Leverage & Cook's Distance: One Point vs a Regression Line

The Flagged Point

Position on X (leverage) 4.0

Position on Y (raw value) -4.0

Background Sample

Scatter noise (σ) 1.2

Sample size (n) 24

Influence Diagnostics

Leverage h_ii

0.00

Cook's D_i

0.00

Δ slope

+0.00

Δ R²

+0.00

ŷ = 0.60 + 0.00·x
without: ŷ = 0.60 + 0.00·x

Low influence

How it works

An ordinary least-squares fit ŷ = a + b·x minimises the sum of squared residuals over every point. Most points barely move it. A few can dominate it — and which few depends on two separate quantities:

Leverage:  h_ii = 1/n + (x_i - x̄)² / Σ(x_j - x̄)²
Residual:  e_i = y_i - ŷ_i
Cook's D:  D_i = e_i² / (p·MSE) · h_ii / (1 - h_ii)²   (p = 2 params)

Leverage measures how extreme a point's x-position is — a point far from the mean predictor can pull the line toward it even with a small residual. The residual alone measures vertical distance from the trend. Cook's distance multiplies the two together, because a point is only dangerous when it is both far out on x and off the trend line: high leverage with a small residual barely moves the fit, and a large residual near x̄ barely moves it either.

Leverage slider — slides the flagged point away from the mean of x, raising h_ii without touching its residual.
Y-value slider — slides the point off the trend line, raising its residual without touching leverage.
The orange line is the fit with the flagged point included; the grey line is the fit computed from the background sample alone. A common rule of thumb flags a point as influential when D_i > 4/n.

Real-world relevance: this is exactly why a data analyst plots residual-vs-leverage before trusting a regression — a single mis-entered data point at an extreme x-value can silently rewrite a model's conclusions.

Background points Flagged point Fit with flagged point Fit without it

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