r measures one thing, and only for lines

The correlation coefficient $r$ measures the strength and direction of the linear association between two quantitative variables. It satisfies $-1 \le r \le 1$:

1. $r = 1$ or $r = -1$: every point falls exactly on a straight line.
2. $r$ near 0: little or no linear association.
3. The sign gives the direction, positive or negative. The magnitude gives the strength.

For the eight students' study hours and exam scores, $r \approx 0.977$: a strong positive linear association.

The sign is not part of the strength, which is the single most common misreading. $r = -0.9$ describes a much stronger association than $r = 0.3$; correlations are ranked by $|r|$, and a strong negative correlation is a strong one. "Negative correlation" means as one variable rises the other tends to fall, not that the relationship is weak or unimportant.

1. Unitless. $r$ has no units and does not change when units change. Converting inches to centimeters, or dollars to euros, leaves $r$ exactly the same.
2. Symmetric. The correlation between $x$ and $y$ equals the correlation between $y$ and $x$. Swapping the axes does not change $r$, even though it changes the regression line.
3. Not the slope. $r$ and the slope share a sign and nothing else. A slope of 3.5 points per hour and $r = 0.977$ describe different things, and $b_1 = r\frac{s_y}{s_x}$ is what connects them.
4. Two quantitative variables only. There is no correlation between eye color and income, or between grade level and club choice. Those are categorical, and Topic 3.14's chi-square handles them.

A fifth property is worth stating negatively: $r$ is not resistant. A single point far from the pattern can move it substantially, which is the subject of the next section.

Two blind spots make the plot mandatory:

Curvature. A clean curved pattern can produce a large $r$. Points falling tightly along a curve that rises then flattens might give $r = 0.9$, and reporting that as evidence of a linear relationship is exactly backwards: the plot shows the pattern is not linear, and $r$ only ever measured how well a straight line would do.

Influential points. Because $r$ is not resistant, a single far-out point can create or destroy a correlation. A formless cloud with one point far up and to the right can produce a large positive $r$ driven entirely by that point; remove it and $r$ collapses toward 0. Quoting $r$ to three decimals for such a plot reports a precision the data do not have.

The working rule: compute $r$ only after looking at the scatterplot, and report it only when the form is approximately linear. A correlation attached to a plainly curved plot is a number about a model nobody should be using.

An association between two variables says they move together. It does not say that one produces the other, and no value of $r$ changes that: $r = 0.99$ from observational data is still an association.

Two alternative explanations always compete with a causal reading:

1. Reverse direction. The response may be driving the explanatory variable rather than the other way round.
2. A lurking variable. A third variable may be driving both. Ice cream sales and drowning deaths correlate strongly because hot weather raises both, and neither one causes the other.

What licenses a causal claim is design, not strength: a randomized experiment, where subjects are assigned to levels of the explanatory variable, rules out lurking variables by making the groups comparable. Observational data can establish that studying and scores go together and cannot establish that studying raises scores, because students who study more may differ in other ways.

Naming a plausible lurking variable is not a formality. "The association may be explained by prior preparation, since better-prepared students may both study more and score higher" is the sentence that shows the alternative was actually considered.