Algebra 1 / Unit 10: Data and Statistics
10.5 Lines of Fit and Residuals
Notes
A line of best fit (least-squares regression line) models the trend in a scatter plot: $\hat{y} = mx + b$.
- The slope is the predicted change in $y$ for each 1-unit increase in $x$.
- Residual = actual $y$ $-$ predicted $y$. Positive residuals are above the line; negative are below.
- A good linear model has residuals scattered randomly around 0 with no pattern.
- Interpolation (predicting inside the data's range) is safer than extrapolation (far outside it).
Practice
A model for a plant's height is $\hat{y} = 2.5x + 4$ ($x$ in weeks, $y$ in cm). The slope means:
Slope = predicted change per week.
The model predicts 30 and the actual value is 26. The residual is:
$26 - 30 = -4$.
A model built from ages 5 to 12 is used to predict height at age 40. This is risky because it is:
Age 40 is far outside the data.
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