Low and High r-squared Values

What “Low” and “High” Mean for \(r^2\)

In “Interpreting \(r\)-squared in Context,” you learned to describe \(r^2\) as the proportion of variation in the response accounted for by its linear relationship with the explanatory variable. This tutorial applies that interpretation to values near the low and high ends of the scale. The central idea is that \(r^2\) summarizes how much of the response’s variation is accounted for by the fitted linear model.

A low \(r^2\), such as \(0.05\), means the linear relationship accounts for a small proportion of the variation in the response. A high \(r^2\), such as \(0.95\), means it accounts for a large proportion. These numbers are proportions, so converting them to percentages makes their meaning easier to express: \(0.05\) is \(5\%\), and \(0.95\) is \(95\%\).

“Fit” here refers to how much variation in the observed response is accounted for by the line, not to whether a prediction for one individual is exactly right. A larger \(r^2\) means the line accounts for more of the response’s variation in the data being described. A smaller \(r^2\) means it accounts for less. Neither value, by itself, identifies why the remaining variation exists.

The complement of \(r^2\) is the proportion of response variation not accounted for by the linear model. For \(r^2=0.05\), the complement is \(1-0.05=0.95\), or \(95\%\). For \(r^2=0.95\), the complement is \(1-0.95=0.05\), or \(5\%\). This complement describes variation not accounted for by the line; it does not say that a particular other variable or cause accounts for all of it.

Interpreting a Low \(r^2\)

Suppose a regression of a response on an explanatory variable has \(r^2=0.05\). The linear relationship accounts for about \(5\%\) of the variation in the response among the observations. The other \(95\%\) is not accounted for by this linear model. That is a careful description of what the statistic says.

It would be too strong to conclude from \(r^2=0.05\) alone that the variables are unrelated. The value describes the strength of the linear model’s accounting for response variation; it does not summarize every possible feature of a relationship. It would also be incorrect to say that the line predicts \(5\%\) of cases correctly. As explained in “\(r\)-squared Is Not Percent Correct Predictions,” \(r^2\) is about variation, not a count of accurate predictions.

A low value often means the fitted line leaves substantial response variation unaccounted for. Whether that makes the line useful for a particular task depends on the purpose and context. For example, a modest account of variation might still be useful in an exploratory setting, while a prediction task requiring high precision may need more. The number alone does not settle that practical decision.

Interpreting a High \(r^2\)

Now suppose \(r^2=0.95\). The fitted line accounts for about \(95\%\) of the variation in the response through its linear relationship with the explanatory variable. This indicates that the linear model accounts for a large proportion of the response variation in the data. The complement is \(5\%\), the proportion not accounted for by the line.

A high \(r^2\) is evidence that the line accounts for a large share of variation in the response; it does not mean the line passes through every observed point. Individual observations can still differ from their predicted values. In “Standard Deviation of the Residuals, \(s\)” and “Using \(s\) to Describe Prediction Accuracy,” you learned about another summary that describes the typical size of prediction errors in the response’s units. The \(r^2\) interpretation is not a replacement for that information.

A high \(r^2\) also does not establish that changes in the explanatory variable cause changes in the response. Describe the relationship in terms of the variables and the variation accounted for, without making a causal claim unless the study design supports one. A high value is about the line’s account of variation in the observed data, not proof of a cause-and-effect mechanism.

Comparing Low and High Values

When interpreting several coefficients of determination, compare the percentages of response variation accounted for by their linear models. State which model accounts for more variation, and keep the comparison tied to the named response. The size of \(r^2\) does not depend on the units used to measure the response, but the interpretation still needs the actual response and explanatory variable to be meaningful.

For a straightforward comparison, the models should be describing the same response in a relevantly comparable setting. If one model predicts a different response, saying that its \(r^2\) is larger does not automatically make it the more useful model overall. Consider the study’s purpose and what counts as a useful prediction. This comparison is about the proportion of response variation accounted for, not every feature of the models.

Common Mistakes and AP Exam Tips

• Leaving out the response variable. “The model explains \(95\%\)” is incomplete. A full-credit interpretation says, for example, “About \(95\%\) of the variation in quiz scores is accounted for by the linear relationship between quiz score and practice problems completed.”
• Calling \(r^2\) a percentage of accurate predictions. An \(r^2\) of \(0.95\) does not mean that \(95\%\) of predictions are correct. It describes the proportion of variation in the response accounted for by the linear relationship.
• Assigning the unexplained percentage to a specific cause. If \(r^2=0.05\), the line does not account for \(95\%\) of response variation. That statement does not show that a particular unmeasured variable accounts for the remaining variation.
• Describing a high value as a perfect fit. A high \(r^2\) means the line accounts for a large proportion of the response variation, not that every observed response equals its prediction.
• Using “no relationship” to describe a low value. A low \(r^2\) says that the linear model accounts for a small proportion of response variation. It does not, by itself, justify a broad claim that the variables have no relationship at all.
• Mixing up percentage and percentage points. When comparing \(40\%\) with \(80\%\), the difference is \(40\) percentage points. State the individual percentages first so the comparison is clear.
• Making a causal claim from \(r^2\). The coefficient of determination describes a linear association in the data. It does not by itself show that the explanatory variable causes changes in the response.

For an AP response, use “about” when the reported value is rounded, name the response, and use “accounted for by its linear relationship with” to make the interpretation precise. For example: “About \(5\%\) of the variation in harvest is accounted for by its linear relationship with sunlight hours.” If discussing the complement, say that variation is “not accounted for by the linear model,” rather than claiming to know its causes.

The next tutorial, “What \(r\)-squared Does Not Tell You,” develops an important caution: \(r^2\) is useful, but it does not summarize everything one might want to know about a regression relationship.