Unit 5 · Topic 5.2 · about 20 minutes

Correlation

Interpret a correlation in context, and say what it can and cannot tell you about two quantitative variables.

Predict first

An engineer tests one car's fuel economy at 13 speeds from 20 to 80 mph. Economy improves up to about 50 mph, then gets worse the faster the car goes. What would you expect the correlation between speed and fuel economy to be?

Fuel economy at 13 test speeds

A strong relationship with $r = - 0.07$ . There is almost no straight-line trend for $r$ to measure.

One number for a linear pattern

The correlation coefficient $r$ summarizes the direction and strength of the linear association between two quantitative variables. You get it from a calculator or software, usually printed alongside the regression line, so the work is in reading it, not computing it.

The sign gives the direction. A positive $r$ goes with a positive association, and a negative $r$ with a negative association.
The size gives the strength. The closer $r$ is to $- 1$ or $1$ , the more tightly the points cluster around a straight line.

The value of $r$ is always between $- 1$ and $1$ , inclusive. At exactly $1$ or $- 1$ , every point sits on one line: a perfect linear association. At $r = 0$ there is no linear association. As the fuel data shows, no linear association is not the same thing as no association.

Daily exercise and resting heart rate for 20 adults

Each dot is one adult. Software gives $r = - 0.65$ for these data.

Worked exampleInterpreting r in context

For the 20 adults plotted above, $r = - 0.65$ between minutes of exercise per day and resting heart rate. Interpret this value in context.

Sign. $r$ is negative, so the association is negative: adults who exercise more tend to have lower resting heart rates.
Size. $- 0.65$ is well away from 0 but also well short of $- 1$ , so the linear association is moderate. The plot agrees: a clear downward drift with plenty of scatter around it.
Form. $r$ describes a linear association, so glance at the plot first. There is no curve here, so a linear description fits.
Put it together in context. There is a moderate, negative, linear association between daily minutes of exercise and resting heart rate for these adults.

Answer.

A moderate, negative, linear association: adults who exercise more per day tend to have lower resting heart rates. The correlation by itself does not show that exercise is what lowers heart rate.

No units, ever

Correlation has no units. Take the used sedans from 5.1 Graphical Representations Between Two Quantitative Variables: with age in years and price in thousands of dollars, $r = - 0.96$ . Measure age in months and price in dollars instead, and $r$ is still $- 0.96$ . Changing units relabels the axes but leaves the pattern of points, and how tightly they hug a line, exactly as it was. That is also why you can compare the strength of two associations measured in completely different units.

Stopping distance at 11 speeds, with a straight line drawn through the points

$r = 0.995$ , yet the points sit above the line at both ends and below it in the middle. That is what a curve looks like when you force a straight line through it.

Correlation is not causation

In an elementary school, students with bigger shoe sizes tend to have higher reading scores, and the correlation is strong and positive. Nobody believes big feet make better readers. Older students have bigger feet, and they have also had more years of reading practice. Age is a confounding variable: it is associated with both shoe size and reading score, so it offers another explanation for the pattern (see observational studies).

The same goes for every correlation, however strong. A relationship between two variables, real or apparent, does not show that changing one would change the other. Evidence of cause and effect comes from a well-designed experiment with random assignment, as in 1.13 Experimental Design.

Lab

Guess the Correlation

Reading strength from a picture takes practice. Play until your guesses usually land within 0.1 of the true $r$ , and notice which values are hardest to tell apart.

Open the full Guess the Correlation lab

Check your understanding

Four studies report these correlations. Which one shows the strongest linear association?

A study of 32 apartments near a university finds $r = - 0.58$ between distance from campus (miles) and monthly rent (dollars). Which interpretation is correct?

A biologist finds $r = 0.74$ between the length and weight of 40 salmon, with length in inches and weight in pounds. She converts every length to centimeters and every weight to kilograms, then recomputes. What is the new correlation?

Records from 200 house fires show a correlation of $r = 0.81$ between the number of firefighters who responded and the dollar amount of damage. Which conclusion is best?

For 25 young trees of one species, a scatterplot of age (years) and height (feet) rises steeply and then levels off. The correlation is $r = 0.91$ . A student says that because $r$ is close to 1, a straight line is a good model for these data. Which response is best?

Course alignment, for teachers

AP Statistics topic 5.2, Unit 5: Regression Analysis.

Skill 4.D: Interpret statistical calculations and results to assess meaning or a claim.