What r cannot see

What you will be able to do

• explain why $r$ near $0$ does not mean "no relationship";
• show how one outlier can change $r$ a lot;
• list what $r$ does and does not tell you, and look at the plot before you trust a number.

A park on hot and cold days

A made-up park counts its visitors at noon on seven days, from a cold day to a very hot one.

The visitors rise and then fall. On a cold day few people come, and on a very hot day few people come too. The relationship is strong. Yet $r = -0.037$, very close to $0$.

Why? The rising half gives positive products, and the falling half gives negative products. They cancel. $r$ looks only for a straight line, and there is no straight line here. So $r$ near $0$ means no straight-line relationship. It does not mean no relationship.

One wrong number

Eight made-up students measure their height and their arm span in centimetres. The two are nearly equal for most people, and the plot is a strong straight line: $r = 0.987$. Then one span of $171\ \text{cm}$ is typed as $17\ \text{cm}$. With that one mistake, $r = -0.465$. One point out of eight has turned a strong positive pattern into a negative one.

An outlier is a point far from the rest of the pattern. It can come from a typing error, a broken instrument, or a real but unusual case. $r$ uses squares of distances, so a point far away has a large pull. What should you do? First check whether it is a mistake, and correct it if it is. If it is real, report $r$ with and without it, and say why.

What r does not tell you

• It does not see curves. A strong curved pattern can have $r$ near $0$, or a high $r$ that hides a bend.
• It is pulled by outliers. One point can raise $r$, lower it, or change its sign.
• It does not give the slope. $r = 0.9$ can belong to a steep line or a nearly flat one. The next course finds the slope.
• It does not prove a cause. Two variables can move together because a third one drives both. The course on correlation and cause is about this.
• It changes with the range. If you look only at students who practise $6$ to $8$ hours, the pattern looks weaker than for the whole class.

So the rule is simple. Draw the scatter plot first. Work out $r$ only for a pattern that looks straight.

Your turn: new numbers

Try it in Python

The program works out $r$ for the park, and for the arm spans with and without the typing mistake.

The usual mistakes

• Saying "$r = 0$, so there is no relationship". Say "no straight-line relationship", and look at the plot.
• Deleting an awkward point. Remove a point only if it is a mistake. Otherwise report both results.
• Comparing $r$ from different ranges. A narrow range of $x$ usually gives a smaller $r$.
• Reading a cause into $r$. A high $r$ shows that two variables move together, nothing more.

Remember these

• $r$ measures straight-line patterns only.
• One outlier can change $r$ a lot, even its sign.
• Look at the scatter plot before you trust $r$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A friend says: "The correlation between hours of sleep and test marks is $0.02$, so sleep has nothing to do with marks." What would you ask first?

2. A scatter plot has $20$ points close to a rising line and one point far below it at the right. If that point is removed, does $r$ go up or down?

3. Can two data sets with the same $r = 0.8$ have very different scatter plots?

4. Why does an outlier far from the means pull $r$ so strongly?

5. A data set has $r = 0.97$. Does that prove that the points lie on a straight line?

References

• OpenStax, Introductory Statistics, Chapter 12 (Linear Regression and Correlation), the sections on the correlation coefficient and on outliers.