Sample size calculator
Math & numbersHow many responses does your survey need?
The number of survey responses you need for a margin of error and confidence level, for any population size.
What you enter stays on your device while you use this tool. Opening a shared link sends the settings in its address to HowToolbox to prepare the page.
More options
Responses needed
385
Completed responses for a margin of error of ±5% at 95% confidence.
n₀ = z² × p(1 − p) ÷ e² = 1.96² × 0.5 × 0.5 ÷ 0.05² = 384.1
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The number of survey responses you need for a margin of error and confidence level, for any population size.
How to use
- Enter the margin of error you can accept and the confidence level you want.
- If you survey a group of known size, such as a company’s staff, open More options, choose Known size and enter it.
- Read how many completed responses you need; you can also set the share you expect under More options if you have a good estimate.
Survey responses at 95% confidence and a 50% expected share
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How it works
A survey of a sample estimates what the whole group thinks. The margin of error is how far the sample's answer may be from the true share, and the confidence level is how often a range that wide would catch the truth if you repeated the survey. Narrower margins and higher confidence both need more responses.
The basic formula is n₀ = z² × p(1 − p) ÷ e², where z is the critical value of the normal distribution for the confidence level (1.96 for 95 %), p the expected share and e the margin as a fraction. For ±5 % at 95 % with p = 0.5 that is 1.96² × 0.25 ÷ 0.05² ≈ 384.2, so 385 responses. For a known population N the finite population correction n = n₀ ÷ (1 + (n₀ − 1) ÷ N) lowers it: 278 for a population of 1,000. The formulas follow the NIST/SEMATECH e-Handbook and Cochran's Sampling Techniques.
The result counts completed responses from a random sample. It cannot fix a biased sample: people who choose to answer may differ from those who don't, so plan to invite more people than the number shown. An expected share of 50 % gives the largest, safest size.
Frequently asked questions
Why does a population of a million need almost the same sample as a population of 10,000?
Precision depends mostly on how many people answer, not on what fraction of the population they are. The correction matters only when the sample is a sizeable part of the whole group.
What confidence level should I use?
95 % is the common choice. 90 % needs fewer responses, and 99 % needs about 70 % more than 95 %.
What should I put as the expected share?
Leave it at 50 % unless an earlier survey tells you the answer will be far from even. 50 % gives the largest sample, so it is never too small.
Does this tell me how many invitations to send?
No, it gives the responses you need. Divide by the response rate you expect: needing 385 responses at a 20 % response rate means inviting about 1,925 people.
Can I use this to estimate an average?
No. This calculator estimates a proportion, such as the share answering yes. Planning a sample for an average also needs an estimate of the values’ spread.
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