Linear regression & loss

Drag points and fit a line. Watch MSE and MAE react to every outlier.

Data point (drag me)
Least-squares line
Residual

Tap empty space to add a point (8/16).

Linear regression finds the line y = mx + c that is “closest” to the data. Each red segment is a residual: how far the line misses a point. A loss function squashes all the residuals into one number.

Least squares picks the line with the smallest mean squared error. Squaring punishes big misses heavily, so a single outlier can drag the line towards it — the mean absolute error is far more relaxed.

ŷ = m·x + c residual r = y − ŷ MSE = (1/n) Σ r² MAE = (1/n) Σ |r| RMSE = √MSE m = Σ(x−x̄)(y−ȳ) / Σ(x−x̄)² c = ȳ − m·x̄

Line

Show errors as

Slope m: 0.47
Intercept c: 1.55
MSE: 0.152
MAE: 0.282
RMSE: 0.390
Points: 8

ŷ = m·x + c residual r = y − ŷ MSE = (1/n) Σ r² MAE = (1/n) Σ |r| RMSE = √MSE m = Σ(x−x̄)(y−ȳ) / Σ(x−x̄)² c = ȳ − m·x̄

Things to try

Note the MSE and MAE, then tap “Add outlier”. Which loss jumps more, and why?
Switch to “Squared errors”: the line minimises the total area of the red squares.
Choose “Fit it yourself” and try to match the dashed best-fit line’s MSE using only the sliders.
Drag points so they form a horizontal line. What happens to the slope? What about a vertical stack?

Test yourself: 5-minute drill Next: Gradient descent