95% Confidence Interval Calculator

A confidence interval is a range of values, derived from sample data, that is likely to contain the true population parameter. For example, a 95% confidence interval means that if you repeated your study 100 times, about 95 of those intervals would contain the true population mean. It reflects the uncertainty inherent in estimating a population parameter from a sample.

To calculate a 95% confidence interval for a mean, use the formula: CI = x̄ ± Z × (σ / √n), where x̄ is the sample mean, Z is the critical z-score (1.96 for 95% confidence), σ is the standard deviation, and n is the sample size. First compute the standard error (SE = σ / √n), then the margin of error (ME = 1.96 × SE), and finally add and subtract ME from the mean to get the lower and upper bounds.

A 95% confidence level means that 95% of confidence intervals constructed using this procedure will contain the true population parameter. It does NOT mean there is a 95% probability that the specific interval you calculated contains the true value — once calculated, the interval either does or does not contain the true value. The confidence level describes the reliability of the estimation method, not a single interval.

At a 95% confidence level, the significance level (alpha) is 0.05, meaning the p-value threshold for statistical significance is p < 0.05. If a hypothesis test corresponds to a 95% confidence interval that excludes the null hypothesis value (e.g., zero for a difference), the result is statistically significant at p < 0.05.

The z-score (critical value) for a 95% confidence interval is 1.96. This value comes from the standard normal distribution and represents the point beyond which only 2.5% of the distribution falls on each side (totaling 5% outside the interval). For other confidence levels: 90% uses 1.645, 99% uses 2.576, and 99.9% uses 3.291.

Larger sample sizes produce narrower (more precise) confidence intervals because the standard error (SE = σ / √n) decreases as n increases. Doubling your sample size reduces the margin of error by a factor of √2 (about 29%). This is why researchers aim for larger samples when they need high precision in their estimates.

You should use the t-distribution when your sample size is small (typically n < 30) and the population standard deviation is unknown. The t-distribution has heavier tails than the normal (z) distribution, producing wider intervals that account for additional uncertainty. For large samples (n ≥ 30), the t and z distributions converge and the difference becomes negligible.

Standard deviation (SD) measures the spread or variability of individual data points within your sample. Standard error (SE) measures the precision of the sample mean as an estimate of the population mean, calculated as SE = SD / √n. The SE is always smaller than the SD (for n > 1) and is the value used directly in confidence interval calculations.