Correlation and linear regression

Read correlation with the scatter · Diagrams

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Causation and equal-coefficient patterns

Read the scatter as well as the coefficient

Compare the patterns below.

Left and right foot size

5 points

The rising pattern shows association. It does not establish that the size of one foot causes the size of the other.

Linear-looking scatter · source r = 0.86

11 points

The points show an approximately linear pattern with scatter. The coefficient is the source report for its original data set.

Curved pattern · source r = 0.86

11 points

A similar reported coefficient can accompany a curved pattern. A straight-line description misses this shape.

One unusual observation · source r = 0.86

11 points

Most points lie close to a line, with one unusual observation. Inspect the whole pattern before interpreting the coefficient.

Ringed point: Unusual observation in this schematic.

Strong correlation alone does not establish causation. In the foot-size example, shared influences such as genes can explain the association.
The source reports r = 0.86 for each of the three original data sets below the foot-size illustration. These reconstructed positions are schematic; they are not the raw data used to calculate that coefficient.
A coefficient alone does not reveal curvature or an unusual observation. Inspect the diagram before interpreting a linear relationship.

A five-point rising foot-size schematic and three eleven-point source-pattern schematics: scattered approximately linear, curved, and an otherwise close linear pattern with one unusual observation. The original source reports r0.86 for each of the three, but supplies no raw data; these reconstructed positions are illustrations rather than numerical datasets.

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