Why ± 2 SE WorksStatLens

There are no numbers on this page, and that is the point. The axis is marked in standard errors away from the parameter, so the empirical rule can be read straight off it: about 95% of samples give a statistic within 2 SE of the parameter. Move the statistic and watch the two verdicts below — they are never allowed to disagree.

Parameter:
Confidence level:

Before you drag: for which positions of the statistic will the interval miss the parameter? Say it in standard errors, then check.

Why ± 2 SE Works — Help

This figure makes the argument behind every statistic ± 2 SE confidence interval, without a single number in it.

The argument. The interval reaches the parameter exactly when the statistic landed within 2 SE of it — the distance between them is the same distance either way. So the interval captures the parameter for exactly the samples in the shaded band, which is about 95% of them. That is all “95% confident” claims.

What it does not say. For any one interval the parameter is either in it or not. The 95% is the success rate of the method across samples, not a probability about the interval in front of you.

In practice SE is not known — it is estimated, from a bootstrap distribution or from a formula. That is the one gap between this picture and the interval a student actually computes, and it is why the real coverage is a little under the advertised rate at small n. CI Coverage measures that; Why the Percentile CI Works makes the matching argument for percentile intervals.

Instructors: there is a printable guide with talking points, and ?mode=present opens the page with the student prompt hidden.

Keyboard