SURVEY RESEARCH / FREE TO USE

Sample size calculator

Plan how many completed survey responses to aim for, with your confidence level and margin of error built in.

No account needed. A little clarity, on the house.

Start with your research plan.

This estimates a sample for one population proportion, not an A/B-test power analysis.

Population size (optional)

Leave blank for an unknown or very large population.

Confidence level

Margin of error (%)

In percentage points, between 0.01 and 99.99.

Expected proportion (%)

Use 50% when unsure; it gives the largest estimate.

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Your result

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Add your numbers and calculate. Your result, breakdown, and a little context will appear here.

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BEHIND THE RESULT

Know what goes into it.

The first formula estimates a sample for a large population. If you supply a finite population N, the second adjusts it for sampling without replacement. The answer is rounded up. Enter margin e and expected proportion p as percentages; they are converted to fractions.

Penn State’s sample-size formulas

THE FORMULA

n₀ = z² × p(1 − p) ÷ e²; n = n₀ ÷ [1 + (n₀ − 1) / N]

A quick example

At 95% confidence, a 5-point margin and 50% expected proportion require 385 responses for a large population, or 278 from a population of 1,000.

Keep in mind

Assumes a simple random sample and a normal approximation for a proportion.
Does not account for design effects, sampling bias, or nonresponse.
Very small or extreme-proportion samples can require a different planning method.

Plan completed responses and invitations separately

Choose a confidence level, margin of error, and expected response proportion. If you have no defensible estimate for the proportion, 50% gives the largest sample under this formula. Enter population size when you know it; leave the field empty for the large-population calculation.

For a large population with a 50% proportion and 95% confidence, the normal approximation gives approximately 385 completed responses for a ±5 percentage-point margin of error, 1,068 for ±3 points, and 97 for ±10 points. These are planning examples under the stated simple-random-sampling assumptions.

If you need 400 completed responses and expect a 25% response rate, plan about 1,600 invitations: 400 ÷ 0.25. Allow for ineligible or unusable responses as well. More invitations do not automatically fix nonresponse bias.

GOOD QUESTIONS

A little more
context.

The useful details behind the numbers and links.

Why use 50% when I do not know the proportion?

The term p × (1 − p) is largest at 50%, so it gives the most conservative sample estimate within this formula.

Is this how many invitations I should send?

No. It is the number of completed responses to aim for. You may need more invitations depending on your expected response rate.

How should I choose confidence level and margin of error?

Choose them before collecting data, according to the uncertainty acceptable for the decision. Higher confidence and a smaller margin require more responses under the same assumptions. Neither setting compensates for biased sampling.

What if I do not know the population size?

Leave population size blank to use the large-population calculation. If you later know a finite eligible population, enter it and review the finite-population correction.

Start with your research plan.

Your result

Ready when you are.

Good answers start with a better survey.

Know what goes into it.

A quick example

Keep in mind

Plan completed responses and invitations separately

A little morecontext.

While you’re here.

Survey response rate calculator

Confidence interval calculator

Statistical significance calculator

A little more
context.