Wiki Page, Math 13 - Perez Sandbox : Module 9 - Linear Regression (1b of 4)

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2026-01-17T21:15:55-08:00

A cautionary note

Before we leave the idea of prediction, we end with the following cautionary note:

Avoid making predictions outside the range of the data.

Prediction for values of the explanatory variable that fall outside the range of the data is called extrapolation. These predictions are unreliable because we do not know if the pattern observed in the data continues outside the range of the data. Here is an example.

EXAMPLE Extrapolation

Crickets chirp at a faster rate when the weather is warm. The scatterplot shows data presented in a 1995 issue of Outside magazine. Chirp rate is the number of chirps in 13 seconds. The temperature is in degrees Fahrenheit.

There is a strong relationship between chirp rate and temperature when the chirp rate is between about 18 and 45.

What form does the data have?

This is harder to determine. A line appears to summarize the data well, but we also see a curvilinear form, particularly when we pay attention to the first and last data points.

Both the curve and line are good summaries of the data. Both give similar predictions for temperature when the chirp rate is within the range of the data (between 18 and 45). But outside this range, the curve and the line give very different predictions. For example, if the crickets are chirping at a rate of 60, the line predicts a temperature just above 95°F. The curve predicts a much lower temperature of about 85°F.

Which is a better prediction?

We do not know which is better because we do not know if the form is linear or curvilinear outside the range of the data.

If we use our model (the line or the curve) to make predictions outside the range of the data, this is an example of extrapolation. We see in this example that extrapolation can give unreliable predictions.

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