Mathfolis

Correlation

Unit 2 · Exploring Two-Variable Data

What AP Stats asks here

The correlation coefficient rr summarizes the linear relationship between two quantitative variables. The recurring AP traps: r≈0r \approx 0 does not mean 'no association' (a strong curve can give r=0r = 0), high rr does not mean causation (lurking variables drive many strong correlations), and rr is not resistant — a single outlier can push it up or down.

Correlation coefficient

Definition
r=1n−1∑i=1n(xi−xˉsx)(yi−yˉsy)r = \dfrac{1}{n - 1} \sum_{i=1}^{n} \left(\dfrac{x_i - \bar{x}}{s_x}\right)\left(\dfrac{y_i - \bar{y}}{s_y}\right)
Range
−1≤r≤1-1 \le r \le 1

Key properties

Unitless
converting units does not change r\text{converting units does not change } r
Symmetric
r(x,y)=r(y,x)r(x, y) = r(y, x)
Linear only
r=0  does NOT imply no associationr = 0 \;\text{does NOT imply no association}
Not resistant
single outliers can move r\text{single outliers can move } r
AP Tip: Use the scatterplot to estimate r before computing. If the plot suggests r ≈ 0.9 and the calculator returns 0.3, the calculator's input is wrong — investigate before reporting.
Caution: Three explanations for high r: (1) X causes Y, (2) Y causes X, (3) a lurking variable Z causes both. Correlation never settles which one is right.
Type 1

Estimate r from a scatterplot description

Match the verbal description to an approximate r. Watch for the parabola trap (r≈0r \approx 0 despite strong curvature) and the flat-line trap (r=0r = 0 when y does not change with x).

Example 1
Estimate r for each scatterplot. (i) Tightly clustered points along an upward line. (ii) Tightly clustered points along a U-shape (parabola). (iii) Perfectly straight horizontal row of points. Which choice gives plausible values? (A) (i) ~0.9; (ii) ~0.9; (iii) ~1.0 (B) (i) ~0.9; (ii) ~0; (iii) 0 (C) (i) ~0; (ii) ~0.5; (iii) ~0 (D) (i) 1.0; (ii) 1.0; (iii) 1.0

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Type 2

Properties of r

rr is unitless and symmetric. Adding an against-trend outlier reduces rr; adding a with-trend outlier inflates it. A reported rr outside [−1,1][-1, 1] is impossible.

Example 2
A study reports r=0.78r = 0.78 between height (cm) and weight (kg). The investigator reconverts the data to inches and pounds. The new rr is: (A) 0.78 unchanged — correlation is unitless. (B) Smaller because inches are smaller units. (C) Larger because pounds are larger numbers than kilograms. (D) Indeterminate without the raw data.

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Type 3

Correlation vs causation

When two variables are both driven by a third (lurking) variable, they move together. The correlation is real, but the causal arrow is not what it looks like.

Example 3
A study finds r=0.92r = 0.92 between monthly ice-cream sales and monthly drowning deaths. Which interpretation is best supported? (A) Eating ice cream causes drowning. (B) Drowning causes ice-cream sales. (C) A lurking variable (warm summer weather) likely drives both ice-cream sales and swimming activity; correlation does not imply causation. (D) The data must be fabricated because rr is too high.

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