Correlation vs Causation Cheat Sheet

Key Facts

• Correlation means two variables are associated, so as one variable changes, the other tends to change in a pattern.
• A positive correlation means both variables tend to increase together, while a negative correlation means one tends to decrease as the other increases.
• The correlation coefficient $r$ measures the direction and strength of a linear relationship, with $-1 \le r \le 1$.
• Values of $r$ near $1$ or $-1$ show a strong linear relationship, while values near $0$ show little or no linear relationship.
• The sample correlation coefficient can be calculated with $r = \frac{1}{n - 1}\sum \left(\frac{x_i - \bar{x}}{s_x}\right)\left(\frac{y_i - \bar{y}}{s_y}\right)$.
• The coefficient of determination $r^2$ gives the fraction of variation in the response variable explained by a linear model.
• Correlation does not prove causation because a lurking variable may affect both variables or the direction of cause may be reversed.
• Strong evidence for causation usually comes from a controlled experiment with random assignment, comparison groups, and careful control of other variables.

Vocabulary

Correlation

A statistical relationship showing how two variables tend to change together.

Causation

A cause-and-effect relationship in which a change in one variable directly produces a change in another variable.

Scatter Plot

A graph of paired data values that helps show the direction, form, and strength of a relationship.

Correlation Coefficient

A number $r$ between $-1$ and $1$ that describes the strength and direction of a linear relationship.

Lurking Variable

An unmeasured variable that may explain or influence the relationship between two studied variables.

Controlled Experiment

A study design that compares groups while controlling conditions so researchers can test for cause and effect.