Correlation and linear regression

Read correlation with the scatter · Diagrams · Complete reading

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Eight scatter patterns

Read shape, direction and scatter

Compare the patterns below.

Linear relationship

6 points

Linear relationship. Inspect the shape as well as the direction.

Curved: inverted U

9 points

Curved: inverted U. Inspect the shape as well as the direction.

Curved: loop

8 points

Curved: loop. Inspect the shape as well as the direction.

Positive linear

5 points

Positive linear. Inspect the shape as well as the direction.

Negative linear

5 points

Negative linear. Inspect the shape as well as the direction.

Strong negative linear

4 points

Strong negative linear. Inspect the shape as well as the direction.

Strong positive linear

6 points

Strong positive linear. Inspect the shape as well as the direction.

Weak/no clear linear

13 points

Weak/no clear linear. Inspect the shape as well as the direction.

Schematic illustrations: positions are not measured observations. No numerical data or calculated correlation is implied.
Look for shape before describing direction or strength.
A clear curved relationship can have little or no linear correlation.

Eight schematic panels, retaining source point counts: positive line 6 points, inverted-U curve 9, loop 8, positive line 5, negative line 5, strong negative 4, strong positive 6, and weak/no clear linear cloud 13. A curved or loop relationship can be strong without being linear. Positions illustrate patterns, not measured data.

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Five coefficient cases

Compare five correlation cases

Compare the patterns below.

Perfect negative · r = −1

3 points

Perfect negative · r = −1. Inspect the shape as well as the direction.

Negative · −1 < r < 0

3 points

Negative · −1 < r < 0. Inspect the shape as well as the direction.

Zero linear correlation · r = 0

8 points

Zero linear correlation · r = 0. Inspect the shape as well as the direction.

Positive · 0 < r < 1

3 points

Positive · 0 < r < 1. Inspect the shape as well as the direction.

Perfect positive · r = +1

3 points

Perfect positive · r = +1. Inspect the shape as well as the direction.

Schematic illustrations: positions are not measured observations. No numerical data or calculated correlation is implied.
The coefficient labels retain the roles stated in the source, not values computed from these schematic positions.
Zero linear correlation does not imply independence or no relationship.

Five schematic panels reproduce the source roles: perfect negative, imperfect negative, zero linear correlation with a curved arrangement, imperfect positive, perfect positive. The imperfect panels show scatter around the drawn line. These are schematic, not supplied numerical observations.

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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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Outlier in a rising scatter

Inspect how an outlier changes the pattern

Compare the patterns below.

Source outlier in a rising pattern

14 points

The other points lie closer to a rising straight-line pattern.

Ringed point: Source outlier (2, 7).

The source identifies (2, 7) exactly. Other point positions in this reconstruction are schematic; no complete numerical data set is supplied.
In this example, the source states that removing (2, 7) raises the positive correlation because the remaining pattern is closer to a rising straight line.
An outlier can increase or decrease a correlation coefficient. Investigate the observation and context before deciding whether exclusion is justified.

Fourteen-point source scatter reconstructed schematically, with the explicitly stated outlier (2,7) above the otherwise rising pattern. Removing it raises the positive correlation in this source example. Remaining coordinates are not supplied as a numerical dataset.

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