Why the Percentile CI Works
The percentile method reads a confidence interval straight off the bootstrap distribution — the middle 95%. This page shows why that captures μ about 95% of the time, by building the true sampling distribution and a bootstrap distribution on the same axis.
The argument in one line
If the bootstrap distribution were exactly the sampling distribution shifted from μ over to x — same shape, same spread, new centre — then the middle 95% of the bootstrap distribution would contain μ exactly when x itself landed in the middle 95% of the sampling distribution, which happens 95% of the time. That is the whole justification. The shift is never exact, so the page lets you check both halves of it every frame and see where it frays.
Stage 1 — the true sampling distribution
The top panel is a population of about 200 values, one circle each. Each draw takes a sample of n , highlights the circles it drew (darker where a circle was drawn more than once), and drops that sample's x into the blue distribution below. Use +1 to watch one at a time, then +1000 to fill it in.
The 200 circles show the population's shape — they aren't a roster of 200 individuals. Samples are drawn as if from a much larger population, which is why a circle occasionally comes up twice even here (about one sample in five at n = 10). Keeping it that way means the sampling distribution's spread is exactly σ/√n , with no finite-population correction to explain away — and it keeps the comparison with the bootstrap honest at every n . In Stage 2 repeats stop being occasional and become constant, because there the resample is drawing n values from only n .
Stage 2 — the bootstrap
One sample is frozen, highlighted in red inside the population and blown up in the Original sample inset. Every resample now comes from those n values only , drawn with replacement — which is why circles go dark: at n = 10 a resample typically uses some values three or four times and skips others entirely. Each x * lands in the red distribution, overlaid on the blue one.
The slider
Original sample mean (x ) picks a different frozen sample by its mean. Sweeping it slides the red distribution across the blue one. Watch the two lines in the verdict flip together as x crosses the edge of the blue band — that agreement is the percentile method's justification.
Where it breaks — and why that matters
The width is estimated. The bootstrap distribution's spread comes from your one sample, so it is not exactly a shift of the sampling distribution. At n = 10 the two verdicts sometimes disagree. Raise n and disagreements get rare.Symmetry is doing work. The clean “if and only if” needs the sampling distribution to be symmetric as well as shifted. Switch the population to Right-skewed to see the percentile interval come out asymmetric — a real advantage over ±2 SE, which is always symmetric — and to see coverage slip below 95% at small n . That gap is what the BCa interval exists to correct.
Reading the lower panel
Blue shaded band — the central 95% of the true sampling distribution: where a single x usually lands.Blue dashed line — μ. Red dashed line and triangle — the frozen sample's x .Red bracket under the axis — the 95% percentile CI. The question is whether it spans the μ line.
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