Residuals Math Example 3

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Example 3

easy
A regression model gives residuals: {3,−2,4,−5,0}\{3, -2, 4, -5, 0\}. Are these residuals consistent with a valid LSRL? Calculate ∑ei\sum e_i and ∑ei2\sum e_i^2.

Solution

  1. 1
    ∑ei=3+(−2)+4+(−5)+0=0\sum e_i = 3 + (-2) + 4 + (-5) + 0 = 0 ✓ (residuals sum to zero — consistent with LSRL)
  2. 2
    ∑ei2=9+4+16+25+0=54\sum e_i^2 = 9 + 4 + 16 + 25 + 0 = 54

Answer

∑ei=0\sum e_i = 0 ✓ (valid LSRL); ∑ei2=54\sum e_i^2 = 54.
The zero-sum property of residuals is a mathematical property of least-squares regression. If residuals don't sum to zero, the line is not the LSRL. The sum of squared residuals (54 here) measures total prediction error.

About Residuals

The difference between an observed value and its predicted value from a regression model: residual=y−y^\text{residual} = y - \hat{y} (observed minus predicted).

Learn more about Residuals →

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