Confidence Interval for Mean Calculator

A confidence interval is a range of values, derived from sample data, that is likely to contain the true population parameter (such as the mean) with a specified level of confidence. For example, a 95% confidence interval means that if you repeated the sampling process 100 times, approximately 95 of the resulting intervals would contain the true population mean.

The confidence level (e.g., 95%) reflects the reliability of the estimation procedure, not the probability that any single computed interval contains the true value. It tells you that, over many repeated samples, the stated percentage of all intervals constructed this way would capture the true population mean.

The margin of error is the half-width of the confidence interval — the amount added to and subtracted from the sample mean to produce the lower and upper bounds. It equals the critical value (z*) multiplied by the standard error of the mean. A smaller margin of error indicates a more precise estimate.

The standard error (SE) measures how much the sample mean is expected to vary from sample to sample. It is calculated as the sample standard deviation divided by the square root of the sample size (SD / √n). Larger samples produce a smaller standard error, leading to a narrower confidence interval.

Increasing the sample size narrows the confidence interval because a larger sample provides a more precise estimate of the population mean. The standard error decreases as n increases (it shrinks proportionally to 1/√n), resulting in a smaller margin of error and a tighter interval.

Common critical values are: 80% → 1.2816, 85% → 1.4395, 90% → 1.6449, 95% → 1.9600, 99% → 2.5758, and 99.9% → 3.2905. These z-scores correspond to the standard normal distribution and are used when the sample size is large enough (typically n ≥ 30) for the Central Limit Theorem to apply.

When your sample size is small (typically n < 30) and the population standard deviation is unknown, it is more accurate to use the t-distribution rather than the standard normal (z) distribution. The t-distribution has heavier tails, producing wider intervals that account for the additional uncertainty from small samples. This calculator uses the z-approximation, which is reliable for most practical cases with n ≥ 30.

This calculator assumes that the sample was drawn randomly from the population, that observations are independent of each other, and that the sampling distribution of the mean is approximately normal (satisfied when n ≥ 30 by the Central Limit Theorem, or when the underlying population is normally distributed). Violations of these assumptions may make the interval unreliable.