Transformation to Achieve Linearity vs Linear Model

When a scatterplot curves, you re-express one or both variables and fit the line to the new pairs. The fit then belongs to the transformed variable, so the equation predicts a transformed value and you undo the function to get back to the original units. Multiplicative growth is the standard case, and a logarithm is the standard tool for it.

A culture is counted every hour for five hours: 12, 25, 48, 103, 195 and 402. A line fitted straight to those counts gives $r = 0.904$ and a residual plot with a clear arc in it. Take $\log_{10} y$ and the six values become 1.0792, 1.3979, 1.6812, 2.0128, 2.2900 and 2.6042, rising by close to a constant amount each hour. Least squares on those gives $\log_{10} \hat{y} = 1.0847 + 0.3038x$ with $r = 0.9998$. Undoing the logarithm, $\hat{y} = 10^{1.0847} \cdot (10^{0.3038})^x = 12.15 \cdot 2.013^x$, so the count is doubling about every hour. At $x = 3$ the model predicts 99 against an observed 103.

The claim to resist is "$r$ went from 0.904 to 0.9998, so the second model predicts the counts better." Those two numbers are about different response variables. The first describes variation in counts, the second variation in the logarithm of counts, and $r$ or $r^2$ cannot rank models whose response has changed underneath them. Judge the two on their residual plots and on predictions returned to the original units.

What a typical error means changes too. Here the standard deviation of the residuals on the log scale is 0.0124, and $10^{0.0124} = 1.029$, so a typical miss is about 3 percent of the predicted count rather than a fixed number of cells.

On the AP course, be precise: the Fall 2026 framework has no topic on transforming data to achieve linearity, since re-expressing data to straighten a curve was the old topic 2.9 that the College Board removed. Detecting a departure from linearity is a different skill and is still assessed, under topic 5.4, Residuals, where curvature in a residual plot is the evidence against a linear model.

A linear model claims that a response is well described by a straight line plus scatter around it. It is written $\hat{y} = a + bx$, where $\hat{y}$ (read y-hat) is the value the line returns, $a$ is the y-intercept, $b$ is the slope, and each observed $y$ differs from $\hat{y}$ by a residual. Any straight line through a scatterplot is a linear model; the least-squares regression line is the particular one that minimizes the sum of the squared residuals.

Six students study 1, 2, 3, 4, 5 and 6 hours and score 60, 72, 65, 78, 71 and 86, and the least-squares model is $\hat{y} = 58 + 4x$. Drop the hat and write $y = 58 + 4x$ and the equation becomes false for all six of them: it asserts scores of 62, 66, 70, 74, 78 and 82 when the students actually scored 60, 72, 65, 78, 71 and 86. The hat is what turns a false equation into an honest prediction.

So the misreading to name is "the data follow the equation." They do not, and no data ever do. A linear model is a summary of the pattern, and the leftover scatter is part of the description rather than a defect in it. A second version of the same slip is treating the model as a fact about the world instead of about these observations: refit on a different sample and $a$ and $b$ come out different.

Nothing in the arithmetic checks whether a line belongs on the data. Feed a clean parabola into a regression routine and it returns a slope, an intercept and a correlation without complaint. The form of the scatterplot and the shape of the residual plot are what justify the model, and the model speaks only for the range of $x$ the data covered.

Linear regression models are topic 5.3 in Unit 5, Regression Analysis.