Coefficient of Determination (R²)

Show your work for each problem.

1.
The correlation coefficient of a data set is $r = 0.6$ . What is $R^2$ ?
1. a)1.2
2. b)0.36
3. c)0.6
4. d)0.06
Answer: 0.36
$R^2 = 0.6^2 = 0.36$ . A common mistake is to keep $0.6$ or to double it; you have to square $r$ .
2.
The correlation between two variables is $r = -0.5$ . What is $R^2$ ? Give your answer as a decimal (like 0.3).
Answer: 0.25
$R^2 = (-0.5)^2 = 0.25$ . $R^2$ is never negative, even when $r$ is.
3.
A regression of plant height on hours of sunlight has $R^2 = 0.72$ . Which statement is correct?
1. a)Sunlight causes 72% of a plant's height
2. b)72% of the data points lie exactly on the regression line
3. c)72% of the variation in plant height is explained by the linear relationship with sunlight
4. d)The slope of the regression line is 0.72
Answer: 72% of the variation in plant height is explained by the linear relationship with sunlight
72% of the variation in plant height is explained by the linear relationship with hours of sunlight. $R^2$ does not count points on the line, does not prove causation and is not the slope.
4.
For a regression line, $SS_{\text{tot}} = 200$ and $SS_{\text{res}} = 50$ . What is $R^2$ ? Give your answer as a decimal.
Answer: 0.75
$R^2 = 1 - \frac{50}{200} = 1 - 0.25 = 0.75$ , so the line explains 75% of the variation.
5.
The correlation between hours of practice and points scored is $r = 0.9$ . Find $R^2$ and the percent of variation explained.
Answer: 81
1. What is $R^2 = 0.9^2$ ? (decimal) 0.81
2. What percent of the variation is explained? (number only, like 25) 81
6.
A bakery's regression of daily bread sales on temperature has $SS_{\text{tot}} = 60$ and $SS_{\text{res}} = 21$ . Find $R^2$ .
Answer: 0.65
1. What is $\frac{SS_{\text{res}}}{SS_{\text{tot}}} = \frac{21}{60}$ as a decimal? 0.35
2. What is $R^2 = 1 - 0.35$ ? 0.65
7.
Which of these values is NOT possible for $R^2$ of a least squares regression line?
1. a)1
2. b)0.5
3. c)0
4. d)1.2
Answer: 1.2
$R^2$ is always between $0$ and $1$ . $0$ (no linear fit) and $1$ (perfect fit) are possible, but $1.2$ would mean more than 100% of the variation is explained, which is impossible.
8.
A regression line has a negative slope and $R^2 = 0.49$ . What is the correlation coefficient $r$ ? Give your answer as a decimal.
Answer: -0.7
$\sqrt{0.49} = 0.7$ , and because the slope is negative, $r = -0.7$ .
9.
A model has $R^2 = 0.64$ . What percent of the variation in $y$ is NOT explained by the linear relationship with $x$ ? Give the percent as a number only (like 25).
Answer: 36
$0.64 = 64\%$ is explained, so $100\% - 64\% = 36\%$ is not explained.
10.
A regression of phone battery charge on hours of use has slope $-2.5$ and $R^2 = 0.36$ . Find the correlation coefficient $r$ .
Answer: -0.6
1. What is $\sqrt{0.36}$ ? 0.6
2. The slope is negative. What is $r$ ? -0.6
11.
The $y$ -values of a data set are $5, 9, 6, 12, 8$ . The regression line on $x$ has $SS_{\text{res}} = 12$ . Find $R^2$ .
Answer: 0.6
1. What is the mean $\bar{y}$ ? 8
2. What is $SS_{\text{tot}} = \sum (y_i - \bar{y})^2$ ? 30
3. What is $R^2 = 1 - \frac{12}{30}$ ? (decimal) 0.6
12.
The residuals of a regression line are $2, -2, 1, 0, -1$ and $SS_{\text{tot}} = 40$ . Find $R^2$ .
Answer: 0.75
1. What is $SS_{\text{res}}$ , the sum of the squared residuals? 10
2. What is $\frac{SS_{\text{res}}}{SS_{\text{tot}}}$ as a decimal? 0.25
3. What is $R^2$ ? 0.75
13.
In a summer study of 30 towns, a regression of sunburn cases on ice cream sales gives $R^2 = 0.85$ . What is the best conclusion?
1. a)The line explains 85% of the variation, but a third variable such as sunny weather may drive both
2. b)Selling less ice cream would prevent sunburns
3. c)Eating ice cream causes 85% of sunburns
4. d)85% of the towns lie exactly on the regression line
Answer: The line explains 85% of the variation, but a third variable such as sunny weather may drive both
The line explains 85% of the variation, but $R^2$ cannot show causation. Sunny, hot weather is a confounding variable that increases both ice cream sales and sunburns.
14.
The residuals of a regression line are $3, -1, -2, 1, -1$ and $SS_{\text{tot}} = 80$ . What is $R^2$ ? Give your answer as a decimal.
Answer: 0.8
$SS_{\text{res}} = 9 + 1 + 4 + 1 + 1 = 16$ , so $R^2 = 1 - \frac{16}{80} = 1 - 0.2 = 0.8$ .
15.
A straight line fitted to some data has $R^2 = 0.95$ , but the residual plot shows a clear U shape. What should you conclude?
1. a)The line is a perfect model because $R^2$ is so high
2. b)The calculation of $R^2$ must be wrong
3. c) $x$ causes 95% of the changes in $y$
4. d)The data follows a curve, so a non-linear model would fit better
Answer: The data follows a curve, so a non-linear model would fit better
A U-shaped residual plot means the data curves, so a non-linear model would fit better, even though $R^2$ is high. $R^2$ cannot replace the residual check, and it never proves causation.
16.
The data $(1, 3)$ , $(2, 2)$ , $(3, 6)$ , $(4, 10)$ , $(5, 9)$ has the regression line $\hat{y} = 2x$ . Calculate $R^2$ .
Answer: 0.8
1. What is the mean $\bar{y}$ ? 6
2. What is $SS_{\text{tot}}$ ? 50
3. What is $SS_{\text{res}}$ ? 10
4. What is $R^2$ ? (decimal) 0.8
17.
Model A has $r = -0.8$ and model B has $r = 0.7$ for the same response variable. Which model explains more of the variation in $y$ , and by how many percentage points?
Answer: 15
1. What is $R^2$ for model A? (decimal) 0.64
2. What is $R^2$ for model B? (decimal) 0.49
3. Model A explains more. By how many percentage points? (number only) 15
18.
A regression has $SS_{\text{tot}} = 250$ and $R^2 = 0.92$ . Find $SS_{\text{res}}$ and the explained variation $SS_{\text{tot}} - SS_{\text{res}}$ .
Answer: 230
1. What is $\frac{SS_{\text{res}}}{SS_{\text{tot}}} = 1 - R^2$ ? (decimal) 0.08
2. What is $SS_{\text{res}}$ ? 20
3. What is the explained variation $SS_{\text{tot}} - SS_{\text{res}}$ ? 230