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How to use
A confidence interval summarizes uncertainty around a population parameter estimate. For a mean this tool uses Student’s t with the sample SD; for a proportion it uses the Wilson interval, which behaves better than the Wald interval near 0% and 100%.
- Choose mean or proportion.
- Select 90%, 95% or 99% confidence.
- Enter the appropriate sample statistics.
- Review estimate, bounds, standard error and critical value.
How it is calculated
Example
For sample mean 72, sample SD 15 and n=25 at 95% confidence, SE=3 and the tool uses a t critical value with 24 degrees of freedom.
Important notes
95% confidence does not mean there is a 95% posterior probability that this one computed interval contains the fixed parameter; it describes long-run coverage of the method across repeated samples. Check sampling assumptions for your context.
Worked examples and interpreting results
A sample mean of 50, standard deviation 10 and size 10 give standard error about 3.1623. A 95% t interval with 9 degrees of freedom has margin about 7.154.
| Case | Calculation | Result |
|---|---|---|
| Standard error | 10 ÷ √10 | 3.162278 |
| Degrees of freedom | 10 − 1 | 9 |
How to check the result
Select mean and 95%. Assume independent observations from a normal population for this small sample. Confidence describes long-run coverage of the interval procedure, not a probability assigned to the fixed mean after calculation.
Frequently asked questions
When should I use t for a mean?
When the standard deviation comes from the sample, which is the usual case.
Why Wilson for a proportion?
It behaves better than the simple Wald interval, especially for small samples or proportions near 0 or 1.
Does higher confidence widen the interval?
Usually yes, because the critical value increases.
Does a 95% confidence interval mean a 95% probability for this interval?
In the frequentist interpretation, the parameter is fixed and this computed interval either covers it or not. The 95% level means the procedure covers the parameter in about 95% of repeated comparable samples.