Math Core

Describing a scatterplot as "strong" or "moderate" is a judgment call, and two people looking at the same plot may disagree. The correlation $r$ turns the strength and direction of a linear relationship into a single number. In this lesson you'll see where that number comes from, how to interpret it, and, just as important, what it cannot tell you.

What r measures

Definition

Correlation

The correlation $r$ measures the direction and strength of the linear relationship between two quantitative variables. For $n$ individuals with values $(x_i, y_i)$ ,

r = \frac{1}{n - 1} \sum \left( \frac{x_i - \bar{x}}{s_x} \right) \left( \frac{y_i - \bar{y}}{s_y} \right) = \frac{1}{n - 1} \sum z_{x_i} \, z_{y_i}

where $\bar{x}$ , $s_x$ , $\bar{y}$ and $s_y$ are the means and standard deviations of the two variables.

In words: standardize every $x$ and every $y$ , multiply each individual's pair of z-scores, add up the products, and divide by $n - 1$ .

Why does this work? An individual that is above average in both variables has two positive z-scores, so its product is positive. An individual below average in both has two negative z-scores, and the product is again positive. Individuals that are above average in one variable and below in the other contribute negative products. So when most points lie in the lower-left and upper-right regions around $(\bar{x}, \bar{y})$ , the sum is positive; when most lie in the upper-left and lower-right, it is negative.

Worked example: Computing r by hand

Find the correlation for the five points $(1, 2)$ , $(2, 4)$ , $(3, 3)$ , $(4, 6)$ , $(5, 5)$ .

First the summary statistics: $\bar{x} = 3$ and $\bar{y} = 4$ . The deviations of $x$ are $-2, -1, 0, 1, 2$ , so $s_x = \sqrt{\dfrac{4 + 1 + 0 + 1 + 4}{4}} = \sqrt{2.5} \approx 1.581$ . The deviations of $y$ are $-2, 0, -1, 2, 1$ , so $s_y = \sqrt{2.5} \approx 1.581$ as well.

$x$	$y$	$z_x$	$z_y$	$z_x z_y$
$1$	$2$	$-1.265$	$-1.265$	$1.6$
$2$	$4$	$-0.632$	$0$	$0$
$3$	$3$	$0$	$-0.632$	$0$
$4$	$6$	$0.632$	$1.265$	$0.8$
$5$	$5$	$1.265$	$0.632$	$0.8$

The products add to $1.6 + 0 + 0 + 0.8 + 0.8 = 3.2$ , so

r = \frac{3.2}{5 - 1} = 0.8.

In practice you'll use a calculator (on a TI-84, turn on diagnostics and run LinReg), but doing one by hand shows exactly what $r$ is adding up.

Interpreting r

The correlation is always between $-1$ and $1$ .

The sign gives the direction. Positive $r$ means a positive association.
The magnitude gives the strength. Values near $-1$ or $1$ mean the points lie close to a line; $r = \pm 1$ only when every point lies exactly on a line. Values near $0$ mean a weak linear relationship.

An interpretation on the AP exam should mention strength, direction, "linear," and the context. For the example above: "The correlation of $r = 0.8$ indicates a fairly strong, positive, linear association between $x$ and $y$ ."

The three plots below show how scatter relates to $r$ .

r ≈ −0.99: points hug a falling line.Open in grapher →

r ≈ 0.59: a rising trend with a lot of scatter.Open in grapher →

r ≈ 0.09: essentially no linear association.Open in grapher →

Common mistake

Strength is about $\lvert r \rvert$ , not the sign. A correlation of $r = -0.85$ is stronger than $r = 0.70$ . Also, $r$ is not a percent and not a slope: $r = 0.5$ does not mean " $50\%$ related," and a nearly flat line can have $r$ close to $1$ .

Properties of correlation

Because $r$ is built from z-scores, several facts follow directly from the formula.

Roles don't matter. Swapping $x$ and $y$ swaps the two factors in each product, which doesn't change the sum. The correlation of height with weight equals the correlation of weight with height.
Units don't matter. Z-scores have no units, so $r$ has no units. Converting inches to centimeters or Fahrenheit to Celsius leaves $r$ unchanged.
Positive linear changes don't matter. Adding a constant to every value, or multiplying every value by a positive constant, does not change any z-score. Multiplying one variable by a negative constant flips the sign of each of its z-scores, so $r$ changes sign.
$r$ is not resistant. It is built from means and standard deviations, which outliers pull strongly. One unusual point can make $r$ much larger or much smaller.

Worked example: Transforming a variable

For a group of adults, the correlation between daily temperature in °F and electricity use is $r = 0.62$ . Find the new correlation if (a) temperature is converted to Celsius using $C = \frac{5}{9}(F - 32)$ , and (b) the analyst instead uses "degrees below $100$ °F," computed as $D = 100 - F$ .

(a) Celsius is a positive multiple of $F$ plus a constant, so every z-score is unchanged: $r = 0.62$ .

(b) $D = -1 \cdot F + 100$ multiplies $F$ by a negative number, which flips the sign of every temperature z-score: $r = -0.62$ . The relationship has the same strength; only the direction of the scale reversed.

Worked example: One point changes r

The eight points $(1, 3)$ , $(2, 4)$ , $(3, 6)$ , $(4, 6)$ , $(5, 8)$ , $(6, 9)$ , $(7, 10)$ , $(8, 12)$ have $r \approx 0.99$ . What happens when the point $(4.5, 15)$ is added?

The point (4.5, 15) sits far above an otherwise tight linear pattern.Open in grapher →

With the new point, a calculator gives $r \approx 0.74$ . The eight original points are still nearly collinear. The new point sits at $x = 4.5$ , exactly at $\bar{x}$ , so its $z_x = 0$ and it adds nothing to the sum of products. But it is far above the other points, so it inflates $s_y$ , which shrinks every other point's $z_y$ . The correlation drops sharply. Always report whether a conclusion depends on one unusual point.

What r can't tell you

Correlation has limits

$r$ only describes linear relationships. A strong curved relationship can have $r$ near $0$ , and a curved relationship can also have $r$ near $1$ . Always look at the scatterplot first.
Correlation does not imply causation. A strong $r$ in observational data can come from a lurking variable that drives both variables. Only a well-designed randomized experiment can establish cause and effect.
$r$ requires two quantitative variables. There is no correlation between gender and favorite sport; use a two-way table instead.

A classic example of the second point: across elementary schools, shoe size and reading level are strongly correlated. Bigger feet don't cause better reading. Age is the lurking variable: older children have bigger feet and read better.

Tip

Estimating $r$ from a plot: first decide the sign from the direction. Then imagine an oval drawn tightly around the points. A thin, pencil-like oval means $\lvert r \rvert$ near $1$ ; a fat, nearly round oval means $\lvert r \rvert$ near $0$ .

Practice

Practice 1

Which correlation indicates the strongest linear relationship?

Practice 2

Find the correlation, to two decimal places, for the points $(2, 9)$ , $(4, 7)$ , $(6, 8)$ , $(8, 4)$ , $(10, 2)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

For $25$ students, the correlation between hours of sleep and reaction time (in milliseconds) is $r = -0.54$ . Which is the best interpretation?

Practice 4

The correlation between the weights of $40$ packages in pounds and their shipping costs in dollars is $r = 0.71$ . Every weight is converted to kilograms (multiply by $0.4536$ ) and a $2 handling fee is added to every cost. What is the new correlation?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

A study finds that the correlation between a car's age $x$ (years) and its value $y$ (dollars) is $r = -0.4$ . What is the correlation if the roles are switched, with value on the $x$ -axis and age on the $y$ -axis?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

A scatterplot shows a clear U-shaped pattern, and a student computes $r = 0.04$ . Which conclusion is correct?

Practice 7

Across $50$ U.S. cities, the number of hospitals and the number of deaths per year have a correlation of $r = 0.93$ . Which statement is most reasonable?

Practice 8

Eight points lie very close to a rising line and have $r = 0.97$ . A ninth point is added far to the right, exactly on the extension of that line. What is the most likely effect on $r$ ?