5.9 Transforming Data for a Linear Model

Why transform data?

When a scatterplot shows a curved (nonlinear) pattern, a straight-line model is inappropriate. Transformations apply a mathematical function (like log, square root, or reciprocal) to x and/or y to straighten the pattern, so least-squares regression on the transformed data becomes valid and residuals become more randomly scattered.

Common transformations

For exponential growth patterns, take the log of y (log(y) vs. x); this works because exponential functions have a constant percent change, and logs turn multiplicative growth into linear growth. For power models, take the log of both x and y (log(y) vs. log(x)); this works because power functions have a constant ratio of log(y) to log(x).

Checking the transformed model

After transforming, refit the linear regression and check the residual plot for the transformed data — it should show no pattern and small, scattered residuals. A higher r² and linear residual scatter confirm the transformation improved the fit. To predict y, remember to back-transform (undo the log) your predicted value.

Key terms

Transformation (in regression)

Applying a mathematical function (log, square root, reciprocal, etc.) to x and/or y to make a nonlinear relationship linear.

Linearizing an exponential model

Plot log(y) vs. x, since exponential growth's constant percent change becomes linear on a log scale.

Linearizing a power model

Plot log(y) vs. log(x), since power functions have a constant ratio between log(y) and log(x).

Checking a transformation's success

Examine the residual plot of the transformed data — it should show random scatter with no pattern.

Back-transforming

Reversing the transformation (e.g., exponentiating a log-based prediction) to express a predicted value in original units.

Try an AP-style question

Data appear to follow y = a · bˣ. Which transformation should make them linear?

1. Plot y² against x
2. Plot log(y) against x
3. Plot y against log(x)
4. Plot 1/y against 1/x