Correlation and regression

Correlation measures how strongly two things move together in a straight line. Regression draws the best line through the data so you can predict one from the other.

Reading r

The correlation r runs from −1 to +1.

• +1: a perfect rising line. −1: a perfect falling line. 0: no straight-line relationship.
• r² is the share of the variation in one variable that the line explains. r = 0.5 means r² = 0.25: the line explains a quarter of the variation.

Three warnings

1. Plot it first. Remember Anscombe: one outlier or a curve can produce the same r as a clean line.
2. r = 0 doesn't mean "no relationship". A U-shaped relationship can have r near zero.
3. Correlation is not causation. Ice cream sales and drownings rise together because both rise in summer.

Regression

A regression line has a slope: how much y changes, on average, when x goes up by one unit. "Each extra hour of study is linked to 2.3 more marks" is a slope. Adding more variables (multiple regression) lets you estimate one factor's link to the outcome while holding others fixed, but only the ones you measured.

Prediction is not explanation

A regression that predicts well does not explain why. Causal claims need a design (experiments, natural experiments) or careful causal reasoning, not just a good fit.

RememberPlot it, read r and the slope, and never read a correlation as a cause without a design that supports it.