An outlier's effect on the regression line

Parametric equations

The main cloud stays fixed. Only the highlighted point $(x_o,y_o)$ moves. Let $b_0$ be the least-squares slope of the main cloud, and $b$ the least-squares slope after that point is added. This page watches the one point. It does not rebuild the cloud.

The leverage drawn here is the horizontal distance $|x_o-{\bar{x}}_0|$ from the main mean. The residual is the vertical gap from the point to the base line, $e_0=y_o-{\hat{y}}_0$, where ${\hat{y}}_0$ is the base line at $x_o$. A point can be isolated on either measure, or far enough on both to rewrite the trend.

Interaction

• High leverage, Large residual, High influence, and Low influence: four places for the highlighted point, to compare how one position changes the fit
• Leverage: show the horizontal distance from ${\bar{x}}_0$. Farther from the main mean means more pull on the slope
• Residual: show the vertical distance to the base line, kept separate from leverage
• Mean: mark $({\bar{x}}_0,{\bar{y}}_0)$, the center of the main cloud and the point the base line passes through
• Drag the highlighted point. Reset returns it to the start

What to notice

• A large residual is not automatically a large change in slope. A point near the main mean usually has a hard time twisting $b$.
• A point far from ${\bar{x}}_0$ can move the slope more than a point near the mean, even when the nearer point has the larger residual.
• The gray line is the trend of the main cloud. The more the new line leaves it, the more that one point may be driving the fit.

Related

• Scatter diagram, correlation, and the regression line
• Percentiles and a box plot
• Data analysis

Further reading

• Simple linear regression (Wikipedia)
• Outlier (Wikipedia)