Correlation

Correlation summarizes linear association; it does not prove that changing one variable causes the other to change.

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.

Read the scatter plot first

Before calculating $r$, inspect a scatter plot for:

• direction;
• form (linear or curved);
• strength;
• outliers and clusters.

A single coefficient can hide nonlinearity, subgroups, or unusual observations.

Positive and negative association

Positive association means larger values of one variable tend to occur with larger values of the other. Negative association means larger values of one tend to occur with smaller values of the other.

“Tend to” is important: association describes a pattern, not a perfect rule for every individual.

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.

Correlation is not slope

The value $r$ is unitless and constrained to $[-1,1]$. Regression slope has output-per-input units and can be any real number.

Changing measurement units changes slope but not correlation. A strong relationship can have a small numerical slope if units are scaled differently.

Outliers and influential points

One unusual point can greatly strengthen, weaken, or reverse correlation. Calculate and visualize with and without a suspected outlier only when there is a defensible reason, and report the decision transparently.

Never remove an observation simply to obtain a preferred result.

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.

Correlation and causation

Association may arise from causation, reverse causation, a lurking variable, selection bias, coincidence, or shared trends. Strong correlation alone cannot identify the mechanism.

Randomized experiments support causal claims more strongly than observational data, when ethics, design, adherence, and analysis are sound.

Restricted range and groups

Correlation can weaken when data cover only a narrow range. Combining distinct subgroups can also create or reverse a pattern, an effect related to Simpson's paradox.

Describe the sampled range and examine groups before generalizing.

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?

Related topics

• Scatter Plots
• Linear Regression
• Bias in Data
• Data Collection