Confidence Interval Calculator

A confidence interval is a range around a sample estimate that should contain the true population value most of the time, under repeated sampling. A 95 percent confidence interval, in plain terms, says that if you repeated the same sampling process many times, about 95 percent of the intervals built this way would capture the real population value. The specific interval you got either contains the truth or does not; the 95 percent describes the procedure, not the single result.

Compute the standard error: for a mean it is the sample standard deviation divided by the square root of n; for a proportion it is the square root of p(1 − p) divided by n. Multiply by the z-score for the confidence level (1.645 at 90 percent, 1.96 at 95 percent, 2.576 at 99 percent) to get the margin of error. Add and subtract it from the sample estimate to get the interval.

Use mean mode when the data is continuous (heights, scores, dollar amounts) and you have a sample mean and standard deviation. Use proportion mode when the data is binary (yes/no, success/failure) and you have a sample proportion. The two modes use different standard-error formulas because the underlying distributions are different.

The z-score approximation is appropriate when the population standard deviation is known or when sample sizes are large (say, above 30 for means). For small samples or unknown population variance, real statistical software typically uses a t-distribution, which produces slightly wider intervals. This calculator uses the z approximation, which is the convention for most planning-stage and intro-stats use.

Use this calculator as a planning estimate or for educational use. Formal research analysis needs full attention to sample design (random, stratified, clustered), non-response, weighting, and the right test for the data type. For published results, use professional statistical software and confirm assumptions with a qualified statistician.

Wider for higher confidence (99 percent intervals are wider than 95 percent), wider for smaller samples, and wider for more variable data. Doubling the sample size cuts the margin of error by about 30 percent. The same data at 99 percent confidence has a margin about 31 percent larger than the same data at 95 percent confidence (2.576 vs 1.96).