Pearson r and what it means

Unit-free, symmetric, linear-only, not robust — computing r from the sums and writing the sentence the paper marks.

Dividing the units out

The covariance has the right sign and useless units. Divide by both standard deviations and the units cancel:

$r = \frac{s _{x y}}{s _{x} s _{y}}$

The Pearson correlation coefficient $r$ always lies between −1 and +1. Its sign is the direction; its distance from 0 is the strength of the linear association; ±1 means every point on one straight line.

Worked example — Northfield, continued

From the sums table: $s_{x}^{2} = \frac{43 , 035 - 8 \times 69.37 5 ^{2}}{7} = \frac{4 , 531.875}{7} = 647.41$ , so $s_{x} = 25.444$ . $s_{y}^{2} = \frac{3 , 801 , 150 - 8 \times 687. 5 ^{2}}{7} = \frac{19 , 900}{7} = 2, 842.86$ , so $s_{y} = 53.319$ .

$r = \frac{1 , 337.5}{25.444 \times 53.319} = \frac{1 , 337.5}{1 , 356.6} = 0.986$

The sentence the paper marks has three parts — direction, strength, linear — and names both variables:

> There is a very strong, positive, linear association between annual income and credit score among these eight clients.

Four properties, each of them examinable

Unit-free. Income in dollars or thousands, score on any scale — $r$ is 0.986 regardless. Adding a constant to every value or multiplying by a positive one leaves $r$ unchanged.
Symmetric. $r_{x y} = r_{y x}$ : correlation does not know which variable is the response. The regression slope (L4.3) does.
Linear only. $r$ measures how close the points are to a straight line. A perfect curve can have $r = 0$ ; a strong $r$ can hide a curve with a trend. Plot first.
Not robust. One point far from the others can drive $r$ from 0.99 to 0.2 (L4.6 does exactly this) or manufacture a correlation where none exists among the rest.

Words for the size

The usual convention: $∣ r ∣ > 0.7$ strong, 0.3 to 0.7 moderate, below 0.3 weak. It is a convention — the paper accepts it, and a question that gives $r = - 0.42$ expects "a moderate, negative, linear association", with the variables named. Never describe $r$ as a percentage, and never say one variable causes the other: these are eight clients who chose their own incomes, and Module 2's confounders — age, years of credit history — sit behind both variables. Association, never effect.