ToolForage.
STATISTICS CALCULATOR

Sample Size Calculator

Plan a sample for estimating a population proportion or mean. Set confidence, margin of error, and any finite population or response-rate adjustment.

Calculator

Use 50% when no prior estimate is available.
percentage points
Absolute percentage points, not a relative percentage.
Use the same measurement unit as standard deviation.
Finite correction assumes simple random sampling without replacement.
YOUR RESULTS
Completed sample needed385
Invitations at expected response rate
385
Before population correction
385
Two-sided z critical value
1.959964

Targets assume simple random sampling. Nonresponse and biased selection can still affect estimates.

Completed responses and invitations needed

The completed sample is rounded upward after the finite-population correction. Invitations are rounded upward separately using the expected response rate. They are planning estimates, not a guarantee of enough representative responses.

After collecting observations, estimate interval bounds with the Confidence Interval Calculator .

The formula

Proportion: n₀ = z²p(1 − p)/E². Mean: n₀ = (zσ/E)². Finite population: n = n₀ / [1 + (n₀ − 1)/N]. Invitations = ceiling(n / response rate).

For a proportion, p and E are converted from percentages to decimals. For a mean, σ and E have the same measurement unit. A zero population input skips finite correction. The minimum completed sample is one; larger results are always rounded upward.

Worked example

At 95% confidence, a 5 percentage-point margin, and expected proportion 50%, an unknown large population needs 385 completed responses. Population 1,000 reduces the target to 278; at an 80% response rate, plan 348 invitations.

How to use this calculator

  1. Choose proportion or mean and the two-sided confidence level.
  2. Enter an absolute error margin and a prior proportion or standard deviation.
  3. Supply population size if known, then use response rate to estimate invitations.

Use percentage points for a survey proportion

A margin of 5 percentage points around 50% refers to roughly 45% through 55%, not 47.5% through 52.5%. Use 50% when you lack a defensible prior proportion because p(1 − p) is largest there. A guess closer to zero or one can reduce the calculated sample but may understate your needs if the guess is wrong.

Sampling design changes the answer

The finite-population correction applies to a simple random sample without replacement from a known population. It does not repair coverage gaps or biased recruitment. Cluster sampling, stratification, weighting, and repeated observations can change precision. This tool does not estimate design effects or plan a hypothesis test’s power.

Recruitment is a separate planning step

Expected response rate converts a completed-sample target into an invitation count. Responses may be uneven across subgroups, so a large total alone does not ensure useful subgroup estimates. If invitations exceed a finite population, review the response-rate assumption or the desired precision. Mean planning needs a reasonable prior population spread, not the standard error of a previous mean.

Assumptions & limitations

What this calculation assumes

  • Normal-approximation planning; simple random independent observations.

What to keep in mind

  • No power analysis, design-effect multiplier, subgroup quotas, or nonresponse-bias adjustment; targets above one billion are rejected.

Common questions

Does 95% confidence mean 95% of people must respond?

No. Confidence describes the repeated-sampling procedure. Response rate is a separate proportion of invited people expected to complete the survey.

Why round upward?

Rounding down would produce fewer observations than the calculated planning requirement.

Sources & further reading