Correlation

The correlation coefficient

Pearson's correlation coefficient $r$ lies between $-1$ and $1$:

The sign gives direction and $|r|$ gives strength of the linear relationship. Values near $1$ indicate strong positive linear association, near $-1$ strong negative association, and near $0$ weak linear association.

Worked interpretation

Interpret the variables, direction, form, and population scope.

Standardized formula idea

Correlation can be viewed as the average product of standardized scores. Paired observations that are both above their means or both below contribute positively; observations on opposite sides contribute negatively.

Because of standardization, $r$ has no units and is unchanged by positive linear changes of units.

Nonlinear relationships

A strong curved relationship can have correlation near zero because positive and negative linear tendencies cancel. Pearson $r$ measures linear association only.

Use a suitable nonlinear model or another association measure when the scatter plot is curved.

Common mistakes

Calling $r=0$ “no relationship.” It means no linear relationship.

Interpreting magnitude without direction. Sign matters.

Treating correlation as a percentage. $r=0.8$ is not “80% correlated.”

Claiming causation from observational correlation. Design and alternatives matter.

Reporting $r$ without a scatter plot. Shape and outliers may make it misleading.

Quick self-check

• Are both variables quantitative and paired?
• What do direction, form, strength, and outliers show visually?
• Is Pearson correlation appropriate for an approximately linear pattern?
• Is $r$ interpreted as unitless association rather than slope?
• Are causal claims supported by the study design?
• Does the conclusion stay within the sampled population and range?