Confidence Interval Calculator
Calculate a confidence interval for one mean, two independent means, one proportion, or the difference between two proportions.
Confidence interval formulas
For one mean, the interval is x̄ ± t* × s/√n. For two independent means, the calculator uses the Welch standard error √(s₁²/n₁ + s₂²/n₂) and Welch-Satterthwaite degrees of freedom. Proportion intervals use Wilson scores because the simple normal approximation can perform poorly with small samples or proportions near 0 or 1.
Checked examples
| Inputs | Method | 95% confidence interval |
|---|---|---|
| Mean 100, SD 15, n = 25 | Student’s t, df = 24 | 93.8083 to 106.1917 |
| Mean 50, SD 10, n = 30 | Student’s t, df = 29 | 46.2659 to 53.7341 |
| Means 10 and 8; SDs 2 and 2; n = 20 each | Welch t, df = 38 | 0.7197 to 3.2803 |
| 50 successes out of 100 | Wilson score | 40.3832% to 59.6168% |
| 0 successes out of 20 | Wilson score | 0% to 16.1125% |
| 60/100 versus 40/100 | Newcombe-Wilson | 6.1411 to 32.8126 percentage points |
How to use this calculator
- Choose the type of estimate.
- Select 90%, 95%, or 99% confidence.
- Enter summary statistics or success counts.
- Select Calculate interval to see the estimate, limits, uncertainty, and method.
Interpretation and limits
A 95% confidence procedure produces intervals that contain the true parameter in about 95% of repeated samples under its assumptions. It does not mean there is a 95% probability that a fixed parameter lies inside one already-computed frequentist interval.
- Samples should be representative and observations should be independent.
- Mean intervals assume the sampling distribution is approximately normal; very small samples with strong skew or outliers need special care.
- The two-mean method is for independent groups, not paired data.
- Counts must be whole numbers and cannot exceed their sample sizes.
- Statistical confidence does not remove sampling bias, measurement error, or model misspecification.
Frequently asked questions
Why does the mean interval use a t value?
The population standard deviation is usually unknown. Student’s t accounts for the extra uncertainty from estimating it with the sample standard deviation.
Why not use p ± z√(p(1-p)/n)?
That Wald interval can extend below 0 or above 1 and can have poor coverage. Wilson and Newcombe-Wilson intervals behave better, especially near the boundaries.
What does a wider confidence interval mean?
It indicates more uncertainty. Higher confidence, greater variability, or smaller samples widen an interval; larger samples generally narrow it.
Can this calculator prove practical importance?
No. The interval quantifies sampling uncertainty. Practical or clinical importance depends on the context and a meaningful effect threshold.
Related calculators
Continue with the T-Test Calculator, P-Value Calculator, Sample Size Calculator, or Standard Deviation Calculator.