Confidence Interval Calculator

A confidence interval is a range of values, derived from sample data, that is likely to contain the true population parameter (such as a mean or proportion). For example, a 95% confidence interval means that if you repeated the sampling process 100 times, approximately 95 of those intervals would contain the true parameter. It reflects estimation uncertainty, not a probability about the true value itself.

A Z interval is used when the population standard deviation (σ) is known, and relies on the standard normal distribution. A t interval is used when the population SD is unknown and you are estimating it from sample data using the sample standard deviation (s). The t-distribution has heavier tails than the normal distribution, producing wider intervals — especially for small sample sizes — to account for extra uncertainty.

A 95% confidence level means the procedure used to construct the interval will produce an interval containing the true population parameter in approximately 95% of repeated samples. It does NOT mean there is a 95% chance the true value falls within any specific computed interval — once calculated, the interval either contains the true value or it does not.

To calculate a confidence interval for a proportion, you need the sample size (n) and the number of successes (x). The sample proportion p̂ = x/n is computed, then the standard error SE = √(p̂(1−p̂)/n), and the interval is p̂ ± z* × SE. This calculator handles all those steps automatically when you select the Proportion mode.

As sample size increases, the standard error decreases, which narrows the confidence interval. A larger sample provides more information about the population, so the estimate becomes more precise. This is why collecting more data generally leads to tighter, more useful confidence intervals.

The margin of error (E) is the half-width of the confidence interval — it equals the critical value multiplied by the standard error (E = critical × SE). The full interval is then: estimate ± E. A smaller margin of error indicates a more precise estimate, achieved by increasing sample size or lowering the confidence level.

Not necessarily. A wider interval often just reflects a smaller sample size, higher variability in the data, or a higher chosen confidence level (e.g. 99% vs 90%). It means your estimate is less precise, not that the data is flawed. You can reduce the width by collecting more data or accepting a lower confidence level.

Use a t interval whenever the population standard deviation is unknown — which is almost always the case in practice. The t interval uses the sample standard deviation (s) as an estimate and applies the t-distribution with n−1 degrees of freedom. As sample size grows large (typically n > 30), the t and Z intervals converge to nearly the same result.