Confidence Interval

Confidence intervals are fundamental to inferential statistics. They express the uncertainty inherent in estimating a population parameter from a sample. A 95% confidence interval means: if you repeated the sampling process 100 times under identical conditions, approximately 95 of those intervals would contain the true population mean. It does NOT mean there is a 95% probability that the true mean lies within your specific interval — the true mean either is or is not in that interval, and you do not know which.

The margin of error (MOE) is the key number reported in opinion polls, market research, and clinical studies — it communicates the precision of the estimate. The standard error (SE) shrinks as sample size increases, which means larger samples produce narrower intervals with more precise estimates. Doubling the sample size reduces the SE by about 29% (1/√2). When the standard deviation is estimated from the same sample, the t-distribution (not the normal) is the correct reference at every sample size, because s is an imperfect estimate of the true population σ; its heavier tails give slightly wider intervals that account for this extra uncertainty.

This calculator therefore uses t with n − 1 degrees of freedom by default and switches to the normal Z only if you tell it σ is known. The interval assumes the data are roughly normal (or n is large enough for the central limit theorem).