Confidence Interval Calculator

Confidence interval for a sample mean, given the sample size, standard deviation, and confidence level

A confidence interval expresses how much uncertainty surrounds a sample-based estimate of a population mean — not a range that the true population mean is guaranteed to fall within, but a range constructed using a method that, applied repeatedly, would capture the true mean a specified percentage of the time.

Why "95% confidence" doesn't mean what it sounds like

A common misinterpretation is that a 95% confidence interval means "there's a 95% probability the true population mean falls in this specific range" — but that's not what confidence level actually describes. It describes the long-run reliability of the method: if the same sampling and interval-construction process were repeated many times, approximately 95% of the resulting intervals would contain the true population mean. Any single, specific interval either does or doesn't contain the true mean — there's no probability attached to that specific instance, only to the method's long-run success rate. The interval's width depends on sample size, standard deviation, and the chosen confidence level (via its associated z-critical value) — larger samples and lower variability produce narrower, more precise intervals, while higher confidence levels (99% versus 95%, for instance) require wider intervals to maintain that higher long-run capture rate.

A worked example

The identical sample data run at a 90% confidence level versus a 99% confidence level produces a narrower interval for the lower confidence level and a wider one for the higher — there's a genuine tradeoff between how confident the method is and how precise (narrow) the resulting range can be, which is exactly why the two numbers move in opposite directions from each other.

How this connects to your other statistical calculations

Confidence intervals are built from a sample's mean and standard deviation, so an accurate Standard Deviation Calculator result feeds directly into a correct confidence interval. For determining how large a sample needs to be to achieve a target margin of error in the first place, the Sample Size Calculator answers that related but distinct planning question.

Common mistakes

Interpreting a specific confidence interval as "there's a 95% chance the true mean is in this range" — rather than the correct interpretation about the method's long-run reliability across repeated sampling — is the most common statistical misconception, one that shows up even in some published research summaries, not just casual explanations.