Sample Size Calculator
Plan completed survey responses, expected invitations and sampling precision. Explore the trade-off between sample size and margin of error.
How many completed responses do you need? Plan the precision of one survey percentage, then explore what your available sample could achieve.
About ±4.99 percentage points at 95% confidence, assuming a 50% share.
1,540 expected invitations at a 25% completion rate
This estimates random sampling error for a simple random sample. A larger sample cannot fix a biased selection or nonresponse. An open-link or volunteer survey does not acquire this margin merely by reaching the count.
What could your sample achieve?
Compare the original target with a number of completed responses you can collect. The curve uses the same population, share and confidence level.
385 responses → about ±4.99 points
Expected invitations: 1,540 at 25% completion.
Start with an example
Illustrative survey plans. Each replaces all settings and remains editable.
Method, assumptions and limits
What this plan means
Margin is measured in percentage points: ±5 points around 50% means 45–55%, not ±5% of 50. The expected share is a planning assumption, not an observed survey result. A prior share other than 50% can reduce the count; if that guess is wrong, the actual precision can be worse.
A 95% confidence procedure is designed to cover the fixed population share in approximately 95% of repeated random samples. It does not say that each survey is 95% representative or that a particular fixed parameter has a 95% probability of being in your interval.
One proportion, simple random sampling
Let p be the expected share, e the target margin as a fraction and z the two-sided standard-normal quantile. For a large population, n₀ = z²p(1 − p)/e². For a known population N sampled without replacement, n = Nn₀/(N − 1 + n₀). Round up once to a whole number of completed responses.
For k completed responses the approximate planning margin is z√[p(1 − p)/k] for a large population, multiplied by √[(N − k)/(N − 1)] for a known finite group. A census has zero sampling error in this model; measurement and nonresponse errors remain. The curve never promises actual interval coverage from non-random respondents.
Expected invitations = round up(responses ÷ expected completion fraction). If this exceeds the known population, the assumed response rate makes the target impractical from that group. Inviting everyone still does not guarantee enough answers, and respondents may differ from nonrespondents.
This is a normal-approximation planning tool, not exact binomial confidence intervals, power for an experiment, sample size for a mean, clustered or weighted sampling, simultaneous confidence for several answers, or a guarantee for subgroups. When either expected count k × p or k × (1 − p) is below 10, an approximation warning appears. Rare events need a suitable model.
Input and numerical limits
Population 2–1,000,000,000; explored responses 1–1,000,000,000 and no more than the known population. Target margin 0.1–25 points; expected share 1–99%; completion rate 0.1–100%. Percentages accept a decimal dot or comma and up to three decimal places; no thousands separators or scientific notation. Blank and malformed inputs are rejected.
Confidence choices are 80%, 85%, 90%, 95% and 99%. Their normal quantiles are rounded upward at 18 decimal places; rational arithmetic determines response and invitation ceilings without intermediate display rounding. Reported margins are approximate. The chart spans a labeled part of the response range and uses at most 101 computed points; all displayed counts use the complete formula.
Sources: Penn State · confidence intervals and sample size, section 2.3; NIST · selecting sample sizes.
