From \(r^2\) to a Sentence About the Variables
In “Defining the Coefficient of Determination \(r^2\)” and “Computing \(r^2\) From \(r\),” you learned what the coefficient measures and how to calculate it. Now the goal is to explain a reported value in context. A correct interpretation names the response variable, identifies the explanatory variable, and describes the proportion of response variation accounted for by their linear relationship.
The order of the variables matters in the sentence. The response is the variable whose variation is being described; the explanatory variable is the variable used to help account for that variation. As in the earlier tutorials on regression, \(x\) is the explanatory variable and \(y\) is the response.
For example, if \(r^2=0.64\), convert the decimal to a percentage: \(0.64(100\%)=64\%\). Then place that percentage and the variables in the sentence. The result is not just “64% is accounted for.” The reader needs to know what varies and which linear relationship is being used to account for it.
The phrase variation in the response is essential. \(r^2\) does not say that a percentage of the observations, people, or predictions is correct. It describes how much of the overall variation among the response values is accounted for by the fitted linear relationship. In the earlier tutorial “Total Variation in the Response Variable,” variation referred to the spread of response values around their mean.
Use the word about when appropriate because reported \(r\) or \(r^2\) values may be rounded. If a problem gives \(r^2=0.64\), for example, “about 64%” is clear and appropriately cautious. The decimal itself is a proportion; converting it to a percentage makes the interpretation easier to read.
Name the Response and Explanatory Variables
Before writing the sentence, check which variable plays each role. The response variable goes after “variation in.” The explanatory variable goes after “relationship with.” This simple check prevents a common mistake: describing variation in the predictor or reversing the roles just because both variables appear in the problem.
Find which variable is the response and which is the explanatory variable in the regression setting.
Write \(r^2\) as a percentage, multiplying the decimal by \(100\%\).
Name the response after “variation in” and the explanatory variable after “relationship with.” Use “accounted for by the linear relationship.”
The percentage has no measurement units: it is a proportion of variation. Still, the variable names should be specific enough to make the interpretation understandable. If the response is “price,” say which price and what it measures. If the explanatory variable is “age,” clarify what is getting older.
The word linear is important. The coefficient of determination describes the variation accounted for by the fitted linear relationship in simple linear regression. It does not summarize every possible pattern between the variables. As discussed in the earlier tutorials on residual plots, a curved pattern or changing spread can matter when judging whether a linear model is useful.
Worked Example: Interpret a Negative Correlation’s \(r^2\)
A fictional group examines the association between the age of used electric scooters, measured in years, and their resale price, measured in dollars. The explanatory variable is age, the response variable is resale price, and the reported correlation is \(r=-0.8\). Interpret the coefficient of determination in context.
State. Find \(r^2\), then write its contextual interpretation.
Plan. Square the reported correlation, keeping the negative sign in parentheses during the calculation. Convert the result to a percentage. Since age is the explanatory variable and resale price is the response, describe variation in resale price.
Do.
As a check, the magnitude is \(0.8\), and \(0.8\times0.8=0.64\). Converting the coefficient to a percentage gives \(0.64(100\%)=64\%\).
Conclude. About 64% of the variation in resale prices of the used electric scooters is accounted for by the linear relationship with scooter age. The negative value of \(r\) indicates a negative direction of association, but the \(r^2\) interpretation itself does not state that direction.
What the Percentage Does—and Does Not—Say
The interpretation is about the response values as a group. It is not a statement about a particular individual or case. For example, \(r^2=0.64\) does not mean that a specific prediction is 64% accurate, that 64% of the response values are predicted exactly, or that 64% of cases lie on the regression line.
Nor does “accounted for” establish that the explanatory variable caused changes in the response. It describes how the linear relationship accounts for variation in the observed response values. A causal conclusion requires more than an \(r^2\) value; do not turn a descriptive regression result into a claim about cause and effect.
The coefficient of determination also does not give the direction of the association. As you saw in “Computing \(r^2\) From \(r\),” squaring removes the sign of \(r\). If direction is relevant, report it separately using the sign of the correlation or the slope. Keep that direction statement distinct from the \(r^2\) sentence.
It is often useful to explain the complement without overinterpreting it. If \(r^2=0.64\), then \(1-0.64=0.36\), or 36%, is not accounted for by the fitted linear relationship. That remainder does not, by itself, identify a single cause. It can include variation associated with other variables, random variation, or patterns not represented by the linear model. In the earlier tutorial “What Variability Explained Means,” you learned how the fitted values and residuals relate to the response’s total variation.
Worked Example: Interpret a Smaller Coefficient of Determination
A fictional school study examines the relationship between the number of optional practice sessions students attended and their score on a later skills assessment. The number of sessions is the explanatory variable, the assessment score is the response, and the reported coefficient of determination is \(r^2=0.25\). Write the interpretation and explain one conclusion the value does not support.
State. Interpret \(r^2=0.25\) in the context of the assessment scores.
Plan. Convert the decimal to a percentage and put the response variable—assessment score—after “variation in.” Put the explanatory variable—number of optional practice sessions—after “relationship with.” Avoid describing a percentage of students or claiming that practice sessions caused score changes.
Do. The conversion is \(0.25(100\%)=25\%\). The response is assessment score, and the explanatory variable is number of optional practice sessions.
Conclude. About 25% of the variation in students’ assessment scores is accounted for by the linear relationship with the number of optional practice sessions they attended. This does not mean that 25% of students received accurate predictions, and it does not establish that attending sessions caused higher scores.
Use the Sentence Even When \(r^2\) Is Given Directly
Sometimes a problem gives \(r\), so you must first square it as in the previous tutorial. Other times a calculator or computer output gives \(r^2\) directly. In either case, the interpretation is based on the coefficient of determination. Do not square a value that is already identified as \(r^2\).
When output includes \(r^2\) but does not make the variable roles obvious, return to the study description or the regression setup before writing. A software label by itself may not tell you which variable is the response. Check which variable was modeled using the other one.
Worked Example: Interpret \(r^2\) Reported in Output
A fictional environmental project models the daily height of a stream as a linear function of the previous day’s rainfall. The response is stream height, measured in centimeters, and the explanatory variable is previous-day rainfall, measured in millimeters. The regression output reports \(r^2=0.64\). Write a complete interpretation and clarify what cannot be inferred from this value alone.
State. Interpret the reported coefficient of determination using the study’s variable roles.
Plan. The value is already \(r^2\), so there is no need to square it. Convert \(0.64\) to a percentage. Name stream height as the response and previous-day rainfall as the explanatory variable.
Do. Since \(0.64(100\%)=64\%\), the percentage for the interpretation is 64%. No correlation sign is supplied here, so \(r^2\) alone does not provide the direction of the association.
Conclude. About 64% of the variation in daily stream heights is accounted for by the linear relationship with previous-day rainfall. This value alone does not indicate whether the relationship is positive or negative, and it does not show that rainfall caused the stream-height variation.
Notice that the sentence names stream height, not rainfall, after “variation in.” Rainfall is the explanatory variable in this model. Reversing the variables would change the interpretation and would not describe the fitted regression of stream height on rainfall.
Common Mistakes and AP Exam Tips
- Putting the explanatory variable after “variation in.” The standard interpretation describes variation in the response. Name the explanatory variable after “relationship with.”
- Leaving out the relationship. “64% of stream height” is incomplete. Say that about 64% of the variation in the response is accounted for by its linear relationship with the explanatory variable.
- Calling it a percentage of cases. \(r^2=0.64\) does not mean that 64% of observations are correctly predicted or lie on the line. It describes response variation.
- Claiming causation. “Rainfall causes 64% of the stream-height variation” goes beyond what \(r^2\) establishes. Use “is accounted for by the linear relationship with.”
- Using \(r^2\) to give direction. A coefficient of determination is nonnegative. Use the sign of \(r\) or the slope to describe direction when asked.
- Interpreting the decimal as a percent without conversion. A value of \(0.64\) corresponds to 64%, not 0.64%. Multiply by \(100\%\) to express the proportion as a percentage.
For a full-credit AP response, make the sentence contextual and complete: include “about,” give the percentage, identify variation in the response, name the explanatory variable, and refer to the linear relationship. If asked for direction or causation, address those separately and only make claims supported by the information given.
Check Your Understanding
For each interpretation, identify the response and explanatory variables before writing the sentence.
- A fictional study has \(r^2=0.49\), with weekly exercise time as the explanatory variable and resting heart rate as the response. Write the contextual interpretation.
- A regression of monthly household water use on household size has \(r^2=0.36\). Identify the response and explanatory variables, then interpret \(r^2\).
- A student says that \(r^2=0.81\) means 81% of observations are predicted exactly. Explain the error and give the correct kind of interpretation.
- If \(r=-0.7\), calculate \(r^2\) and write the interpretation for a model predicting resale price from the age of an item. State what the squared value does not tell you about direction.
- A model has \(r^2=0.25\). What percentage of the response variation is not accounted for by the fitted linear relationship, and why should you not assign that remainder to one specific cause based on \(r^2\) alone?