mathematics//statistics//estimation//confidence interval

A confidence interval is a range computed from data by a procedure that, repeated over many datasets, would contain the true value of a parameter a stated share of the time, and it is how an estimate is reported with its margin: a pump's mean vibration of 4.2 mm/s becomes 3.9 to 4.5 mm/s at 95 %. For a mean it is the estimate plus or minus about two standard errors.

A confidence interval is a range computed from data by a procedure that, repeated over many datasets, would contain the true value of a parameter a stated share of the time, and it is how an estimate is reported with its margin: a pump's mean vibration of 4.2 mm/s becomes 3.9 to 4.5 mm/s at 95 %. For a mean it is the estimate plus or minus about two standard errors.

With the pump's 25 readings (mean 4.2 mm/s, standard deviation 0.8, standard error 0.16) the 95 % interval uses 2.06 instead of 1.96, from Student's ttt distribution, which widens the interval a little because σ\sigmaσ itself was estimated from the same 25 points; with a few hundred points the difference vanishes. So 4.2±2.06×0.164.2\pm2.06\times0.164.2±2.06×0.16, from 3.9 to 4.5 mm/s.

The 95 % belongs to the procedure: of all the intervals built this way, 95 % contain the true mean. A given interval either contains it or does not. The statement a reader usually wants, the mean is between 3.9 and 4.5 with 95 % probability, is a credible interval, a statement about belief in the parameter that comes from a posterior (Bayesian inference). With plenty of data and a vague prior the two come out nearly equal; with few data and real prior knowledge they differ.

The interval of the mean is the wrong tool for the next reading. The mean's interval narrows as data accumulate; the spread of the next measurement does not. The next reading of that pump falls, with 95 % probability, in 4.2±2.06⋅0.81+1/25≈4.2±1.74.2\pm2.06\cdot0.8\sqrt{1+1/25}\approx4.2\pm1.74.2±2.06⋅0.81+1/25​≈4.2±1.7 mm/s, a prediction interval five times wider, and it stays that wide however many readings are taken.

A quantity without a formula (a percentile, a maximum, a ratio of two means) gets its interval from the bootstrap.

The interval covers random error only. It says nothing about a sensor offset, a biased sample or a model that does not fit (selection bias), and with correlated readings it is far too narrow unless built on the effective sample size.

Alarm thresholds26 · oct 09Read noteDo not put an alarm threshold at the edge of the confidence interval of the mean. Every healthy reading has a wide chance of landing outside it, so the alarm fires on normal noise with impeccable statistical rigour; set it from the prediction interval, or from costs (detection threshold).