1.7 Transformations to Achieve Linearity
Section Overview
- Example 1.7.1
- Gallery 1.7.1 - Exp. & Power Reg on Calc
- Gallery 1.7.2 - Example 1.7.1
- Pop Quiz 1.7.1
Nonlinear Modeling
So far, we have focused on using linear regression for our data sets. In this section, we will introduce situations where our data, when plotted, turns out to be nonlinear. When this happens, we can sometimes apply logarithmic transformation techniques to our data so it reverts back into the good old linear variety that we know and love!
There are two types of models that result from logarithmic transformations:
- The Power Model, which takes the form: and transforms into
- The Exponential Model which takes the form: and transforms into ,
The model that is most appropriate for your data is the one with which you can best plot your data on a line.
Example 1.7.1
Example 1.7.1 a) — recorded answer
a) To decide whether a power model or an exponential model is more appropriate, we need to perform find out which model best transforms the dataset into a line. The fastest way we can do this is to use our calculator.
Using our calculator, we can see if the ExpReg function gives a higher correlation coefficient than PwrReg. The two functions are located on the same screen as the LinReg function. Before this is done, however, we can refer to Gallery 1.5.3 to learn how to turn diagnostics on for our calculator and to learn where to access the LinReg function.
After performing both types of regression, it is clear that PwrReg wins out with its compared to the ExpReg output of . These steps can be found in Gallery 1.7.1. We can also look at Gallery 1.7.2 to see how the power model provides a better linear solution than the exponential transformation.
Gallery 1.7.1 Example 1.7.1 (Performing Exponential and Power Regression on TI-83/84)
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Choosing the List Editor under the EDIT Menu. Open full-size image S0133Frame 2 of 4
Filling in the x values into L1 and y values into L2 from our given data table Open full-size image S0134Frame 3 of 4
Results from performing Exponential Regression Open full-size image S0135Frame 4 of 4
Results from performing Power Regression Open full-size image S0136
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Filename capture times, with no recorded timezone. Frames above follow the original gallery positions.
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The displayed regression outputs are transcribed with their source ellipses:
| Output | Exponential Regression | Power Regression |
|---|---|---|
| Model | y = a × b^x | y = a × x^b |
| a | 1.5266… | .9419… |
| b | 1.1351… | 1.4473… |
| r² | .8037… | .9984… |
| r | .8965… | .9992… |
The models are and respectively.
Example 1.7.1 b) — recorded answer
b) Since we now know to use the power model, we can use the equation:
When we performed the PwrReg function, we should have also been able to find the intercept to be and the slope to be . Plugging these numbers as well as the new data point into the equation, we get:
It would take 902 Earth years for this 10th planet to make a full revolution around its sun!
Gallery 1.7.2 Example 1.7.1
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The linear plot for Example 1.7.1. Note how the plot has a noticeable curvature. We need to perform either Power or Exponential Regression on the plot in order to make the dataset more linear. Open full-size image S0137Frame 2 of 3
The exponential plot for Example 1.7.1. Note how finding the natural log for the y-values this has not transformed the data into a linear function. Open full-size image S0138Frame 3 of 3
The power plot for Example Problem 1.7.2. It linearly transforms the data better than the exponential plot. Open full-size image S0139
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Filename capture times, with no recorded timezone. Frames above follow the original gallery positions.
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Screenshot 2026-10-05 at 3.46.57 PM.png - · S0139 · displayed frame 3
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Pop Quiz 1.7.1
Glossary and deeper reading · 1 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Least-squares regression
Least-squares regression fits a line predicting a response from an explanatory variable. The fitted line minimizes the sum of squared vertical residuals between observed and predicted responses. For example, a fitted line can predict a plant’s height from its age, within the range of observed ages.
Read more: Least-squares regression in the captured Statistics reading →