How to use
- Choose a calculation and a confidence level (95% by default).
- Mean CI: enter the sample mean, standard deviation and sample size. Use t when the population standard deviation is unknown (the usual case) and z when it is known.
- Proportion CI: enter the number of successes x and the sample size n to get both the Wilson and Wald intervals.
- Sample size: enter the margin of error you want and get the required sample size, rounded up.
Formulas
| Quantity | Formula |
|---|---|
| Mean CI (t) | x̄ ± t(α/2, n−1) × s ÷ √n |
| Mean CI (z) | x̄ ± z(α/2) × σ ÷ √n |
| Proportion — Wald | p̂ ± z × √(p̂(1 − p̂) ÷ n) |
| Proportion — Wilson | (p̂ + z²/2n ± z√(p̂(1 − p̂)/n + z²/4n²)) ÷ (1 + z²/n) |
| Sample size (proportion) | n = z² × p(1 − p) ÷ E² |
| Sample size (mean) | n = (z × σ ÷ E)² |
| Finite population correction | n = n₀ ÷ (1 + (n₀ − 1) ÷ N) |
Common z values: 90% 1.6449, 95% 1.9600, 99% 2.5758. The t critical value depends on the degrees of freedom; instead of looking it up in a table, the tool computes it from the incomplete beta function (for example, df 24 at 95% → 2.0639).
Examples
- Mean: 25 students average 50 points with a standard deviation of 10 → standard error 10 ÷ √25 = 2, t = 2.0639 → 95% CI 45.87 to 54.13 (margin ±4.13).
- Proportion: 520 of 1,000 respondents say yes → p̂ = 52%, Wald margin 1.96 × √(0.52 × 0.48 ÷ 1,000) = ±3.1 points — the “margin of error ±3.1 percentage points at 95% confidence” often quoted for 1,000-person polls.
- Sample size: at 95% confidence, a ±5-point margin needs 385 respondents and ±3 points needs 1,068. If the whole population is only 1,000 people, ±5 points needs 278.
Things to keep in mind
- A 95% CI does not strictly mean “a 95% chance the true value is inside.” It means that if you repeated the sampling many times, about 95% of the intervals built this way would contain the true value.
- With small samples or proportions near 0% or 100%, the Wald interval is too narrow or spills outside 0–100%. Use the Wilson interval there.
- The margin of error assumes a random sample. It does not account for non-response, wording effects or a biased sampling frame.
FAQ
When should I use t instead of z?
Use t whenever you estimate the standard deviation from the sample itself — nearly every real analysis. With samples in the hundreds, t and z are almost identical.
What expected proportion should I use for sample size?
Use 50%. p(1 − p) is largest at p = 0.5, so it gives the most conservative sample size that holds whatever the result.
Where do I get the mean and standard deviation?
Paste your raw data into the average calculator to get the mean and sample standard deviation s first.