What “95% confidence” really means
Scroll down through the story — the picture above stays in view and rebuilds itself as you go. (Prefer to drive it yourself? Open the interactive version.)
Here is a whole real population — the living areas of 2,930 houses in Ames, Iowa. Because we have all of them, we know the one thing you almost never know in real life: the true mean μ (the dashed line).
In real life you can't measure everyone. So you take a sample — here, 50 houses (the ticks) — and from it you build a 95% confidence interval: a range that says “the true mean is plausibly in here.” This one caught μ.
One interval is luck. The real claim of “95% confidence” is about the long run. So we repeat the whole process 100 times — 100 samples, 100 intervals.
Count them: about 95 of 100 intervals contain the true mean (teal, solid). The rest miss (orange-red, dashed).
Those misses aren't mistakes — they're the ~5% the method is allowed. That is what “95% confidence” means: the method's long-run capture rate across many samples — not a 95% probability about any one interval you've already drawn.
The same simulation, in editable Python, is the flagship lesson: What a 95% confidence interval really means → — or drag the sliders yourself.