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

What this topic is asking

Standards A1.S.ID.C.8 and A1.S.ID.C.9 finish the data strand. C.8: interpret the correlation coefficient $r$ of a linear fit, a number describing the direction and strength of a linear relationship. C.9: distinguish correlation from causation, knowing that a relationship between two variables does not prove one causes the other.

Reading the correlation coefficient

$r$ packs two pieces of information into one number between $-1$ and $1$:

• Sign = direction. $r > 0$ is a positive association (line rises); $r < 0$ is negative (line falls).
• Magnitude = strength. $|r|$ near $1$ means a strong linear pattern (tight cluster); $|r|$ near $0$ means weak or no linear pattern.

So $r = 0.9$ is strong positive, $r = -0.85$ is strong negative, and $r = 0.1$ is weak. A value of exactly $\pm 1$ means the points lie perfectly on a line.

Correlation is not causation

A strong $r$ tells you two variables move together, not why. Three explanations are possible, and only the first is causation:

1. $x$ really does cause $y$.
2. A lurking variable causes both (the ice cream and drowning example: hot weather drives both).
3. Coincidence in the sample.

Because the data alone cannot tell these apart, you must not leap from a correlation to "$x$ causes $y$." Establishing causation requires a controlled experiment, not just observed association.

How TNReady examines this topic

• Multiple choice. Interpret an $r$ value (direction and strength), or choose the valid conclusion about causation.
• Inline choice. Complete a statement about whether a relationship is causal.
• Multiple select. Choose all correct interpretations of a correlation.

A clarifying idea is that $r$ describes the same line you fit in scatter plots and linear models: a high $|r|$ means the points hug that line of best fit, so the model predicts well, but it still says nothing about cause.

Why the causation distinction matters

This is one of the most important ideas in the whole statistics strand because it guards against a tempting but wrong inference. People naturally read a strong correlation as "doing more of $x$ will change $y$," and that mistake leads to bad decisions: cutting firefighters to reduce fire damage, or banning ice cream to prevent drownings. The fix is to ask, every time, could a third factor explain both? Hot weather, town size, age, or income are common lurking variables that make two unrelated things rise together. The standard wants you to recognize that observational data, no matter how strong the correlation, supports only an association, and that a controlled experiment (randomly assigning the treatment) is what can establish cause. On the EOC, the safe answer to "what does this strong correlation prove" is almost always that the variables are associated, not that one causes the other, and the best distractor to avoid is the one that asserts causation.

Try this

Q1. A data set has $r = 0.2$. Describe the linear relationship. [1 point]

• Cue. Weak positive (small magnitude, positive sign).

Q2. Shoe size and reading ability are positively correlated in children. What likely explains this? [1 point]

• Cue. A lurking variable, age: older children have bigger feet and read better.