Key takeaway: Correlation vs causation is the difference between two variables that move together in data and one variable actually producing a change in the other; a link alone never proves the cause.
Grade 9 – Algebra 1S-ID.C.9Statistics and Data
Definition of Correlation vs. Causation
The idea of correlation vs causation starts with a scatter plot: when two variables rise and fall together, they are correlated. Causation means a change in one variable really produces the change in the other. The first does not guarantee the second.
A third variable, called a lurking variable, can drive both quantities. Two variables can also be tied by coincidence. Only a well-designed experiment, or strong evidence ruling out other explanations, supports a claim of cause and effect.
Formula and notation
There is no formula for causation. Correlation is described with a scatter plot and the correlation coefficient \(r\), where \(-1 \le r \le 1\). We write:
\[ \text{strong correlation} \;\not\Rightarrow\; \text{causation} \]
Say it in words: “\(x\) is associated with \(y\)” is a safe statement, while “\(x\) causes \(y\)” needs more evidence.
Key facts
- A strong correlation can come from a lurking variable that affects both quantities.
- Coincidence alone can produce a high value of r in a small data set.
- A randomized experiment is the standard way to test whether one variable causes another.
- Causation usually produces correlation, but correlation does not work backward to prove causation.
Correlation vs. Causation example
A correlation vs causation question: a town records that months with more ice cream cones sold also have more drowning rescues at its lake. The correlation coefficient is \(r = 0.88\). A councilman proposes banning ice cream near the lake. Is that reasonable?
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The value \(r = 0.88\) shows a strong positive association between cone sales and rescues.
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Ask what else changes by month. In July the temperature is high, so more people buy ice cream and more people swim.
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Temperature is a lurking variable that affects both counts. Nothing in the data shows cones lead to rescues.
The ban is not justified: the data show correlation, and warm weather explains both variables.
Common mistake
Watch out
Priya finds \(r = 0.93\) between a student’s shoe size and reading score across grades 1 to 8 and says bigger feet make kids read better.
Why it is wrong
Age is the hidden variable: older students have both larger feet and higher reading scores. A high \(r\) describes a pattern in the data only. To claim a cause, you need an experiment or other evidence that rules out lurking variables.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 1 vocabulary · Statistics and Data