Linear regression

Objective. Use regression lines for justified interpolation.

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2. Worked example
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Learn: the key idea

Linear regression fits a straight line y = a + bx through bivariate data by minimising the sum of squared vertical residuals, giving the least-squares regression line of y on x. The gradient b equals Sxy divided by Sxx, and the line always passes through the mean point (x̄, ȳ), so a = ȳ − bx̄. This line should only be used for interpolation, predicting y-values within the range of the given x-data, since extrapolating beyond it assumes the trend continues, which may be false. For instance, a line fitted for x between 10 and 50 should not be used to predict at x = 200.

Worked example

Given Sxy = 30, Sxx = 10, x̄ = 4, ȳ = 15, find the equation of the linear regression line of y on x.

1 $b = Sxy \div Sxx = 30 \div 10 = 3$
2 $a = ȳ - bx̄$
3 $a = 15 - 3(4) = 15 - 12 = 3$
4 $Line: y = 3 + 3x$

Answer: y = 3 + 3x

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Common mistake. Students use the regression line to predict far outside the data range, e.g. extrapolating a line built from x = 0–10 to predict at x = 1000. Only interpolate within the observed range, since the linear trend may not continue.

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