Interpret the correlation coefficient as a measure of the strength and direction of a linear association, distinguish correlation from causation, and use residuals to assess the fit of a linear model (MA.912.DP.2.6, MA.912.DP.2.8, MA.912.DP.2.9).

What this topic is asking

MA.912.DP.2 asks you to interpret the correlation coefficient $r$, to understand why correlation does not imply causation, and to use residuals to judge whether a linear model fits. These are conceptual Statistics points on the B.E.S.T. Algebra 1 EOC, tested mostly with multiple choice and short interpretation.

The correlation coefficient

The correlation coefficient $r$ is a single number summarizing a linear association:

• Sign: positive $r$ means upward trend; negative $r$ means downward.
• Magnitude: $|r|$ near $1$ is strong (points hug a line); near $0$ is weak or no linear association.

So $r = 0.9$ is strong positive, $r = -0.3$ is weak negative, and $r = 0.05$ is essentially no linear association. Note $r$ measures only linear association; a strong curve can have $r$ near $0$.

Correlation is not causation

A strong correlation means two variables move together, but it does not prove one causes the other. The classic reason is a lurking variable, a third factor driving both.

Only a controlled experiment (randomly assigning a treatment) can establish causation; observational correlation cannot.

Residuals and fit

A residual is the difference between an actual data value and the model's prediction:

A positive residual means the point is above the line; negative means below. A residual plot graphs residuals against $x$. If the residuals scatter randomly around zero with no pattern, the linear model is a good fit. If they form a clear pattern (a curve or a funnel), a line is not the right model.

How the B.E.S.T. EOC examines this topic

• Multiple choice. Interpret $r$ (direction and strength), or identify a correlation-causation error.
• Short interpretation. Explain a lurking variable, or why correlation is not causation.
• Residual items. Compute a residual or read a residual plot to judge fit.

A clarifying idea: $r$ and the line of best fit describe the same linear trend, $r$ scores how tightly the points follow the line, while the line gives the trend's equation. A high $|r|$ means predictions from the line are more trustworthy.

Why correlation cannot establish causation

The reason a correlation alone never proves cause is that several different real situations produce the same observed pattern, and the data cannot tell them apart. Variable $X$ might cause $Y$; $Y$ might cause $X$; a lurking variable $Z$ might cause both; or the link might be coincidence in a small sample. Ice-cream sales and drownings rise together not because either causes the other, but because summer heat ($Z$) lifts both. Since the scatter plot only records that $X$ and $Y$ move together, it is consistent with all four explanations, so concluding causation overreaches the evidence. To actually pin down cause, you need a controlled experiment that randomly assigns the supposed cause and holds other factors fixed, ruling out lurking variables and reverse causation. This is why the EOC rewards naming a plausible third variable and stating plainly that correlation does not imply causation.

Try this

Q1. Interpret $r = 0.85$. [1 point]

• Cue. A strong positive linear association.

Q2. A residual plot shows a clear U-shaped pattern. Is a line a good model? [1 point]

• Cue. No; a pattern in the residuals means a line does not fit well.