Calculate the confidence interval for a population mean (using Z or t-distribution) or a proportion. Enter your sample data and get the interval with margin of error.
A confidence interval is a range of values that, with a certain probability (confidence level), contains the true population parameter. A 95% CI means: if you repeated the study many times, 95% of the calculated intervals would contain the true value.
If your 95% CI for mean exam score is [72, 78], you can say: "We are 95% confident the true population mean is between 72 and 78." This does not mean there is a 95% probability the true mean is in this specific interval.
Margin of error = z × (standard deviation / √n). It represents half the width of the confidence interval. Larger samples reduce the margin of error. Doubling sample size reduces margin of error by a factor of √2 ≈ 1.41.
A confidence interval estimates where the population mean lies. A prediction interval estimates where a single new observation will fall — it is always wider than the confidence interval for the mean.
Use t-distribution when the population standard deviation is unknown and sample size is small (n < 30). For large samples, the t and z distributions converge. This calculator uses the z approximation, which is appropriate for n ≥ 30.
A narrower CI means more precision — you have a more specific estimate of the population parameter. Achieved by: larger sample size, lower confidence level, or less variability in the data.
Clinical trials report confidence intervals for treatment effects. If a 95% CI for a drug effect does not include zero (for absolute risk reduction) or one (for relative risk), the effect is statistically significant at p < 0.05.