9.3.1: Interpretation of r-squared

We now know how to calculate and interpret \( r \), the correlation coefficient. But statisticians often report a closely related number,\( r^2 \), called the coefficient of determination. In many ways, \( r^2 \) is even more useful for communicating how well a linear model fits the data.

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What Is \( r^2 \)?

The coefficient of determination, denoted \( r^2 \), is the square of the correlation coefficient. It tells us the proportion of variability in the response variable \( y \) that is explained by the linear relationship with the explanatory variable \( x \).

• \( r^2 \) is always between 0 and 1 (or equivalently, 0% and 100%)
• A value of \( r^2 = 0.85 \) means that 85% of the variation in \( y \) is explained by its linear relationship with \( x \)
• The remaining percentage (here, 15%) is variation left unexplained by the model due to other variables, randomness, or non-linearity

Notice that because we are squaring \( r \), \( r^2 \) is always non-negative. In a similar manner to standard deviation, it tells us how much of the variation is explained, but not the direction. You still need \( r \) (or the slope of the regression line) to determine whether the association is positive or negative.

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Where Does "Explained Variation" Come From?

To understand what \( r^2 \) is actually measuring, it helps to think about two different ways of predicting \( y \).

Without the regression model: If someone asked you to predict a student's exam score and you had no information about study hours, your best guess would simply be the mean: \( \bar{y} \). You'd be wrong by varying amounts for each student, and we can measure that total spread as the total variation in \( y \).

With the regression model: Once we know a student's study hours, we can use \( \hat{y} = a + bx \) to make a better prediction. The model accounts for some of that spread. The variation that is still unexplained after using the model is captured by the residuals.

\( r^2 \) is the fraction of the total variation that the model does explain:

\[ r^2 = \frac{\text{variation explained by the model}}{\text{total variation in } y} = 1 - \frac{\text{unexplained variation}}{\text{total variation in } y} \]

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Example: Study Hours and Exam Scores

From our worked example in Section 9.3, we found \( r \approx 0.99 \). Squaring this:

\[ r^2 = (0.99)^2 \approx 0.98 \]

Interpretation: About 98% of the variation in exam scores among these 10 students is explained by their linear relationship with study hours. Only about 2% of the variation in scores is left unexplained — due to factors our model doesn't account for, such as prior knowledge, test anxiety, or sleep.

This is a very high \( r^2 \), which makes sense given how tightly the points hugged the regression line in our scatterplot.

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More Examples: Interpreting \( r^2 \) in Context

Interpreting \( r^2 \) in Context

Context	\( r \)	\( r^2 \)	Plain-language interpretation
Study hours → exam score	0.99	0.98	98% of variation in scores is explained by study hours. Strong model.
TV hours → GPA	−0.62	0.38	38% of variation in GPA is explained by TV hours. Moderate — many other factors matter.
Height → arm span	0.97	0.94	94% of variation in arm span is explained by height. Very tight linear relationship.
Shoe size → math score	0.01	0.0001	Less than 0.01% of variation in math scores is explained by shoe size. Essentially no relationship.

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\( r \) vs. \( r^2 \): Which Should You Report?

Both \( r \) and \( r^2 \) are useful, but they answer slightly different questions:

When to Use \( r \) vs. \( r^2 \)

Use \( r \) when…	Use \( r^2 \) when…
You want to describe the direction and strength of the linear association	You want to describe how much of the variability in \( y \) the model accounts for
You are comparing two correlations to see which is stronger	You are explaining the model to a non-technical audience ("our model explains 80% of the variation")
Direction (positive or negative) matters to your conclusion	You want to assess the practical usefulness of a prediction model

In regression output from calculators and software, you will usually see both reported. Get comfortable reading and interpreting each one.

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A Common Mistake: Confusing \( r \) and \( r^2 \)

Watch out: It is easy to accidentally swap \( r \) and \( r^2 \) when writing interpretations. Here are two statements about the same data — only one is correct:

• ❌   "The correlation \( r = 0.80 \) means that 80% of the variation in \( y \) is explained by \( x \)."
• ✅   "The correlation \( r = 0.80 \), so \( r^2 = 0.64 \), meaning 64% of the variation in \( y \) is explained by \( x \)."

The "proportion of variation explained" language always belongs to \( r^2 \), never to \( r \) directly.

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Visualizing \( r^2 \): What Does "Explained Variation" Look Like?

The interactive plot below shows a fixed regression line and a set of student data points from our study hours example. For any point you select, it illustrates the two components of variation:

• Total deviation (purple): the distance from the point to \( \bar{y} \) — how far off the mean alone would have been
• Explained deviation (green): the distance from \( \hat{y} \) to \( \bar{y} \) — how much the regression line improved the prediction
• Residual (red): the distance from the point to \( \hat{y} \) — what's left unexplained

Notice that:   Total deviation = Explained deviation + Residual

Click on any data point to see its deviation breakdown.

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How Good Is "Good Enough"?

There is no single universal threshold for what counts as a "good" \( r^2 \). It depends entirely on the field and context:

• In physics or engineering, where variables have precise, controllable relationships, \( r^2 \geq 0.99 \) is common and expected.
• In education or psychology, where human behavior involves many unpredictable factors, \( r^2 \approx 0.40 \) might be considered a meaningful result.
• In economics or social science, values anywhere from 0.20 to 0.70 are routinely reported and considered informative.

The key question is not "Is \( r^2 \) high?" but rather "Does this model explain enough variation to be useful for our purpose?"

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Think About It:

• If \( r = -0.90 \) and \( r = 0.90 \), which model explains more variation in \( y \)? How do you know?
• A researcher reports \( r^2 = 0.25 \) for a study predicting depression scores from hours of sleep. Is this model useless? What does it tell us?
• Why can't we determine the direction of a linear relationship from \( r^2 \) alone?
• Using the interactive visualization above, find a point whose explained deviation is larger than its residual. What does that tell you about how the model is performing for that student?

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With \( r \) and \( r^2 \) in hand, we have a complete toolkit for describing a linear relationship: its direction, its strength, and how much of the story it tells. In the next section, we turn to the full least-squares regression line — using these ideas to build a model we can use for prediction.