50 80 92 99
200 66 132 169 196
300 73 169 235 291
500 81 218 341 476
750 86 255 441 696
1,000 88 278 517 906
1,500 91 306 624 1,298
2,000 92 323 697 1,656
3,000 94 341 788 2,287
5,000 95 357 880 3,289
7,500 95 366 935 4,212
10,000 96 370 965 4,900
15,000 96 375 997 5,856
20,000 96 377 1,014 6,489
30,000 96 380 1,031 7,276
50,000 96 382 1,045 8,057
75,000 96 383 1,053 8,514
100,000 96 383 1,056 8,763
250,000 96 384 1,063 9,249
500,000 97 384 1,065 9,423
1,000,000 97 384 1,066 9,513
10,000,000 97 385 1,067 9,595
100,000,000 97 385 1,068 9,603
Unlimited 97 385 1,068 9,604

Tips

FAQ

What sample size is statistically significant?

There is no single number. Significance depends on the effect you want to detect and the precision you need. For estimating a percentage, 385 responses give ±5% at 95% confidence from any large population, and 1,068 give ±3%, which is why many national polls use about a thousand people.

Why does the sample barely grow for larger populations?

The precision of a random sample depends on the number of responses, not the share of the population. Beyond about 100,000 people the corrected and uncorrected answers are almost identical, so surveying a city or a whole country needs roughly the same sample size.

Does the calculator work for small populations?

Yes. Enter the population size and the finite population correction reduces the sample. For a club of 200 members, ±5% at 95% needs 132 responses rather than 385, because each reply covers a much larger share of the whole group.

See also: all Statistics tools, Arithmetic, Number Theory, Algebra and Geometry, Trigonometry and Calculus.

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