What r-squared Does—and Does Not—Say
In “r-squared and Strength of Association,” you compared \(r\), which includes direction and strength, with \(r^2\), which summarizes the strength of a linear relationship. This tutorial focuses on wording that can turn a correct \(r^2\) value into an incorrect interpretation. In particular, \(r^2\) is not the slope, and it does not show that the explanatory variable causes the response to vary.
In simple linear regression, \(r^2\) describes the fraction of variation in the response accounted for by its linear relationship with the explanatory variable. As covered in “Interpreting r-squared in Context,” a strong interpretation identifies the response and describes its variation. The word “accounted for” describes a statistical summary of the observed data; it does not automatically mean “caused by.”
The slope \(b\) and \(r^2\) answer different questions. The slope describes the predicted change in the response for a one-unit increase in the explanatory variable. Its units are response units per explanatory-variable unit. By contrast, \(r^2\) has no units and summarizes the fraction of response variation accounted for by the linear relationship. A slope might be 2 points per practice hour, while \(r^2\) might be 0.64, or 64%.
A second common error is to say that \(r^2\) is the percentage of variation in the explanatory variable that is accounted for. The response is the variable whose variation is described in the standard interpretation. Keep the roles straight: the explanatory variable is used to predict, and the response is what the model predicts.
A third error is turning “accounted for” into a causal claim. A high \(r^2\) does not prove that changing the explanatory variable would change the response. In an observational study, other variables or the way people entered the study may help explain the association. Even when a study is designed to investigate cause and effect, the design and evidence—not the \(r^2\) value alone—must support a causal conclusion.
A Quick Audit for an r-squared Interpretation
When reading or writing a statement about \(r^2\), check three things: the statistic being described, the variable whose variation is described, and the strength of the claim. This short audit catches many errors before they reach a final answer.
If the statement gives a rate of predicted change per unit of the explanatory variable, it is describing the slope, not \(r^2\). If it gives a fraction or percentage of variation, it may be describing \(r^2\).
The standard interpretation of \(r^2\) refers to variation in the response. Make sure the sentence does not switch the response and explanatory variable.
Use wording such as “accounted for by the linear relationship.” Do not replace it with “caused by” unless the study’s design and evidence justify a causal conclusion.
These checks are related but separate. A sentence can name the correct response and still make an unsupported causal claim. It can also avoid causal language but incorrectly call \(r^2\) a slope. Audit each part instead of relying on a single phrase.
Worked Examples: Correcting Common Errors
Worked Example: r-squared Is Not the Slope
A fictional study relates the number of weekly practice hours \(x\) to a skills-assessment score \(y\), measured in points. The fitted line is \(\hat{y}=18+2x\), and the reported coefficient of determination is \(r^2=0.64\). A student writes, “For every additional practice hour, the score increases by 64%.” Identify the error and give correct interpretations of the slope and \(r^2\).
State. The student has treated \(r^2\) as though it were the slope. The slope is the coefficient of \(x\) in the fitted line, while \(r^2\) describes a fraction of response variation.
Plan. Read the slope from the fitted equation and attach its units: score points per practice hour. Convert \(r^2\) to a percentage and interpret it for the response, assessment score, using noncausal wording.
Do. In \(\hat{y}=18+2x\), the slope is \(b=2\). Its units are assessment-score points per weekly practice hour. The coefficient of determination as a percentage is
The value 2 describes the predicted change in score for a one-hour increase in weekly practice time. The value 0.64 does not describe a change per hour; it is a proportion without units. It describes the fraction of variation in assessment scores accounted for by the linear relationship with weekly practice hours.
Conclude. The slope interpretation is: for each additional weekly practice hour, the model predicts an assessment score that is 2 points higher, on average. The \(r^2\) interpretation is: about 64% of the variation in assessment scores among the students in this study is accounted for by the linear relationship between assessment score and weekly practice hours. Neither statement says that practice hours caused a particular student’s score to rise.
Worked Example: Same r-squared, Different Slope Numbers
Suppose a fictional analysis relates weekly operating hours \(x\) of a community garden’s water pump to water use \(y\). With water use recorded in gallons, the fitted line is \(\hat{y}=10+2x\), and \(r^2=0.64\). The same water-use measurements are then recorded in tenths of a gallon, so every response value is multiplied by 10. A student argues that the \(r^2\) must change because the slope changes. Is that argument correct?
State. The slope’s numerical value changes when the response is recorded in a different unit. That change does not require \(r^2\) to change. We will compare the fitted equations and check what happens to the association summary.
Plan. Multiplying every response by 10 multiplies both the intercept and slope by 10. It changes the units and numerical scale of the response, but not the pattern’s strength as summarized by \(r^2\). Interpret each slope using the response unit in its equation.
Do. The original equation predicts gallons, so its slope is 2 gallons per pump hour. Recording each response in tenths of a gallon gives
The new slope is 20 tenths of a gallon per pump hour, which is the same rate expressed in a different unit: \(20\) tenths of a gallon equals \(2\) gallons. The intercept changes from 10 gallons to 100 tenths of a gallon for the same reason. The data have not gained or lost a linear pattern just because the response unit changed, so \(r^2\) remains 0.64.
As a check on the interpretation, both versions account for the same percentage of the response variation: \(100(0.64)=64\%\). The number 2 or 20 is a slope, with units; 0.64 is \(r^2\), a unitless fraction.
Conclude. The student’s argument is incorrect. A change in the slope’s numerical value can result from changing measurement units, while \(r^2\) remains the same. About 64% of the variation in water use is accounted for by its linear relationship with pump operating hours, whether water use is recorded in gallons or tenths of a gallon.
Worked Example: “Accounted For” Does Not Mean “Caused By”
In a fictional observational study, a group of students reports how many hours they spend with a tutor each month. Researchers also record scores on a course assessment. The regression output gives \(r^2=0.49\). One report says, “Tutoring caused 49% of the differences in assessment scores.” Explain why this conclusion goes beyond what \(r^2\) establishes and rewrite it appropriately.
State. The reported value is a coefficient of determination for the observed association between tutoring hours and assessment scores. The study is observational, so the value alone does not show that tutoring caused differences in scores.
Plan. Convert \(r^2\) to a percentage, identify the response variable, and describe the fraction of its variation accounted for by the linear relationship. Then distinguish that description from a claim about causation.
Do. Converting the coefficient of determination gives
The response is assessment score, so it is score variation—not variation in tutoring hours—that the interpretation describes. Also, students were not described as randomly assigned to tutoring amounts. Students who seek more tutoring may differ in prior preparation, available study time, or other relevant ways. The observed relationship could reflect several influences; \(r^2\) does not separate them or show what would happen if the same students were assigned different tutoring hours.
Conclude. A suitable interpretation is: about 49% of the variation in assessment scores among the students in this study is accounted for by the linear relationship between scores and reported tutoring hours. The statement that tutoring caused 49% of score differences is not justified by this \(r^2\) value. It changes a description of association into a causal claim.
Common Mistakes and AP Exam Tips
- Calling \(r^2\) a slope. A slope is a predicted response change per one-unit increase in the explanatory variable, with units. \(r^2\) is a unitless fraction or percentage of variation in the response.
- Putting the percentage on the wrong variable. Do not say that \(r^2\) describes variation in the explanatory variable. Name the response whose variation is being summarized.
- Replacing “accounted for” with “caused.” An \(r^2\) value by itself does not establish that changing the explanatory variable causes a change in the response.
- Calling \(r^2\) a prediction success rate. As discussed in “r-squared Is Not Percent Correct Predictions,” it is not the percentage of cases predicted correctly or the percentage of predictions within a certain distance.
- Leaving out context. A statement such as “64% is explained” does not identify what varies or what the linear relationship is between. Name both variables and keep the response in focus.
For full credit, first identify what the statistic means, then connect it to the situation. A clear structure is: “About [percentage] of the variation in [response] among [the observed individuals or items] is accounted for by the linear relationship between [response] and [explanatory variable].” Avoid causal wording unless the design and evidence support it. If the question also asks about predicted change per unit, use the slope and its units instead of \(r^2\).
Check Your Understanding
For each item, decide what the reported value can and cannot say. Keep the response variable, units, and causal language in mind.
- A fitted line predicts delivery time in minutes from distance in kilometers. Its slope is 4 and \(r^2=0.36\). Which value describes predicted change per kilometer, and which describes a fraction of response variation?
- A student says, “The \(r^2\) of 0.81 means 81% of the explanatory-variable variation is accounted for.” Identify the wording error and state which variable’s variation belongs in the interpretation.
- A report says that a linear relationship between weekly reading time and vocabulary score has \(r^2=0.25\). Write a cautious interpretation that identifies the response.
- An observational study finds a large \(r^2\) between hours of screen use and bedtime. Can the value alone establish that screen use causes later bedtimes? Explain briefly.
- A response is converted from meters to centimeters, changing the numerical slope. Must \(r^2\) change as a result? Explain the distinction between the two statistics.