Topic 2.5 Correlation: calculate and interpret the correlation coefficient r, understand its properties (range, unit-free, resistance), and recognize what it can and cannot tell you.

What this topic is asking

The College Board (Topic 2.5) wants you to calculate and interpret the correlation coefficient $r$, to know its properties (its range, that it is unit-free and symmetric, and that it is not resistant), and to understand exactly what $r$ measures and what it misses, including the correlation-causation caution.

What r measures

You will almost always get $r$ from technology rather than this formula, but the formula reveals two things: $r$ is built from how the two variables' standardized deviations move together, and because standardizing strips out units and scale, $r$ is unit-free, unaffected by changing units, and symmetric in $x$ and $y$.

The properties of r

A few consequences are worth stating plainly. Because $r$ is symmetric, "the correlation of height with weight" equals "the correlation of weight with height," unlike a regression line, whose slope depends on which variable is the response. Because $r$ is not resistant, you should always look at the scatterplot before trusting $r$, since one stray point can inflate or deflate it. And because $r$ captures only linearity, it is silent about curvature.

What r cannot tell you

Two limitations are examined relentlessly. First, correlation is not causation: a large $|r|$ shows that two variables move together linearly, but a lurking variable, reverse causation, or coincidence can produce that pattern, so you may never conclude cause from $r$ alone. The classic examples (churches and crime, ice cream and drowning) all feature a lurking variable that drives both. Second, $r$ measures only linear association, so a value near zero does not mean the variables are unrelated; a strongly curved relationship (a U-shape, say) can have $r \approx 0$ while clearly being a tight relationship, just not a straight-line one. The reverse trap also exists: a high $r$ confirms a strong linear fit only if the scatterplot is genuinely linear; computing $r$ for a curved cloud and reporting "strong relationship" is wrong. The discipline that protects you from both traps is the same one from Topic 2.4: always describe the scatterplot's form first, and only interpret $r$ once you have confirmed the pattern is roughly linear. The College Board returns to these two cautions, no causation and linear-only, in almost every regression question, so internalising them now pays off across the whole unit.

Reading the size of r

It helps to attach rough verbal labels to magnitudes, while remembering they are conventions, not hard rules. An $|r|$ around $0.9$ or above is usually called strong, around $0.5$ to $0.8$ moderate, and below about $0.3$ weak, with the exact cut-offs unimportant compared with reading the scatterplot. What matters on the exam is interpreting $r$ in context and pairing the number with the picture: "$r = 0.85$ indicates a strong positive linear association between distance and fuel used, consistent with the tight upward-sloping scatterplot." A common follow-up is the relationship between $r$ and $r^2$ (the coefficient of determination of Topic 2.8): $r^2$ is the fraction of the variation in $y$ explained by the linear model, so a correlation of $0.85$ corresponds to $r^2 = 0.7225$, meaning about $72\%$ of the variation in $y$ is accounted for by the linear relationship with $x$. Holding $r$ and $r^2$ as related-but-different ideas, one a measure of linear strength and the other a proportion of explained variation, prepares you for the regression topics that follow.

Try this

Q1. State what the sign and the magnitude of $r$ each tell you. [2 points]

• Cue. The sign gives the direction of the linear association (positive or negative); the magnitude (closeness of $|r|$ to $1$) gives its strength.

Q2. A scatterplot is strongly U-shaped, and $r = 0.02$. Does this mean no relationship? Explain. [1 point]

• Cue. No; $r$ measures only linear association, so $r \approx 0$ here reflects the lack of a linear trend, not the absence of the strong non-linear (curved) relationship.