Correlation, Regression, and Causation

What is correlation?

Correlation describes how two variables move together.

These bands are rough guides. What counts as "strong" depends on the field.

• Positive: larger values of one go with larger values of the other.
• Negative: larger values of one go with smaller values of the other.
• Pearson measures linear relationships; Spearman measures monotonic (rank-based) relationships and is less sensitive to outliers.
• r ≈ 0 does not rule out a non-linear relationship. Always look at a scatter plot.

Why correlation is not causation

Two variables can move together because:

Simpson's paradox: a trend can appear in every subgroup but reverse when the groups are combined, because the group sizes differ. Always check relevant segments.

Regression basics

Regression estimates the relationship between an outcome (dependent variable) and one or more inputs (independent variables).

Simple linear regression: y = a + b·x

• Multiple regression includes several inputs, e.g., ad spend, price, and season.
• The fitted relationship depends on assumptions (linearity, independent errors, no extreme influential points).
• Regression alone does not establish causation.

Example

If ad spend and sales are correlated, more spend may be associated with more sales. But other factors could also drive sales:

• Seasonality (festive periods)
• Discounts running at the same time
• Brand awareness built over years
• Overall market demand

Stronger causal evidence requires:

• A controlled experiment (A/B test with random assignment), or
• A suitable causal design: difference-in-differences, regression discontinuity, or natural experiments

Key takeaways

• Use correlation to find relationships worth investigating, not as proof of cause.
• Always check context, confounders, study design, and whether the conclusion goes beyond the evidence.
• Plot the data before trusting a coefficient.

Try it: Find two variables that might be correlated in a business dataset. List at least one third factor that could affect both.