Teacher Guide: Coefficient of Determination (R²)

Learn what R² tells you about a regression line, how to calculate it from r or from sums of squares, and why a high R² does not prove causation.

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Learning Objectives

Calculate $R^2$ from the correlation coefficient using $R^2 = r^2$
Calculate $R^2$ from $SS_{\text{res}}$ and $SS_{\text{tot}}$ using $R^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{tot}}}$
Interpret $R^2$ in context as the percent of variation in $y$ explained by the linear relationship with $x$
Find $r$ from $R^2$ using the sign of the slope
Explain why a high $R^2$ does not prove causation or a correct model, and use residuals to check the fit

Prerequisites

• Correlation Coefficient: what $r$ measures and why $-1 \le r \le 1$
• Linear Regression: finding and using a line of best fit $\hat{y} = mx + b$
• Residuals: observed minus predicted values and residual plots
• Scatter Plots and squaring decimals, square roots and percents

Discussion Starters

1. Two models have $R^2 = 0.64$ , one with $r = 0.8$ and one with $r = -0.8$ . How are they the same and how are they different?
2. Why do you think we square the residuals before adding them instead of just adding them?
3. If a model has $R^2 = 0.30$ , is it useless? When could it still be helpful?
4. Ice cream sales and sunburns have a high $R^2$ . Does ice cream cause sunburn? What else could explain it?
5. How could curved data still give a high $R^2$ for a straight line?

Common Misconceptions

$R^2$ is the percent of points that lie on the regression line

Remediation: Show a data set where no point is exactly on the line but $R^2 = 0.81$ . Explain that $R^2$ compares leftover variation with total variation, it does not count points.

$R^2 = r$ , so an $r$ of $0.7$ means 70% is explained

Remediation: Have students compute $0.7^2 = 0.49$ and compare 49% with 70%. Repeat with $r = 0.3$ ( $R^2 = 0.09$ ) to show how quickly the explained share drops.

A high $R^2$ means $x$ causes $y$ and the line is the correct model

Remediation: Use the curved data set $y = x^2$ (high $R^2$ , U-shaped residuals) and a confounding example such as ice cream and sunburns. Ask students what $R^2$ cannot see.

Differentiation Ideas

For Struggling Students:

• Start with $R^2 = r^2$ only, using a calculator for the squares
• Use a sentence frame: "___% of the variation in ___ is explained by the linear relationship with ___"
• Give $SS_{\text{res}}$ and $SS_{\text{tot}}$ directly before asking students to compute them

For On-Level Students:

• Calculate $SS_{\text{tot}}$ , $SS_{\text{res}}$ and $R^2$ for small data sets of 5 points
• Go from $R^2$ back to $r$ using the sign of the slope
• Compare two models of the same data and decide which explains more variation

For Advanced Students:

• Show that $SS_{\text{tot}} = SS_{\text{res}} + SS_{\text{reg}}$ for the least squares line and that $R^2 = \frac{SS_{\text{reg}}}{SS_{\text{tot}}}$
• Explore how one outlier changes $R^2$
• Fit a curve to the data $y = x^2$ and compare its $R^2$ and residuals with the straight line

Standards Alignment

HSS.ID.C.8 (CCSS.MATH.CONTENT.HSS.ID.C.8)
Compute (using technology) and interpret the correlation coefficient of a linear fit
HSS.ID.B.6.B (CCSS.MATH.CONTENT.HSS.ID.B.6.B)
Informally assess the fit of a function by plotting and analyzing residuals
HSS.ID.C.9 (CCSS.MATH.CONTENT.HSS.ID.C.9)
Distinguish between correlation and causation

Lesson Resources

visualStrong vs Weak Fit Scatter Plots
Two scatter plots with trend lines and residuals: one with $r = 0.9$ ( $R^2 = 0.81$ ) and one with $r = 0.3$ ( $R^2 = 0.09$ )
activityExplained or Not?
Students collect class data (arm span and height, shoe size and test score), find $R^2$ with a calculator and argue which relationship is stronger
worksheetR² Practice
Convert between $r$ and $R^2$ , calculate $R^2$ from sums of squares and write interpretations in context

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

The coefficient of determination, written

R^2

, tells you how much of the variation in the response variable

y

is explained by the linear relationship with the explanatory variable

x

For simple linear regression (one

x

, one

y

), it is the square of the correlation coefficient:

R^2 = r^2

It can also be calculated from two sums of squares:

R^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{tot}}}

$SS_{\text{tot}} = \sum (y_i - \bar{y})^2$ is the total variation of the $y$ -values around their mean
$SS_{\text{res}} = \sum (y_i - \hat{y}_i)^2$ is the variation that is left over after using the regression line (the sum of squared residuals, written SSR in the Residuals lesson)

R^2

is always between

0

and

1

. If

R^2 = 0.81

, then **81% of the variation in

y

is explained by the linear relationship with

x

**, and the other 19% is not explained by the line.

Worked Examples

For a group of students, the correlation between daily screen time (

x

, hours) and hours of sleep (

y

) is

r = -0.7

. Find

R^2

and interpret it.

Square the correlation coefficient

R^2 = r^2 = (-0.7)^2 = 0.49

→

R^2 = 0.49

Write it as a percent

0.49 = 49\%

→ 49%

Interpret in context

49% of the variation in hours of sleep is explained by the linear relationship with screen time. The other 51% comes from other factors. → 49% explained, 51% not explained

Answer: $R^2 = 0.49$ : 49% of the variation in sleep is explained by the linear relationship with screen time. Squaring removes the minus sign, so $R^2$ is never negative.

Common Mistakes

Using $r$ as the percent: " $r = 0.5$ , so 50% of the variation is explained"

Why it's wrong: The percent of explained variation comes from $R^2$ , not from $r$ . You must square first.

Correct: $R^2 = 0.5^2 = 0.25$ , so only 25% of the variation is explained.

Forgetting the sign when going from $R^2$ back to $r$

Why it's wrong: $\sqrt{0.49} = 0.7$ is always positive, but $r$ can be negative. $R^2$ hides the direction of the relationship.

Correct: Take the sign from the slope: if the slope is negative, $r = -\sqrt{R^2}$ , for example $r = -0.7$ .

Thinking a high $R^2$ proves that $x$ causes $y$

Why it's wrong: $R^2$ only measures how well a line fits the data. A confounding variable can drive both $x$ and $y$ .

Correct: Say "81% of the variation in $y$ is explained by the linear relationship with $x$ ", never " $x$ causes 81% of $y$ ". Causation needs a controlled experiment.

Thinking a high $R^2$ means a straight line is the right model

Why it's wrong: Curved data can still give $R^2 = 0.96$ . $R^2$ does not show patterns in the residuals.

Correct: Always look at the residual plot. Random scatter supports the line; a U shape or another pattern means a different model fits better.

Why It Matters

Once you have a regression line, the next question is: how good is it?

R^2

answers this with one number.

Science: researchers report $R^2$ to show how well a model describes their measurements
Business: analysts compare sales models and keep the one that explains more of the variation
Sports: coaches check how much of the variation in sprint times is explained by training hours
Medicine: doctors see how strongly a dose is linked to a change in blood pressure

Knowing what

R^2

does not say is just as important: a high

R^2

does not prove that

x

causes

y

, and it does not prove that a straight line is the right model. You still need to look at the residuals.

Real World Applications

Sports Analytics

Coaches compare training data with race results to see how much of the difference in performance is linked to training.

Example:

If the correlation between weekly training hours and 100 m sprint time is $r = -0.85$ , then $R^2 = 0.7225$ : about 72% of the variation in sprint times is explained by the linear relationship with training hours.

1Try It Yourself

For a swimming club, the correlation between weekly training hours and 50 m race time is $r = -0.9$ .

What is $R^2$ ? Give it as a decimal.

Step 1: Write the mathematical expression

Square the correlation coefficient:

Real Estate Pricing

Real estate analysts build regression models for house prices and use $R^2$ to judge whether a model is good enough to use for valuations.

Example:

A model of price on floor area with $R^2 = 0.70$ explains 70% of the price variation; the other 30% comes from location, condition and other factors.

2Try It Yourself

For a model of apartment prices on floor area, the total variation is $SS_{\text{tot}} = 500$ and the sum of squared residuals is $SS_{\text{res}} = 75$ (both in squared thousands of dollars).

What is $R^2$ ? Give it as a decimal.

Step 1: Write the mathematical expression

Use $R^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{tot}}}$ :

Medicine and Dosage

In clinical studies, researchers check how strongly the dose of a medicine is linked to a change in a measurement such as blood pressure.

Example:

If a higher dose goes with lower blood pressure (negative slope) and $R^2 = 0.36$ , then $r = -0.6$ : a moderate negative linear relationship.

3Try It Yourself

A study finds that a higher dose goes with lower blood pressure. The regression line has a negative slope and $R^2 = 0.64$ .

What is the correlation coefficient $r$ ?

Step 1: Write the mathematical expression

Take the square root and choose the sign from the slope:

Key Takeaways

1The coefficient of determination $R^2$ is the share of the variation in $y$ explained by the linear relationship with $x$
2For simple linear regression, $R^2 = r^2$ , so squaring the correlation coefficient gives $R^2$
3 $R^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{tot}}}$ compares the leftover variation with the total variation
4 $R^2$ is between $0$ and $1$ : $R^2 = 0.81$ means 81% of the variation in $y$ is explained
5To go back from $R^2$ to $r$ , take $\sqrt{R^2}$ and give it the sign of the slope
6A high $R^2$ does not prove causation or a correct model: always check the residual plot

Frequently Asked Questions

Can R² be negative?

Not for a least squares regression line like the ones in this lesson:

SS_{\text{res}}

is never larger than

SS_{\text{tot}}

, so

R^2

stays between

0

and

1

. Squaring

r

also always gives a number that is

0

or more.

What counts as a good R²?

It depends on the field. In physics experiments, values above

0.95

are common. When studying people (behavior, health, economics), an

R^2

0.30

can already be useful, because people are affected by many factors.

Is R² always equal to r²?

Yes for simple linear regression with one explanatory variable, which is what this lesson covers. With several explanatory variables there is no single

r

, but

R^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{tot}}}

still works.

Why does R² say nothing about the direction of the relationship?

Because squaring removes the sign:

r = 0.8

and

r = -0.8

both give

R^2 = 0.64

. To know the direction, look at the slope of the regression line or at

r

itself.

Glossary

Coefficient of determination ( $R^2$ ): The share of the variation in $y$ that is explained by the linear relationship with $x$ ; a number from $0$ to $1$
Total sum of squares ( $SS_{\text{tot}}$ ): $\sum (y_i - \bar{y})^2$ : the total variation of the $y$ -values around their mean
Sum of squared residuals ( $SS_{\text{res}}$ , also SSR): $\sum (y_i - \hat{y}_i)^2$ : the variation left over after using the regression line
Residual: The difference between an observed and a predicted value: $y_i - \hat{y}_i$
Explained variation: The part of the total variation in $y$ that the regression line accounts for: $SS_{\text{tot}} - SS_{\text{res}}$

Formula Card

R² from r

R^2 = r^2

For simple linear regression, square the correlation coefficient

R² from sums of squares

R^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{tot}}}

One minus the share of the variation that is left over

Total sum of squares

SS_{\text{tot}} = \sum (y_i - \bar{y})^2

Squared distances of the y-values from their mean

Sum of squared residuals

SS_{\text{res}} = \sum (y_i - \hat{y}_i)^2

Squared distances of the y-values from the regression line

r from R²

r = \pm\sqrt{R^2}

Use the sign of the slope of the regression line

Teacher Guide: Coefficient of Determination (R²)

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Sports Analytics

Real Estate Pricing

Medicine and Dosage

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

R² from r

R² from sums of squares

Total sum of squares

Sum of squared residuals

r from R²

More in This Topic

Introduction to Regression

Linear Regression

Scatter Plots

Line of Best Fit