Interactive 縄跳び

The jumping rope

Regression is a rope held at the points you have measured. Where observations are close together the rope is pinned and barely moves. Where they are far apart it thrashes — and that thrashing is the uncertainty in your predictions.

Click the panel to pin a point

Slack in the rope

±0.00

2 observations

This visualisation needs JavaScript. The short version: with two observations the fitted curve is almost unconstrained between them and swings wildly; each new observation shortens the leash until the curve barely moves at all.

Rope stiffness How hard the rope resists bending between pins. Pin tightness How firmly each observation holds the rope down.

What you are looking at

The ropes

Seven red strands, each a curve that is fully consistent with the observations. They are not decoration: every frame is an honest draw from the posterior, so wherever the strands separate, the data genuinely does not say which one is right.

The band

The ±2σ envelope — the region the rope stays inside about 95% of the time. Watch it pinch shut at each pin and balloon in the gaps. The number in the corner is its average width, which is the one-line summary of how much you still do not know.

The dark line

The posterior mean — the single curve a normal regression would hand you. It is the least interesting object on the screen. Reporting it alone is exactly the habit this picture is meant to break.

Things worth trying

Reset to two points and watch the middle. The rope is pinned at both ends and still swings almost as freely as if you had measured nothing at all: at the midpoint those two observations close the band by under a tenth. Two points say very little about what happens between them.
Add points one at a time and watch the slack number fall. It drops fast and then barely at all: averaged over many runs, the fifth observation buys about thirty times as much certainty as the twentieth. Most of what a dataset can tell you arrives early.
Pin several points close together on one side, then look at the far side. Clustered data collapses uncertainty locally and does almost nothing for the region you never measured. This is what extrapolation costs.
Now push stiffness up without touching the data. The slack collapses — on a fixed set of four points it runs from about ±0.43 at the loosest setting to ±0.05 at the stiffest. You did not measure anything to earn that. A stiff rope is just a strong prior that the truth is smooth, and if it is not, the narrow band is false confidence. Note the rope stays curved throughout: stiffness buys smoothness, never straightness.

Teaching