r vs r-squared: what is r squared and what r means

By Jude Wallis · Published July 28, 2026

r, the correlation coefficient, measures the direction and strength of a linear relationship between two quantitative variables and always falls between -1 and 1. r-squared, the coefficient of determination, is the fraction of the variation in y that the linear relationship with x explains.

AP Statistics: Unit 5 (topics 5.2 Correlation, 5.5 Least-Squares Regression). Correlation (r) is Unit 5 topic 5.2 and the coefficient of determination (r-squared) appears in topic 5.5, and Unit 5 is 10 to 20% of the multiple-choice section, where you interpret both with a calculator rather than computing them by hand.

What the correlation coefficient r means

The correlation coefficient, written r, is a single number that summarizes the direction and strength of a linear relationship between two quantitative variables. The direction comes from the sign: a positive r means the two variables tend to increase together, and a negative r means one tends to decrease as the other increases.

The value of r is always between -1 and 1, inclusive. The closer r sits to -1 or 1, the stronger the linear association. An r of exactly -1 or 1 means the points fall perfectly on a straight line, and an r of 0 means there is no linear association at all.

r has no units. Whether you measure height in inches or centimeters, r comes out the same, because it is built from standardized deviations rather than from raw values. That is why you can compare r across completely different pairs of variables.

One property to watch is that r is not resistant to outliers. A single unusual point can pull r sharply up or down, so always look at the scatterplot before you trust the number. You can see that effect in the regression influential point interactive.

What is r squared

r-squared (written $r^2$ ) is exactly what its name says: the correlation coefficient multiplied by itself. It is called the coefficient of determination, and it answers a different question than r does.

Where r measures how tightly the points cluster around a line, $r^2$ measures how much of the variation in the response variable your linear model accounts for. In AP terms, x is the explanatory variable (the predictor) and y is the response variable (the outcome you are trying to explain).

The value of $r^2$ is the proportion of the variation in the response variable that is explained by the linear relationship with the explanatory variable. It runs from 0 to 1, and you usually report it as a percentage.

Use this template sentence every time you interpret it: About ___ percent of the variation in the response variable y is explained by the linear relationship with the explanatory variable x. Filling that template with your own variable names is what earns credit on the AP exam.

r versus r-squared side by side

The two numbers describe the same relationship from different angles. r puts direction and strength on one scale; $r^2$ converts strength into a share of variation explained.

Feature	r (correlation coefficient)	r-squared (coefficient of determination)
Range	-1 to 1	0 to 1
Shows direction	yes, from the sign	no, the sign is lost
Shows strength	yes	yes, as variation explained
Units	none	none
Reads as	strength and direction of a linear association	percent of the variation in y explained by the linear relationship with x

How to read r values

A few anchor points are worth memorizing, and they come straight from the course:

r = 1 is a perfect positive linear association, with every point on a rising line.
r = -1 is a perfect negative linear association, with every point on a falling line.
r = 0 is no linear association.

Between those anchors, strength is a judgment call. The closer r gets to -1 or 1, the stronger the linear association, and the closer it gets to 0, the weaker. There is no official cutoff for "strong" versus "moderate," so describe strength in words and support it with the scatterplot rather than quoting a fixed threshold.

Keep in mind that an r near 1 or -1 still does not prove a line is the right model. A curved pattern can produce a large r, so a high value is a hint, not a guarantee that a linear model fits.

Going from r to r-squared, and why you cannot go back

Getting $r^2$ from r takes one step: square it. If r = 0.8, then $r^2 = 0.64$ . If r = -0.6, then $r^2 = 0.36$ .

Going the other direction is trickier, because squaring throws away the sign. From $r^2 = 0.64$ you know r is either 0.8 or -0.8, but $r^2$ by itself cannot tell you which. Direction lives in r, not in $r^2$ .

So report both when you can. r gives you direction and strength on a -1 to 1 scale, and $r^2$ gives you the share of variation explained on a 0 to 1 scale.

What r and r-squared do not tell you

Both numbers share the same two blind spots.

First, they only describe linear relationships. A strong curved pattern can have an r near 0, and a scatterplot with an obvious bend can still post a high r. Neither number knows the shape of your data, so a residual plot is the real test of whether a line fits. See how to interpret residual plots.

Second, correlation is not causation. A large r between two variables does not mean one causes the other, because a lurking variable or pure coincidence can produce the same number. This is a favorite exam trap, covered in full at correlation vs causation.

When you are ready to fit the actual line, the mechanics live in the guide to the least-squares regression line, and you can produce r, $r^2$ , and the equation together with the regression calculator.

Where this fits in AP Statistics

Correlation is topic 5.2 in Unit 5, Regression Analysis, and the coefficient of determination appears in topic 5.5, Least-Squares Regression. Unit 5 is 10 to 20% of the multiple-choice section.

On the exam you are expected to interpret r and $r^2$ in context rather than derive them by hand, since a graphing calculator produces both. That work falls under Practice 4, Interpret Results, in the course framework. Points come from the interpretation sentence, so name your variables and say the word "linear."

You can review the whole unit at Unit 5: Regression Analysis.

Given r, find r-squared and interpret both

A study of students reports a correlation of r = -0.6 between hours of screen time before bed (x) and hours of sleep that night (y). Find r-squared, then interpret both numbers.

Read off the correlation. Here x is the explanatory variable and y is the response variable, and the study gives r = -0.6.
Square r to get the coefficient of determination: $r^2 = (-0.6)^2 = 0.36$ .
Interpret r: because r is negative and moderate in size, there is a moderate negative linear association, so students with more screen time tend to get less sleep.
Interpret r-squared: about 36% of the variation in hours of sleep is explained by the linear relationship with screen time. The remaining 64% of the variation in sleep is not explained by the linear relationship with screen time.
Watch the sign: $r^2 = 0.36$ is positive even though r is negative. Squaring removes the sign, so r-squared alone cannot tell you the direction of the association.

r-squared = (-0.6)^2 = 0.36, so about 36% of the variation in hours of sleep is explained by the linear relationship with screen time. The negative direction comes from r, not from r-squared.

Compute r and r-squared from a small dataset

Five students report hours spent on practice problems (x) and points earned on a 5-point quiz (y): x = 1, 2, 3, 4, 5 and y = 1, 3, 2, 5, 4. Find r and r-squared and interpret them.

List the data with n = 5. x (hours on practice problems) is the explanatory variable and y (quiz points) is the response variable.
Find the means. The mean of x is $\bar{x} = (1+2+3+4+5)/5 = 15/5 = 3$ , and the mean of y is $\bar{y} = (1+3+2+5+4)/5 = 15/5 = 3$ .
Find each deviation from the mean. For x the deviations are -2, -1, 0, 1, 2, and for y they are -2, 0, -1, 2, 1.
Multiply the paired deviations and add them: $(-2)(-2) + (-1)(0) + (0)(-1) + (1)(2) + (2)(1) = 4 + 0 + 0 + 2 + 2 = 8$ .
Add the squared deviations for each variable: $\sum (x_i - \bar{x})^2 = 4+1+0+1+4 = 10$ and $\sum (y_i - \bar{y})^2 = 4+0+1+4+1 = 10$ .
Apply the correlation formula: $ $r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2}} = \frac{8}{\sqrt{10 \cdot 10}} = \frac{8}{10} = 0.8.$ $
Square r to get the coefficient of determination: $r^2 = 0.8^2 = 0.64$ .
Interpret r: r = 0.8 is positive and fairly strong, so there is a fairly strong positive linear association between practice hours and quiz points.
Interpret r-squared: about 64% of the variation in quiz points (the response variable) is explained by the linear relationship with practice hours (the explanatory variable).

r = 0.8, a fairly strong positive linear association, and r-squared = 0.64, so about 64% of the variation in quiz points is explained by the linear relationship with practice hours.

Frequently asked questions

Is r-squared just r squared?

Yes. For a simple linear regression with one explanatory variable, the coefficient of determination is the correlation coefficient multiplied by itself, so $r^2 = r \times r$ . The one thing squaring changes is the sign, since r can be negative but $r^2$ is always between 0 and 1.

Can r-squared be negative?

No, not in the simple linear regression you study in AP Statistics. Because it is r squared, its smallest possible value is 0 (no variation explained) and its largest is 1 (all variation explained), so you report it as a percent from 0% to 100%.

What counts as a good r-squared value?

It depends on the context, and there is no official cutoff. A physics measurement might need an r-squared above 0.99 to be useful, while an r-squared of 0.30 can be meaningful for messy human behavior. Judge it against the subject, not a fixed number.

Does a high r mean x causes y?

No. A high r only means the points cluster tightly around a line, and it says nothing about cause. A lurking variable or coincidence can create a strong correlation, which is why correlation does not imply causation. See the correlation vs causation guide.