About the confidence interval calculator
A confidence interval gives a range of plausible values for a population mean or proportion based on your sample. This calculator builds the interval for a mean using either the z or Student's t distribution, and for a proportion using the normal approximation, and shows the critical value, standard error and margin of error.
Choose a confidence level from 80% to 99.9%.
Who should use the confidence interval calculator
- Students and researchers reporting results.
- Analysts summarising survey or experiment results.
- Anyone wanting to express uncertainty about an estimate.
How to use the confidence interval calculator: what each input means
- Estimate
- A mean or a proportion.
- Confidence level
- 80%, 85%, 90%, 95%, 98%, 99% or 99.9%.
- Sample size (n) and mean or successes
- From your sample.
- Standard deviation
- For a mean. Choose whether it comes from the sample (t) or is known for the population (z).
How the confidence interval calculator works: method and formula
The interval is estimate ± margin of error. For a mean, the margin is the critical value times the standard error s ÷ √n, using t with n − 1 degrees of freedom when the standard deviation comes from the sample. For a proportion it is z × √(p̂(1 − p̂) ÷ n).
- Mean
- x̄ ± t* × s ÷ √n
- Proportion
- p̂ ± z* × √(p̂(1 − p̂) ÷ n)
How to read your confidence interval calculator result
A 95% interval means that if you repeated the sampling many times, about 95% of the intervals built this way would contain the true value. It does not mean there is a 95% chance the true value is in this one interval. Larger samples and lower confidence levels give narrower intervals.
Confidence Interval Calculator worked example
A sample of 40 with mean 52 and standard deviation 8, at 95% confidence (sample standard deviation).
- Standard error = 8 ÷ √40 = 1.2649.
- t critical value for 39 degrees of freedom = 2.0227 (z would be 1.9600).
- Margin of error = 2.0227 × 1.2649 = 2.5585.
- Interval = 52 ± 2.5585, from 49.441 to 54.559.
- For a proportion of 56 successes in 100, p̂ = 0.56 and the margin is 9.73 points.
95% confidence interval for the mean: 49.44 to 54.56.
Practical uses of the confidence interval calculator
- Reporting the precision of an average.
- Summarising poll results.
- Comparing the uncertainty of two estimates.
Assumptions and limitations of this calculator
- Assumes a random sample and roughly normal data (or a large sample) for means.
- The proportion interval uses the normal approximation, which is unreliable with fewer than about 10 successes or failures.
- Does not compute intervals for differences between groups or for small non-normal samples.