Residuals and Residual Plots

Define a residual as the vertical gap between an observed data value and the value predicted by a regression line, compute residuals from a regression equation and data, and use residual plots to judge whether a linear model is appropriate.

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Tutorial

Defining the Residual

A regression line gives a predicted value $\hat{y}$ for each value of $x$ . The actual observed value $y$ usually differs from $\hat{y}$ , and that difference is called the residual.

The residual for a data point $(x_i, y_i)$ is the observed value minus the predicted value:

e_i = y_i - \hat{y}_i

Geometrically, $|e_i|$ is the vertical distance from the point to the regression line.

For example, suppose the regression line is $\hat{y} = 3x - 1$ and one of the observations is $(2, 7)$ . The predicted value at $x=2$ is

\hat{y} = 3(2) - 1 = 5,

so the residual is

e = 7 - 5 = 2.

The observation sits $2$ units above the regression line.