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Sample Size Calculator

Estimate survey response counts from confidence, precision, and population assumptions.

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Monthly active users, beta testers, customers. Use 1,000,000+ for "infinite" populations.

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Estimate of the population share with the trait. Leave at 50% for the most conservative sample.

Outreach planning (optional)

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Use a response-rate assumption based on similar outreach, if available.

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Required sample

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Enter population, confidence, and margin, then Calculate.

Why this matters

More responses improve precision under the sampling assumptions. Choose a margin narrow enough for your decision, then check recruitment and question wording. Sample size alone does not establish representativeness.
n₀ = (Z² × p × (1−p)) ÷ E²; n = n₀ ÷ (1 + (n₀−1)/N)

p = 0.5 gives the largest normal-approximation sample. N is population size; for an effectively infinite population, n = n₀.

Planning a survey sample

Choose the precision the decision needs, then check recruitment, time, and cost. Sample size affects sampling uncertainty; it does not make a biased or unrepresentative survey reliable.

The formula

For a large population, n₀ = Z² × p(1−p) / E², rounded up.

  • Z is the confidence critical value: about 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%.
  • p is the expected proportion. Use 0.5 when unknown for the largest normal-approximation sample.
  • E is the margin as a proportion: five percentage points is 0.05.

For sampling without replacement from a finite population N, use n = n₀ / (1 + (n₀−1)/N). The calculator treats populations of one million or more as effectively infinite.

NIST explains the normal-approximation sample formula. The calculation plans a single proportion; use the A/B planner to compare two variants.

Confidence and margin of error

A method producing 95% confidence intervals covers the true population proportion in 95% of repeated samples under its assumptions. Higher confidence needs a larger sample at the same margin.

The margin describes precision. A result of 60% with a five-percentage-point margin gives an interval from 55% to 65%. Choose a margin narrow enough for your decision rather than assigning one automatically to pricing, features, or exploratory work.

Responses, invitations, and time

If you need 400 responses and assume a 20% response rate, plan about 2,000 invitations. Use response history from a similar audience and method; the actual rate may differ. The timeline is a scenario based on the tool's collection-rate assumptions, not a guarantee of when responses will arrive.

Check who the survey represents

More responses do not fix selection bias, nonresponse bias, or misleading questions. AAPOR recommends reporting the sampling method and whether precision estimates account for the study design.

If you need to compare subgroups, plan their precision separately. A 400-person sample split equally across four groups gives only 100 responses per group. This formula also does not set a required number of qualitative interviews.

What is Survey sample size?

The response count needed to estimate a survey proportion at a chosen confidence level and margin of error. This model assumes suitable sampling; a larger sample does not remove selection or nonresponse bias.

Z: confidence critical value; p: expected proportion; E: margin as a proportion; N: population

n₀ = Z² × p(1−p) ÷ E²; n = n₀ ÷ (1 + (n₀−1)/N)

Calculated survey example

For a large population, 95% confidence, ±5 percentage points and p = 0.5 require 385 completed responses under the normal approximation.

NIST: planning a sample for a proportion
Planning an A/B comparison? Use the A/B planner→

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Common questions

What does this sample-size calculator plan?
It plans completed survey responses for estimating a proportion, such as the share of customers who select an answer. Enter confidence, margin of error, expected proportion, and population size. For a difference between A/B variants, use the separate A/B test planner.
How is the survey sample size calculated?
For a large population, n = Z² × p(1−p) ÷ E², rounded up. Z comes from the confidence level, p is the expected proportion, and E is the margin expressed as a proportion. For example, 95% confidence, p = 0.5, and E = 0.05 require 385 responses before finite-population correction.
What does a 95% confidence level mean?
Under the sampling assumptions, a method producing 95% confidence intervals covers the true population proportion in 95% of repeated samples. It does not mean that 95% of respondents answer correctly, or that the survey is free from bias.
What margin of error should I choose?
Choose the precision needed for the decision. A result of 60% with a five-percentage-point margin gives an interval from 55% to 65%. Narrower margins require more responses; they do not correct biased questions or unrepresentative recruitment.
How do population size and response rate affect the plan?
For sampling without replacement from a finite population, the calculator can reduce the required sample with a finite-population correction. The response-rate assumption estimates invitations needed to obtain that many responses. For example, 400 responses at an assumed 20% response rate need about 2,000 invitations; the actual response rate can differ.
Does reaching the sample size make my survey representative?
No. Sampling error is only one source of uncertainty. Check who can be selected, who responds, and whether the questions measure what you need. Opt-in or convenience samples do not automatically justify a conventional margin of sampling error.