Instructor Guide: Why ± 2 SE Works

Companion notes for the StatLens conceptual figure — the capture argument for a ± 2 SE confidence interval

Overview ~8 min

One figure, no numbers. The axis is marked in standard errors either side of the parameter (p−2SE, p−SE, p, p+SE, p+2SE), so the empirical rule can be read straight off it. A draggable marker is one sample's statistic; the bar under the axis is that sample's interval. The page reports two verdicts and they are never allowed to disagree — that is the argument.

|statistic − parameter| ≤ 2·SE ⇔ parameter is inside [statistic ± 2·SE]

which is one fact about a distance, read from either end. The symbolic axis is what makes it visible: with numbers on the axis it looks like arithmetic a student has to take on trust, and in SE units the two arrows are plainly the same length.

Why only 68 / 95 / 99.7? Because at exactly those levels the shaded band's edges land on tick marks (±1, ±2, ±3 SE). A 90% level would put the edge at 1.645 SE, between two ticks, and the student would be back to reading a number off an axis built to avoid them.

Running it ~8 min

1. Establish the axis before anything moves ~2 min

Point at the ticks and read them aloud: p, p plus one SE, p plus two SE. Make sure the room agrees that the curve is the sampling distribution of the statistic — one dot per possible sample — and not the distribution of the data.

Ask: “What is one point on this curve?” Wanted answer: the statistic from one sample. If anyone says “one observation”, stop and fix it here — the rest will not land otherwise.

2. Drag the statistic inside the band ~2 min

Move it around while staying inside the shading. Both verdicts stay ✓. Draw attention to the bar reaching past the dashed line at p every time.

Ask: “What would have to happen for the bar to fall short?”

3. Cross the edge ~2 min

Drag slowly out past p+2SE. Both verdicts flip at the same instant and the bar turns orange. That simultaneity is the lesson — stop and say so.

Ask: “Which samples give an interval that misses?” Wanted answer: exactly the ones outside the band — about 5% of them.

4. Hand it to chance ~2 min

Press Draw a sample twenty or thirty times. The tally creeps toward the advertised level. Students should see that no sample knows whether it is one of the lucky ones.

Ask: “Which of the intervals we just drew contains p?” The honest answer is that in real life you cannot tell — that is precisely why we quote a success rate for the method rather than a probability for the interval.

Common misconceptions

“There is a 95% chance p is in this interval.” The figure is the cleanest refutation available: p is drawn as a fixed dashed line and never moves. The interval is what moves. For any one interval the answer is already yes or no.

“Wider interval, more confident in my answer.” Switch to 99.7%. The band widens, the bar lengthens, misses become rare — and the interval now says much less about where the parameter is. Confidence is bought with precision.

“The 5% is the chance the data were bad.” Nothing is wrong with a sample that lands in the tail. It is an ordinary sample from the same curve; it just happened to be unusual, and ordinary methods applied to it give a wrong interval.

The gap to be honest about

This figure assumes SE is known. In practice it is estimated — from a bootstrap distribution, or from a formula with the sample's own numbers in it — and that is the one place the clean picture is optimistic. It is why real coverage runs a little under the advertised rate at small n. Do not paper over it; it is the reason the course keeps going after this slide.

Link recipes

StatLens — statlens home · This page is designed to print. Built from Todd Will's Mathematica demonstration of the same argument.