Definition: Correlation means two variables move together; causation means a change in one produces a change in the other, and correlation alone cannot prove it.
Why is the difference important?
Two quantities can rise together for reasons that have nothing to do with each other. Reading a strong r as proof of cause leads to wrong decisions.
What explains a correlation without cause?
- A hidden third variable drives both.
- The direction is reversed from what you expect.
- Pure coincidence in a small sample.
Example. Over twelve months, a town records umbrella sales and the number of muddy shoe complaints, and finds r = 0.91. Umbrellas do not cause mud. Rainfall is the hidden variable behind both.
How is causation shown?
Usually through a designed experiment with random assignment, where one group gets a treatment and the other does not. Observational data can suggest a link but rarely settles it. A common mistake is to write that x causes y just because the scatter plot looks tight. A safer sentence is: x and y are strongly associated.