Interpreting correlation means reading both the sign and the size of $r$ correctly. On the AP exam, precise language matters because the number describes only linear association and carries no units.

What the correlation coefficient tells you

The correlation coefficient, written as $r$, summarizes the direction and strength of a linear association between two quantitative variables. It is a single number, so its meaning depends on careful interpretation rather than on the value alone.

When you interpret correlation, focus on two features at the same time:

A set of eight scatterplots shows how the sign of $r$ determines direction (upward vs. downward trend) while the magnitude $|r|$ determines strength (tight vs. diffuse clustering around a line). The endpoints $r=1$ and $r=-1$ display perfect linear patterns with no scatter, while values near $0$ show weak linear association. Source

• the sign of $r$, which tells the direction of the linear association

• the magnitude of $r$, which tells how strong the linear association is

A correct interpretation should always use the word linear.

Four scatterplots illustrate how $r$ captures linear association: positive, negative, and zero. The final panel shows a strong non-linear pattern with $r=0$, emphasizing that correlation measures only straight-line structure (not “relationship” in general). Source

Correlation does not describe every possible relationship between variables; it specifically describes how well the points follow a straight-line pattern.

Unit-free and bounded values

One of the most important facts about correlation is that it is unit-free. This means the value of $r$ does not depend on the measurement units used for the variables. If height is measured in inches instead of centimeters, or time in minutes instead of seconds, the correlation stays the same.

Because correlation is unit-free:

• it has no label

• it is not measured in dollars, inches, percentages, or any other unit

• it can be compared across different settings more easily than many other statistics

Correlation is also always between $-1$ and $1$, inclusive. That means:

• $-1 \le r \le 1$

• values outside this interval are impossible

• a value closer to either endpoint represents a stronger linear association

Interpreting the sign of $r$

The sign of correlation tells the direction of the association.

• If $r$ is positive, higher values of one variable tend to go with higher values of the other variable.

• If $r$ is negative, higher values of one variable tend to go with lower values of the other variable.

• If $r$ is zero, there is no linear association.

The sign does not tell you how strong the relationship is. A correlation of $-0.90$ is not weaker than a correlation of $0.60$ just because it is negative. In fact, $-0.90$ shows the stronger linear association because its magnitude is farther from $0$.

To compare strength, look at the absolute value of correlation, written as $|r|$:

• larger $|r|$ means a stronger linear association

• smaller $|r|$ means a weaker linear association

Interpreting the magnitude of $r$

The magnitude of $r$ tells how closely the data follow a straight-line pattern.

• Values of $r$ close to $1$ or $-1$ indicate a strong linear association.

• Values of $r$ close to $0$ indicate a weak linear association.

There are no universal cutoffs that work in every situation, so your interpretation should be cautious. Words such as weak, moderate, and strong are acceptable, but they should match the size of $r$ and the context.

A larger magnitude means the points would be expected to fall more closely around a straight line. A smaller magnitude means the points would be more scattered and less clearly linear.

Special values: $1$, $-1$, and $0$

Some values have especially important interpretations.

• $r=1$ means a perfect positive linear association. Every point lies exactly on a straight line that rises from left to right.

• $r=-1$ means a perfect negative linear association. Every point lies exactly on a straight line that falls from left to right.

• $r=0$ means no linear association.

The word perfect matters for $r=1$ and $r=-1$. These values mean the data fit a straight line exactly, with no scatter at all.

The phrase no linear association also matters for $r=0$. It does not justify saying the variables have “no relationship” in every possible sense. Correlation only addresses linear patterns.

Writing a complete interpretation

On AP Statistics questions, a complete interpretation should include:

• the direction: positive or negative

• the strength: weak, moderate, or strong

• the fact that the association is linear

• the names of the two variables in context

Good interpretations are specific. Instead of writing “the variables are strongly related,” write that there is a strong positive linear association or a weak negative linear association between the variables named in the problem.

Keep your wording statistical rather than causal. Correlation describes association; it does not by itself show that one variable causes changes in the other.

Common interpretation mistakes

Several errors appear often:

• Ignoring the word linear. Correlation measures linear association, so that word should appear in interpretations.

• Treating the sign as strength. The sign gives direction, not strength.

• Giving units to $r$. Correlation is unit-free, so it should never be reported with units.

• Saying $r=0$ means no relationship at all. It means no linear association.

• Assuming values near $1$ or $-1$ are perfect. Only exactly $1$ or exactly $-1$ represent perfect linear association.

Accurate interpretation depends on reading both parts of the number: the sign and the magnitude. When both are stated clearly, the meaning of correlation becomes precise and statistically correct.