Residuals
A residual is observed y minus predicted y. Positive residuals mean the point is above the line. Least squares residuals always add up to zero. Plotting them against x reveals problems with the model.
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Analogy
The residual is the leftover after the line has explained what it can.
Example
Line: 2.2 + 0.6xx=1: 2.8, residual 2-2.8 = -0.8x=2: 3.4, residual +0.6x=3: 4.0, residual +1.0x=4: 4.6, residual -0.6; x=5: 5.2, -0.2Sum = 0
In practice
A pattern in the residuals, such as a curve, tells you a straight line is the wrong model.
Practice
Flashcards
- Define a residual.Observed y minus predicted y.
Questions people ask
What should a good residual plot look like?
A shapeless cloud of points around zero with roughly constant height. A curve means the relationship is not a straight line. A funnel that widens means the spread changes with x. Either pattern suggests the model needs work, such as a curve term or a transformed variable.
Why do residuals sum to zero?
The least squares line passes through the point of means, and the best-fitting intercept balances the points above against those below. As a result, the positive and negative gaps cancel exactly. This holds for any least squares line with an intercept.