Confidence Interval Calculator

A confidence interval is a range of values that, with a given probability (e.g. 95%), contains the true population parameter (e.g. the mean). It does not mean that 95% of observations fall within the range; it means that the construction method produces a correct interval in 95% of repeated experiments.

A 95% confidence level means that if we repeatedly drew samples and computed confidence intervals, about 95% of those intervals would contain the true parameter. It is the standard level in social sciences and medicine.

The CI for a mean uses CI = x +/- z * (sigma / sqrt(n)) and is used when estimating a quantitative mean (e.g. height, income). The CI for a proportion uses CI = p +/- z * sqrt(p*(1-p)/n) and is used for percentage shares (e.g. voter turnout).

The margin of error is half the width of the confidence interval: z * (sigma / sqrt(n)). It tells you how much the estimate may differ from the true value. The larger n or the smaller sigma, the smaller the margin.

Increase the sample size n — the width of the interval decreases proportionally to 1/sqrt(n). To halve the margin, you need to quadruple the sample. Alternatively, choose a lower confidence level (e.g. 90% instead of 99%), which gives a narrower interval at the cost of lower certainty.

A p-value tests the null hypothesis — it tells you whether a result is statistically significant. A confidence interval gives the range of plausible values for the parameter. Both are complementary: a CI provides more information because it also reveals the magnitude of the effect.

These are quantiles of the standard normal distribution: z=1.645 for 90% confidence, z=1.96 for 95%, z=2.576 for 99%. They are used when the population standard deviation is known or the sample is large (n>=30). For small samples with unknown sigma, use the t-distribution.

The calculator uses the normal distribution (z-values), which is appropriate for large samples (n>=30). For small samples (n<30) with unknown population standard deviation, the t-distribution with n-1 degrees of freedom should be used. Results from this calculator are then an approximation.

The proportion p is the estimated fraction in the sample, expressed as a number between 0 and 1 (e.g. 0.4 means 40%). The widest interval occurs at p=0.5. The calculator automatically clamps results to [0, 1], since a proportion cannot be negative or greater than 1.

No — the calculator is an educational and auxiliary tool. In scientific, clinical or business research you should also consider the data distribution type, effect size, Bessel correction for small samples and the substantive context of the results.