Correlation (Pearson r)

Measure of the linear relation between two variables — value between −1 (perfectly opposed) and +1 (perfectly aligned).

Also known as: Pearson correlation, correlation coefficient, Pearson r

The Pearson correlation coefficient $r$ measures how strongly and in which direction two continuous variables move together. Unlike regression, it delivers only a single number, no prediction equation — and makes no distinction between independent and dependent variable.

r = \frac{\sum _{i} ( x _{i} - x ˉ ) ( y _{i} - y ˉ )}{\sum _{i} ( x _{i} - x ˉ ) ^{2} \cdot \sum _{i} ( y _{i} - y ˉ ) ^{2}}

|r| < 0.3 — weak association
0.3 ≤ |r| < 0.7 — moderate association
|r| ≥ 0.7 — strong association
$r^{2}$ — share of shared variance; equal to the R² of the simple linear regression

Used in

In the Algorithm Lab

regression/correlation-analysis

Sources

Fahrmeir, L., Künstler, R., Pigeot, I. & Tutz, G. — Statistik: Der Weg zur Datenanalyse, Springer
Rodgers, J. L. & Nicewander, W. A. — Thirteen Ways to Look at the Correlation Coefficient, The American Statistician, 42(1), 1988