The Plumbline — the Least Squares bench

The line is forced. Punish error by its square and exactly one line is best — and three strangers who share no code (the normal equations, a from-scratch gradient descent, a perturbation jury) all swear to the same slope. Drag the cloud; Σr² falls to a floor no nudge can get under.

The cloud

Scatter Add noise Tidy Watch GD converge Drag an outlier

Drag a point to re-snap the line live · click empty space to add · right-click a point to remove. The line always equals the closed form — it is recomputed every frame.

Σr² — the floor

Σr² —

The gold fill is the total squared error — the sum of the literal squares on the plane. The tick is the analytic minimum. At rest the fill sits on the floor; nudge the line and it can only rise above it.

R² as geometry

explained

R² —

One bar = the total variance SS_tot. The gilt fraction is the variance the line explains — that gilt fraction literally is R². The red remainder is what's left unexplained (SS_res).

R² = 1 − SS_res/SS_tot—

= r² (Pearson²)—

✓ identical

Pearson r (signed)—

best-fit line—

The teeth — best is a choice

L2 · Σr²

L1 · Σ|r|

both

'Best' was a choice. The square is what makes the line unique and gives the closed form — but it is not robust. Flip to L1 (Σ|r|) and the line moves; drag an outlier and L2 chases it while robust L1 holds.

Show

The literal squares

Residual stems

Mean baseline ȳ (SS_tot)

Variance ghosts

Legend

L2 best-fit (solid slate) — the answer

L1 / LAD (dashed violet) — a choice

mean ȳ (dotted grey) — the null

a squared residual — Σ of these is the floor

a residual (vertical: point→line)

Residuals are vertical — OLS minimises vertical error, x is assumed known (this is not perpendicular/Deming regression). Each squared residual is hung as a true square; the total gold area is Σr².

↗ the cross

↗ The Rydberg Constant Fit a line to light: the same Σr² minimum reads the Rydberg constant off four smudges of starlight — slope −R, intercept R/4.