Interpret the correlation coefficient of a linear fit and distinguish correlation from causation (LA A1: S-ID.C.8, S-ID.C.9).

What this topic is asking

Standards A1: S-ID.C.8 and S-ID.C.9 ask you to interpret the correlation coefficient of a linear fit and to distinguish correlation from causation. On LEAP 2025 these are Type I and Type II items in the Additional and Supporting Content category. The calculator reports $r$ on the calculator sessions; you interpret its sign, size, and meaning.

What the correlation coefficient measures

The correlation coefficient $r$ summarizes how well a straight line fits a scatter plot. It lives between $-1$ and $1$.

So $r = 0.9$ is a strong positive linear relationship, $r = -0.85$ a strong negative one, and $r = 0.1$ almost no linear relationship.

Reading r from a scatter plot

A loose, shapeless cloud would have $r$ near $0$; a perfectly straight line would have $r = \pm 1$.

Correlation is not causation

A strong correlation means two variables move together, not that one causes the other. S-ID.C.9 stresses three reasons a correlation can mislead:

• Lurking variable: a hidden third factor drives both (ice cream sales and drownings both rise with hot weather).
• Coincidence: unrelated trends happen to track each other over a period.
• Reverse causation: the supposed effect actually drives the supposed cause.

To establish causation, you need a controlled experiment that isolates the variable, observational correlation alone cannot.

How LEAP examines this topic

• Multiple choice. Interpret the sign and size of $r$, or match $r$ to a scatter plot.
• Type II reasoning. Explain why a correlation does not prove causation, naming a lurking variable.
• Equation response. Read or compare correlation coefficients from calculator output.

A clarifying idea: $r$ measures linear fit only. A scatter plot with a strong curved pattern can have an $r$ near $0$, because the relationship, though real, is not linear.

Why correlation cannot establish causation

The gap between correlation and causation is one of the most important ideas in all of statistics, and S-ID.C.9 places it in Algebra I because it is so easily misused. Correlation is a purely descriptive measure: it quantifies how tightly two variables move together in the data you happened to observe. It says nothing about why they move together, and there are several distinct explanations that all produce the same strong $r$. A lurking variable is the classic culprit: hot weather raises both ice cream sales and swimming (and thus drownings), so the two outcomes correlate strongly even though neither causes the other. Reverse causation can masquerade as causation in the wrong direction. And with enough variables compared, some will correlate by sheer coincidence. The only reliable way to establish that one variable causes another is a controlled experiment, where you deliberately change the suspected cause while holding other factors fixed and observe the effect, which observational data simply cannot do. This is why careful writing uses "associated with" rather than "causes," and why a headline claiming a cause from a mere correlation should be treated skeptically. Understanding this protects you from a pervasive error and is exactly the reasoning the standard wants you to articulate, often by naming a plausible lurking variable for a given correlation.

Try this

Q1. A linear fit has $r = 0.88$. Describe the relationship. [2 points]

• Cue. A strong positive linear relationship (close to $1$, positive sign).

Q2. Shoe size and reading ability are positively correlated in children. Name a lurking variable. [1 point]

• Cue. Age: older children have bigger feet and read better; age drives both.