Confidence Interval Calculator
Build a confidence interval for a mean or a proportion from your sample.
About the Confidence Interval Calculator
A confidence interval gives a range of plausible values for a population figure. A 95% interval means that if you repeated the sampling many times, 95% of the intervals built this way would contain the true value.
For a mean, the t-distribution is used because the population standard deviation is estimated from the sample. For a proportion, both the standard (Wald) interval and the more reliable Wilson interval are shown.
The formula
Mean: x̄ ± t × s ÷ √n; Proportion: p̂ ± z × √(p̂(1 − p̂) ÷ n)
- t
- t critical value with n − 1 degrees of freedom
- z (95%)
- 1.95996
Worked example
Example: 95% confidence interval = 51.655694 to 58.344306
- Inputs: Interval for a: Mean; Sample mean: 55; Sample standard deviation: 11; Sample size (n): 44; Confidence level: 95%.
- Formula: Mean: x̄ ± t × s ÷ √n; Proportion: p̂ ± z × √(p̂(1 − p̂) ÷ n)
- Standard error SE = s ÷ √n = 11 ÷ √44 = 1.658312
- Margin of error t × SE = 2.0167 × 1.658312 = 3.344306
- Interval 55 ± 3.344306
- Result: 95% confidence interval = 51.655694 to 58.344306
- Margin of error: ± 3.344306
- Standard error: 1.658312
- t critical value: 2.016692
Questions people ask
How do I make the interval narrower?
Increase the sample size (width shrinks with √n) or accept a lower confidence level.
Is 95% the chance the true value is in my interval?
Strictly, no: it's the long-run success rate of the method. In practice it's read as a plausible range.