Confidence Interval Calculator
This confidence interval calculator covers a population mean and a population proportion. For a mean you can use a t interval with the sample standard deviation, or a z interval when σ is known. For a proportion the default is the Wilson interval, which behaves better than the textbook Wald interval, especially with smaller samples or proportions near 0 or 1. A separate, older tool on this site still computes a mean-only interval if you want that specialized page.
Confidence interval calculator
Compute a confidence interval for a mean (t or z) or for a proportion (Wilson or Wald).
Working
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Confidence interval formulas
Mean, σ unknown: x̄ ± tα/2, n−1 × s/√n.
Mean, σ known: x̄ ± zα/2 × σ/√n.
Proportion, Wald: p̂ ± zα/2 × √(p̂(1−p̂)/n). This interval can fall outside [0, 1].
Proportion, Wilson: the center is (p̂ + z²/(2n)) / (1 + z²/n), and the margin uses √(p̂(1−p̂)/n + z²/(4n²)) in the numerator, divided by 1 + z²/n.
Wilson is the default here because it has better coverage than Wald in common classroom and research settings.
Worked example
A sample of n = 16 observations has mean 50 and sample SD 8. For a 95% t interval, df = 15 and t0.025, 15 ≈ 2.131. The standard error is 8/√16 = 2, so the margin of error is about 4.263, and the interval is [45.737, 54.263].
For a proportion, 12 successes in 50 trials give p̂ = 0.24. The 95% Wilson interval is about [0.142, 0.374]. The Wald interval is about [0.122, 0.358] and is slightly over-optimistic in the lower tail.
Related calculators and guides
Frequently asked questions
Use the t interval whenever σ is unknown and you are using the sample standard deviation s. Use the z interval only when σ is treated as known.
The Wald interval can be too short, can go outside 0 and 1, and can be inaccurate for small n or for p̂ near 0 or 1. The Wilson interval has better coverage in those cases and is a standard recommended default.
In repeated sampling, about 95% of intervals computed this way would contain the true parameter. It is not the probability that this particular interval contains the parameter after you have seen the data.
Yes. The Confidence Interval Calculator for the Population Mean is a mean-only tool. This page adds a proportion interval and keeps the mean interval in one place.
Yes, for a mean interval. Paste the data in the raw-data box and leave n, the mean, and s blank. The calculator will compute those from the list.