Find a confidence interval for a mean or a proportion.
Mean ± critical value × standard error. For n = 25, mean 50 and s = 10 the 95% interval is (45.87220288, 54.12779712).
A confidence interval gives a range likely to contain the true population value. For a mean, enter the sample size, mean and standard deviation and choose the t-distribution (σ unknown) or z (σ known). For a proportion, enter the successes and sample size to get both the Wilson and Wald intervals. Each result shows the margin of error and critical value.
Wider intervals mean more uncertainty; larger samples narrow them — plan with the Sample Size Calculator. Related: Z-Score Calculator, P-Value Calculator, Standard Deviation Calculator and Statistics Calculator.
The calculator applies the standard method step by step.
| Item | Value |
|---|---|
| Sample mean | 50 |
| Standard error (s ÷ √n) | 2 |
| Distribution used | t with 24 degrees of freedom |
| Critical value | 2.0638986 |
| Margin of error | 4.127797123 |
| Lower bound | 45.87220288 |
| Upper bound | 54.12779712 |
| Sample size | 95% interval |
|---|---|
| n = 10 | (42.84643094, 57.15356906) |
| n = 25 | (45.87220288, 54.12779712) |
| n = 50 | (47.15803145, 52.84196855) |
| n = 100 | (48.01578305, 51.98421695) |
| n = 400 | (49.01703635, 50.98296365) |
| Confidence level | α | Critical z (two-tailed) |
|---|---|---|
| 80% | 0.2 | 1.2816 |
| 85% | 0.15 | 1.4395 |
| 90% | 0.1 | 1.6449 |
| 95% | 0.05 | 1.9600 |
| 98% | 0.02 | 2.3263 |
| 99% | 0.01 | 2.5758 |
| 99.9% | 0.001 | 3.2905 |
| Degrees of freedom | 95% (α = 0.05) | 99% (α = 0.01) |
|---|---|---|
| 1 | 12.7062 | 63.6567 |
| 2 | 4.3027 | 9.9248 |
| 5 | 2.5706 | 4.0321 |
| 10 | 2.2281 | 3.1693 |
| 20 | 2.0860 | 2.8453 |
| 30 | 2.0423 | 2.7500 |
| 60 | 2.0003 | 2.6603 |
| 120 | 1.9799 | 2.6174 |
| ∞ (z) | 1.9600 | 2.5758 |
A range of values, computed from a sample, that is likely to contain the true population value at a stated confidence level.
If you repeated the study many times, about 95% of the intervals would contain the true value.
Use t when the population standard deviation is unknown (almost always); z when it is known or the sample is very large.
Half the width of the interval: critical value × standard error.
The Wilson interval is more reliable than the simple Wald interval, especially for small samples or proportions near 0 or 1.
Increase the sample size, accept a lower confidence level, or reduce variability.
It uses the standard method: CI = x̄ ± t* × s ÷ √n proportion p̂ ± z* √(p̂(1 − p̂) ÷ n) The full working and a worked example are shown on this page.
Decimal results use standard double-precision arithmetic (about 15–16 significant digits). Tools that work with whole numbers — big numbers, factors, bases, combinations — use exact integer arithmetic, so no digits are lost.
Yes for checking your work and learning the method. The steps and formula are shown so you can understand the answer, but follow your teacher's rules about calculators in exams.
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