From statistical completes to field invitations through two attrition gatesUpdates with every input
Math & Statistics
Sample Size Solver Calculator
Solve a proportion-survey recruitment target from confidence, margin of error, expected prevalence, finite population, design effect, eligibility, and response assumptions—without confusing completed responses with invitations.
Infinite-population base n-
After finite correction-
Required completed responses-
Eligible contacts needed-
Invitations needed-
Sampling fraction-
Two-sided z critical-
Extra n versus p=50%-
RECRUITMENT ARCHITECTURE
From statistical completes to field invitations through two attrition gates
A precision curve fixes the analytic target while a recruitment funnel separates design effect, eligibility loss, and nonresponse.
MARGIN SENSITIVITY
How the completed-response target reacts to tighter precision
Confidence, prevalence, population, and design effect stay fixed while margin changes around the entered plan.
| Margin | Base n0 | Finite n | Design-adjusted completes | Invitations | Change vs plan |
|---|
SURVEY SETUP
Freeze the estimand and fieldwork rates before solving
- Define the proportion and target population.
- Use an expected proportion from credible prior evidence, or 50% for the conservative variance maximum.
- Enter an absolute—not relative—margin.
- Use a defensible design effect for clustering or weighting.
- Keep completed responses, eligible contacts, and invitations as separate quantities.
DESIGN LOGIC
Precision and recruitment answer different planning questions
The statistical formula concerns the number of usable completed observations. It cannot predict how many people must be approached.
Eligibility and response gross-ups are operational assumptions. They should be stress-tested and updated from field monitoring rather than hidden inside the margin of error.
PROPORTION SAMPLE DESIGN
Solve statistical precision before grossing up fieldwork
Cochran's large-population formula sets the variance target. The finite-population correction reduces n when sampling fraction is material. Design effect inflates completed responses; eligibility and response assumptions then convert completes into invitations.
Detailed calculation process and general formulas
n0 = z^2 p(1-p) / e^2nF = n0 / [1 + (n0-1)/N]nC = ceil(DEFF x nF)eligible contacts = ceil(nC / rE)invitations = ceil(nC / (rE x rR))Symbols, meanings, and units
- p
- expected population proportiondecimal
- e
- absolute margin of errorproportion points
- N
- finite population sizepeople or units
- DEFF
- variance inflation from the sample designratio
- rE, rR
- eligibility and response ratesdecimal
DECISION CHECKS
Three controls prevent a false sense of certainty
The result is only as credible as its design and field assumptions.
Variance peak
-A 50% expected proportion produces the largest binomial variance.
Population correction
-The finite correction matters only when the uncorrected sample is not tiny relative to N.
Recruitment risk
-Track eligibility and response separately so corrective action has a clear target.
Decision takeaway: Report both required completes and planned invitations, with every gross-up assumption stated.
Applied decisions
Survey designs this solver supports
Customer prevalence study
A finite customer file is sampled to estimate the share using a feature.
What the result clarifies: The finite correction and response gross-up belong to different stages.
Clustered community survey
Households are selected through geographic clusters.
What the result clarifies: Design effect protects the completed-response target from within-cluster similarity.
Worked current scenario
Substitution, intermediate values, and reconciliation
Method references
Sources for this calculator's specific method
Scope and limitations
This formula targets simple two-sided precision for one proportion. It does not replace stratified allocation, complex-weight variance analysis, power analysis, rare-event planning, subgroup minimums, multiplicity control, or an ethical recruitment plan.
Sample Size Solver Calculator | Finite-Population Proportion Survey FAQ
Why does 50% require the largest sample?
The binomial variance p(1-p) is maximized at p=0.5.
Should response rate change the margin of error?
No. It changes invitations needed, while achieved valid completes determine sampling precision.
Can design effect be below one?
Efficient stratification can produce a value below one, but it should come from a defensible design analysis.