Topic 5.2 Notes – Correlation

What Correlation Measures

The correlation coefficient $r$ describes the direction and strength of a linear association between two quantitative variables. It comes from paired data $(x,y)$, so each $x$-value is matched with its own $y$-value.

A few facts you need cold:

• $-1 \le r \le 1$
• Sign tells direction
  • $r>0$ means positive association
  • $r<0$ means negative association
  • $r=0$ means no linear association
• Magnitude $|r|$ tells strength
  • close to 1 means strong linear association
  • close to 0 means weak linear association

The endpoints have exact meanings:

• $r=1$ means all points lie exactly on an upward-sloping line
• $r=-1$ means all points lie exactly on a downward-sloping line

Two properties students forget:

• Unit-free means changing inches to centimeters or pounds to kilograms does not change $r$.
• Symmetry means switching $x$ and $y$ does not change $r$.

That tells you what the number means. Now you need to connect it to what the graph looks like.

What Different Correlations Look Like

A scatterplot lets you estimate $r$ by looking at two things:

1. Does the trend rise or fall?
2. How tightly do the points hug a straight line?

These three plots show the pattern you should have in mind when you match a graph to a correlation value.

Scatterplots with positive, near-zero, and negative correlation

Quick matching:

• Tight upward band $\rightarrow r$ near $1$
• Tight downward band $\rightarrow r$ near $-1$
• Loose cloud with little linear pattern $\rightarrow r$ near $0$

There are no official cutoffs for weak, moderate, and strong. Your description should match both the value of $r$ and the actual graph.

One common mistake is mixing up steepness with correlation. A steep line and a shallow line can have the same $r$. Correlation measures how tightly points follow a line, not how fast $y$ changes.

How to Interpret Correlation in Context

On a quiz or FRQ, don’t stop at “strong positive.” Say what the variables are and who the data are about.

A solid template is:

There is a [weak/moderate/strong] [positive/negative] linear association between ___ and ___ for ___; as ___ increases, ___ tends to [increase/decrease].

Example:

There is a strong positive linear association between weekly study time and quiz score for these students; as study time increases, quiz score tends to increase.

Correlation can support:

• statements about overall linear association
• wording like “tends to increase” or “is associated with”

Correlation cannot support:

• the slope or amount of change
• percent explained $(r^2$ is later)
• perfect prediction
• causation

That last point matters a lot because $r$ leaves out many things the graph can show.

What Correlation Does Not Tell You

Correlation measures only linear association. You can have a strong relationship and still get $r=0$ if the pattern is curved.

So always look at the scatterplot, because $r$ does not show:

• curvature
• clusters
• gaps
• unusual observations

Also, correlation is not resistant. One influential outlier can change the sign and size of $r$ a lot.

In the second graph, the five original points have $r \approx 1.0$, then one far-right low point pulls the overall correlation way down.

Clusters can also fool you. Two groups may each have their own pattern, and one overall $r$ can hide that.

One more technical case. If all $x$-values are the same, or all $y$-values are the same, then $r$ is undefined.

Correlation Is Not Causation

A correlation does not mean one variable causes the other to change.

Possible explanations for a correlation:

• $x$ may affect $y$
• $y$ may affect $x$ (reverse causation)
• a third variable affects both
• chance or data-collection problems create the pattern

A lurking variable is a variable related to both measured variables that helps explain the association. Confounding means the effect of one variable is mixed up with another so you can’t separate them.

Classic example: Ice cream sales and drownings may be positively correlated, but hot weather is the lurking variable.

On the AP exam, use association language unless the study design justifies causation. Say “tends to” or “is associated with.” Even a correlation near $1$ or $-1$ does not prove cause-and-effect.