Mean 95 Confidence Interval Calculator

95% confidence interval for the mean

A 95% confidence interval for the population mean is built around the sample mean with IC = x̄ ± z·s/√n, using z = 1.96 whenever σ is known or n is large (that's the Central Limit Theorem at work). When the sample is small (n < 30) and σ is unknown, swap 1.96 for Student's t critical value with n − 1 degrees of freedom. Notice that the standard error s/√n shrinks as n grows, so the interval gets tighter. One thing trips people up, so worth flagging. Important caveat: the frequentist reading is "if we repeated the sampling many times, 95% of the resulting intervals would contain the true mean", and not "there is a 95% chance the true mean lands inside this particular interval". Worked example: x̄ = 50, s = 8, n = 30 → SE = 8/√30 ≈ 1.461; IC = 50 ± 1.96·1.461 ≈ [47.14, 52.86].

Applications

You'll find it behind election polls (that famous "margin of error of ±3 points" is just a 95% CI), clinical trials reporting drug efficacy, A/B tests, the validation of analytical methods under ABNT NBR ISO/IEC 17025 (laboratory accreditation), quality control charts, and pretty much any scientific paper that reports an average drawn from a sample.

FAQ

Why 95% and not 99%? Mostly convention. 95% strikes a decent balance between a narrow interval and real confidence. 99% runs wider and shows up when a mistake is expensive, like drug approval or aviation.

When should I use t instead of z? Reach for t when the population σ is unknown and n is small, with n < 30 as the usual rule of thumb. Once n gets large the t distribution converges to the normal anyway.

Does the population need to be normal? With large n the CLT keeps the sample mean approximately normal no matter how the population itself is distributed. With small n, though, the population's normality starts to matter.

Confidence interval vs prediction interval? A CI estimates the mean. A prediction interval estimates a single future observation, and it comes out wider because it has to account for individual variability on top of that.