Calculators
Confidence Interval Calculator
Calculate a confidence interval for a mean (t-distribution) or a proportion (Wilson score interval), with the margin of error.
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The tool
Enter your numbers below. Results update as you type, and nothing leaves your browser.
Everything you enter stays on your own machine. Nothing is sent to us or to anyone else, which you can confirm in your browser network tab.
What people use it for
- Reporting survey and experiment results
- Quality control
- Academic research
- Business metrics with uncertainty
About this tool
What the Confidence Interval Calculator does
A confidence interval gives a range of plausible values for a population value. For a mean, it is the sample mean ± t × standard error, where t comes from the t-distribution with n − 1 degrees of freedom. For a proportion, the Wilson score interval is more reliable than the simple formula, especially with small samples or proportions near 0% or 100%.
Formula
- Mean: x̄ ± t(n − 1) × s ÷ √n
- Proportion (Wilson): (p̂ + z²/2n ± z√(p̂(1 − p̂)/n + z²/4n²)) ÷ (1 + z²/n)
Worked example
Using the starting values in the calculator above:
| Input | Value |
|---|---|
| Interval for | A mean |
| Sample mean | 52.3 |
| Sample standard deviation | 8.1 |
| Sample size | 40 |
| Successes (for a proportion) | 18 |
| Confidence level | 95% |
| Result | Value |
|---|---|
| Confidence interval | 49.7095 to 54.8905 |
| Margin of error (±) | 2.5905 |
| Point estimate | 52.3 |
| Standard error | 1.2807 |
| Critical value (t or z) | 2.0227 |
How to read the result
A 95% interval does not mean a 95% chance the true value is inside this particular interval; it describes the method’s long-run success rate.
To halve the margin of error you need about four times the sample.
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Questions
Questions about this tool
The things clients actually want to know — costs, timelines, ownership and what happens after launch.
Take the estimate and add and subtract the critical value times the standard error.
For a mean when the population standard deviation is unknown, which is almost always.
It stays accurate for small samples and extreme proportions, where the simple formula can fail.
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