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Correlation

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The correlation coefficient $r$ measures the strength and direction of a linear association between two quantitative variables, with $-1 \le r \le 1$, the sign giving direction and the magnitude giving strength. It is unitless, unchanged by unit conversions, symmetric in the two variables, and it is not the slope: $b_1 = r\frac{s_y}{s_x}$ connects them. For the eight students, $r \approx 0.977$. It is not resistant, so a single far-out point can create or destroy it, and a large $r$ can accompany a clearly curved pattern, which is why the plot comes first.

The sign gets read as part of the strength, so $r = -0.9$ is called weaker than $r = 0.3$ when it describes a far stronger association. $r$ gets treated as changing with units, or as the slope, or as computable between a categorical and a quantitative variable. A large $r$ gets offered as proof of linearity when the plot shows a curve, and a correlation driven by one influential point gets quoted to three decimals. And an observational association gets converted into a causal claim, when only random assignment rules out the reverse direction and the lurking variables that make ice cream sales track drowning deaths.

the sign is direction, the magnitude is strength r = 0.98 r = -0.92 r = 0.30 -0.92 is far STRONGER than 0.30: rank by |r| negative means as one rises the other falls, not that it is weak
The middle plot is nearly as tight as the left one and its correlation is negative. Ranking by the raw value instead of the absolute value puts it below a loose cloud.
two plots where r alone would mislead r = 0.90, and CURVED a large r is not evidence of a linear pattern one point r = 0.85, driven by ONE point remove it and r collapses toward 0: r is not resistant look at the plot first, then decide whether r is worth reporting
Both plots return a correlation a summary table would call strong. Only the scatterplot shows that one measures a curve and the other measures a single point.

The work

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Lesson
Correlation ›

Defines r as the strength and direction of a linear association, separates its sign from its magnitude, works through unitlessness, symmetry, and its distinctness from the slope, then shows the curvature and influential points it cannot see and why design rather than strength licenses a causal claim.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items on correlation: negative values read as weak, r treated as unit-dependent or as the slope, large r offered as proof of linearity, single points driving a correlation, and observational associations read as causes. Take it cold to find your habit, or after the lesson to check it is gone.

Not started · 10 items · ~15 min
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Targeted Practice
Drill a single misconception ›

Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.

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