Calculator
Example data table
| Scenario | n | Population N | Design effect | Point estimate | Notes |
|---|---|---|---|---|---|
| Binary outcome (Satisfied = Yes) | 200 | — | 1.00 | 118 successes (p̂ = 0.59) | Compare Wald vs Wilson for stability. |
| Rating scale (1–5 satisfaction) | 200 | 1,500 | 1.20 | x̄ = 3.90, s = 0.80 | Finite correction tightens the interval. |
Formula used
Wilson: score-based center ± adjusted half-width
Agresti–Coull: p̃ ± z · √(p̃(1−p̃)/ñ) · FPC
c = t(df=n−1) for unknown σ; c = z for known σ
Confidence level selection
Survey reports commonly use 90%, 95%, or 99% confidence. Moving from 90% to 95% increases the critical value from about 1.645 to 1.96, widening the interval by roughly 19%. Use higher confidence when decisions are costly, and lower confidence when rapid directional insight is acceptable. For one-sided statements, the same confidence concentrates all error in a single tail, producing a tighter bound than a two-sided interval.
Margin of error drivers
For proportions, the standard error scales with √(p̂(1−p̂)/n). It is largest near p̂=0.50 and smaller near 0 or 1. Doubling sample size reduces margin of error by about 29% because the √n term grows slowly. When planning, p=0.50 is a conservative choice if no prior estimate exists. Always record n and the selected confidence level with the estimate.
Design effect and effective n
Complex designs, clustering, and unequal weights inflate variance. Design effect (DEFF) converts a nominal sample to an effective sample, n_eff = n/DEFF. For example, n=600 with DEFF=1.5 behaves like n_eff=400, increasing uncertainty even though fieldwork was larger. Weighting effects can raise DEFF when a few respondents carry large weights. If DEFF is unknown, sensitivity testing with 1.2–2.0 helps bound risk across likely design scenarios.
Finite population correction
When sampling without replacement from a small population, uncertainty shrinks. The finite population correction is FPC=√((N−n)/(N−1)). If N=1,500 and n=300, FPC is about 0.894, tightening the interval by roughly 11%. Apply FPC only when the sampling frame is well defined and coverage is high. If n is under 5% of N, the correction is close to 1 and usually negligible, so omitting it simplifies reporting without changing decisions.
Choosing an interval method
For proportions, Wald intervals are fast but can be unstable for small n or extreme p̂, sometimes producing bounds near 0 or 1 that misrepresent uncertainty. Wilson and Agresti–Coull intervals are typically better behaved. For means, use t critical values when σ is unknown; t approaches z as n grows. Document method choice, corrections, and rounding so the same inputs reproduce the same interval in audits, dashboards, or published briefs. Include the point estimate and interval together; margins alone can hide skewed or bounded metrics.
FAQs
Which proportion method is best for most surveys?
Wilson is a strong default because it stays stable for small samples and extreme proportions. Agresti–Coull is also robust. Wald is acceptable for large n with p̂ away from 0 and 1.
When should I use a design effect above one?
Use DEFF when the sample is clustered, heavily weighted, or stratified in ways that inflate variance. If you have a prior DEFF from methodology reports, enter it. Otherwise, run a sensitivity range to see impact.
Can I enter a percentage instead of successes?
Yes. Choose the sample proportion option and enter p̂ as a decimal, like 0.62 for 62%. The tool will infer an approximate success count as round(p̂·n) for calculations and exports.
How do one-sided confidence intervals differ?
One-sided intervals place all allowable error in one tail. You get either a lower bound or an upper bound at the chosen confidence level. They are useful for threshold claims, such as “at least 70%.”
When does population size matter for the correction?
Enter N when sampling without replacement from a well-defined, relatively small population and n is a meaningful share of N. If n is tiny relative to N, the correction is near 1 and can be skipped.
Why does the mean interval use t instead of z?
When the population standard deviation is unknown, replacing it with the sample standard deviation adds uncertainty. The t distribution accounts for that, especially at small n. As n grows, t and z become very similar.
How to use this calculator
- Select an estimate type: proportion for yes/no, mean for scores.
- Choose the confidence level and interval type (two-sided is typical).
- Enter sample size, and optionally population size for finite correction.
- If the design is clustered or weighted, set a design effect above one.
- Provide proportion inputs (x or p̂) or mean inputs (x̄ and s).
- Press Calculate to view results and export as CSV or PDF.