Sample Size Calculator
Estimate the sample size for a population proportion using confidence, margin of error, and an optional finite population. Choose a preset or enter your own positive Z; an entered Z is not assigned an invented confidence percentage. Enter response rate separately from estimated proportion. All numeric inputs start empty.
Choose a preset or enter your own positive Z; an entered Z is not assigned an invented confidence percentage. Enter response rate separately from estimated proportion. All numeric inputs start empty.
Required completed observations:
Before finite population correction:
Unrounded large-population n₀:
Unrounded corrected n:
Standard-normal z:
People to invite:
p=0% or 100% makes the variance term and formula sample size zero. This mathematical degeneracy does not establish certainty, guarantee precision or justify doing no survey. The normal approximation is unsuitable at these endpoints.
Invitations exceed the finite population. That many distinct people cannot be invited from this population at the entered expected response rate; the result is not capped.
Approximate planning result; always rounded upward.
Method and rounding order
The three columns retain existing 90%, 95% and 99% two-sided normal Z approximations (more digits than the source’s rounded values). They use your proportion and population, independent of selected Z and response rate. Custom mode adds your Z as a separate column.
For one population proportion under simple random sampling with independent observations. Confidence uses a two-sided standard-normal z value. Margin is an absolute percentage-point margin, not a relative percentage.
Convert margin and proportion to fractions: n₀ = z²p(1−p)/e². For finite population N: n = Nn₀/(N+n₀−1). First completed = ceil(n), then invitations = ceil(completed/(response rate/100)). For p = 0 or 1, n₀ and n are 0 by the degenerate formula; no precision guarantee follows.
Approximate planning for simple random sampling, not a guarantee of precision or an exact binomial interval. Invitation gross-up models an expected response percentage; it does not correct nonresponse bias. Clustering, weighting, design effects and hypothesis-test power are excluded. Means, rare proportions, small samples and separately reported subgroups require suitable methods.
A QUICK WALKTHROUGH
How to use this tool
- Choose a two-sided confidence level.
- Enter the margin in percentage points and an estimated proportion; use 50% when unknown.
- Optionally enter the finite population size, then calculate.
- Choose the confidence input mode and supply a level or positive Z.
- Enter your expected response rate and 1–20 comparison margins.
- Review unrounded sizes, rounded completed counts and invitations; copy or download the frozen report.
Two-sided confidence level
For one population proportion under simple random sampling with independent observations. Confidence uses a two-sided standard-normal z value. Margin is an absolute percentage-point margin, not a relative percentage.
n₀ = z²p(1−p)/e²
Convert margin and proportion to fractions: n₀ = z²p(1−p)/e². For finite population N: n = Nn₀/(N+n₀−1). First completed = ceil(n), then invitations = ceil(completed/(response rate/100)). For p = 0 or 1, n₀ and n are 0 by the degenerate formula; no precision guarantee follows.
Approximate planning result; always rounded upward.
Approximate planning for simple random sampling, not a guarantee of precision or an exact binomial interval. Invitation gross-up models an expected response percentage; it does not correct nonresponse bias. Clustering, weighting, design effects and hypothesis-test power are excluded. Means, rare proportions, small samples and separately reported subgroups require suitable methods.
Completed-observation comparison
The three columns retain existing 90%, 95% and 99% two-sided normal Z approximations (more digits than the source’s rounded values). They use your proportion and population, independent of selected Z and response rate. Custom mode adds your Z as a separate column.
Expected response rate (%)
Choose a preset or enter your own positive Z; an entered Z is not assigned an invented confidence percentage. Enter response rate separately from estimated proportion. All numeric inputs start empty.
GOOD TO KNOW
Common questions
Why use 50%?
p = 0.5 gives the largest variance and most conservative size for this formula. Count completed observations; invitations may need to be higher for nonresponse.
Are inputs uploaded?
No. The calculation runs locally in your browser.