The standard Cochran formula
n = z²p(1 − p) ÷ e², the same formula behind the sample size tables you see quoted in research briefs.
Tools / Sample Size Calculator
Sample Size Calculator
Enter the margin of error you can live with, pick a confidence level, and get the number of completed responses you need before you start fielding.
n = z² × p(1 − p) ÷ e², with an optional finite population correction
Enter values to see the result
- 1
Set your margin of error
How many percentage points of swing you can accept. 5% is a common default, 3% is tighter and much more expensive.
- 2
Pick a confidence level
95% is the standard. 90% needs fewer responses, 99% needs many more.
- 3
Read the required n
That is completed responses, not invitations. Divide by your expected response rate to plan the send.
Why calculate sample size first
Fielding a survey and then discovering the result swings by 8 points is an expensive way to learn this. Size the sample before the invitations go out.
Finite population correction
Surveying 200 employees rather than the open web? Enter the population and the required sample drops accordingly.
Conservative by default
The expected proportion starts at 50%, which produces the largest sample requirement. Safe when you do not know the answer yet.
When to size a sample
Planning a survey wave
Decide how many invitations to send once you know your usual response rate.
Research proposals and class projects
Justify n with a stated confidence level and margin instead of a round number.
Employee and member surveys
Use the population field when the group is known and small, so you do not over-collect.
The formula
n₀ = z² × p(1 − p) ÷ e² · n = n₀ ÷ (1 + (n₀ − 1) ÷ N)
z comes from the confidence level (1.96 at 95%), p is the expected proportion, and e is the margin of error as a decimal. The second step is the finite population correction, applied only when you enter a population size N.
Sample size at 95% confidence, p = 50%
±1%
9,604 responses
±3%
1,068 responses
±5%
385 responses
±7%
196 responses
Sample size vs margin of error
Sample size (this page)
- You know the precision you want
- Output is the number of responses
- Use it before fielding
Margin of error calculator
- You know how many responses you will get
- Output is the ± band
- Use it when n is fixed
Related calculators
Frequently asked questions
How do you calculate sample size for a survey?
Use the Cochran formula for a proportion:
n = z² × p(1 − p) ÷ e²
z is the critical value for your confidence level (1.96 at 95%), p is the expected proportion (use 0.5 when unknown), and e is the margin of error as a decimal (0.05 for ±5%).
At 95% confidence, ±5% margin, and p = 0.5, that gives 385 responses.
How many responses do I need for a 5% margin of error?
385 completed responses at 95% confidence, assuming a large population and a 50% expected proportion.
At 90% confidence the same margin needs about 271. At 99% it needs about 664. Tightening the margin costs far more than raising confidence: ±3% at 95% needs about 1,068.
Why is 385 the magic number?
It is what the formula returns for the most commonly quoted combination: 95% confidence, ±5% margin, and the conservative p = 0.5.
Because the population term drops out once the population is large, 385 is the answer whether you are surveying a city or a country. That is why the same figure shows up in so many research summaries.
Does population size matter?
Only when the population is small relative to the sample. Surveying 200 employees at ±5% needs about 132 responses, not 385, because the finite population correction applies.
Once the population is above roughly 20,000, the correction changes the answer by less than a response or two, so you can leave the field blank.
What expected proportion should I enter?
Leave it at 50% unless you have a solid prior. p(1 − p) is largest at 0.5, so that assumption gives the biggest sample requirement and never leaves you short.
If you already know the rate is near 10% or 90%, entering it will cut the required sample substantially.
Is this the number of invitations to send?
No. It is the number of completed responses you need. Divide by your expected response rate to get the send size: 385 completes at a 25% response rate means roughly 1,540 invitations.
The response rate calculator has published rates by channel if you need a starting estimate.
What do I do after the survey closes?
Use the actual counts with the confidence interval calculator to get a Wilson interval around each percentage. Planning assumes p = 0.5; after fielding you know the real rate, so the interval is usually tighter than the plan.
Are my numbers stored?
No. Everything runs locally in your browser.
How do I actually collect the responses?
Any survey tool works. Formms builds the form from a prompt and gives you a short link and QR code to share, then you can bring the counts back to these calculators.
Collecting fresh survey responses? Try Formms for a form with a short link and QR.
Collect the responses on your phone next.
Formms builds the survey from a prompt, then you can paste counts back into these calculators anytime.
View a demo form →