The correlation coefficient $r$ is a single number that describes how closely a scatter plot follows a line. You saw it next to lines of fit. It is useful, and it is easy to over-read.
- $r$ is always between $-1$ and $1$.
- The sign gives the direction: positive when $y$ tends to rise with $x$, negative when it tends to fall.
- The size gives the strength: near $\pm 1$ the points hug a line; near $0$ there is no linear trend.
- $r$ measures only linear association: points on a U-shaped curve can have $r$ near $0$.
Why it works
Correlation does not show causation. Two variables can move together because a third variable, a lurking variable, drives both: ice cream sales and drowning accidents both rise in summer heat, but ice cream does not cause drowning. They can also move together by coincidence, or the cause can run the other way. The way to show cause and effect is a controlled experiment: change one variable on purpose, at random, and watch the other.
Key idea
$r$ tells the direction and strength of a linear relationship. A strong correlation alone never proves that one variable causes the other.
Worked examples
Worked example: Read r
Which is stronger, $r = 0.4$ or $r = -0.85$?
- Strength is the size of $r$: $|-0.85| = 0.85 > 0.4$.
Answer: $r = -0.85$: a strong negative relationship.
Worked example: A lurking variable
Across many towns, the number of churches and the number of crimes are strongly correlated. Why?
- Larger towns have more of almost everything, including churches and crimes.
- Population is a lurking variable driving both.
Answer: Town size, not churches, explains the correlation.
Common mistake
Concluding that one variable causes the other because they are strongly correlated. Before claiming a cause, ask: could a third variable push both? Could the cause run the other way? Was there an experiment? Without one, describe the result as an association.