What is causation in statistics?
Causation means a change in one variable directly produces a change in another: the first is the cause and the second is the effect. Two variables can be correlated without causation, because a lurking variable or chance may link them. A well-designed randomized experiment is the standard way to show causation.
Causation in a picture
Formal definition and notation
Variable X causes variable Y when changing X, with everything else held fixed, changes Y. In a study, X is the explanatory variable and Y the response variable. Evidence of causation requires an experiment in which subjects are randomly assigned to values of X, so that lurking variables are spread evenly across the groups; an observational study can show correlation but not causation.
| Symbol | Read it as | Meaning |
|---|---|---|
| X → Y | X causes Y | an arrow in a diagram shows the direction of cause and effect |
| r | r | the correlation coefficient: measures linear association, not cause |
Key facts
- Correlation does not imply causation. A strong r shows that two variables move together, not why.
- A lurking (confounding) variable affects both variables and can create a correlation between them.
- Reverse causation: the effect may run the other way. Cities with more police officers have more crime, partly because high crime leads cities to hire more police.
- Random assignment to treatment and control groups is what lets an experiment show cause and effect.
- An observational study (a survey or records) only watches; it can find an association but cannot prove a cause.
Causation examples, step by step
Example 1: Spot the lurking variable
For 5 students in grades 1 to 5, shoe size (1, 2, 3, 4, 5) and reading score (20, 35, 45, 60, 70) are strongly correlated. Do bigger feet cause better reading?
- Compute or estimate the correlation: r ≈ 0.998, a very strong positive association.
- Ask what else changes with both: age. Older students have bigger feet and have had more years of reading practice.
- Age is a lurking variable, so the data do not show that feet cause reading skill.
Answer: No. The correlation (about 0.998) comes from age, a lurking variable, not from shoe size causing reading skill.
Example 2: Read a randomized experiment
200 volunteers are randomly split into two equal groups. 24 of the 100 who take a new vitamin catch a cold, compared with 40 of the 100 who take a look-alike placebo. What is the difference in cold rates, and can it be called an effect of the vitamin?
- Rates: 24100 = 24% and 40100 = 40%.
- Difference: 40% − 24% = 16 percentage points.
- The groups were formed by random assignment, so lurking variables such as age or habits should be balanced between them.
- With a large enough difference, the experiment supports a cause-and-effect claim for people like these volunteers.
Answer: The vitamin group had 16 percentage points fewer colds; because of random assignment, the difference can be attributed to the vitamin.
Example 3: Observational data on a test
A survey of 400 adults finds that those who exercise daily have an average resting heart rate of 64 beats per minute, while the others average 72. What is the difference, and which conclusion does the survey support?
- Difference: 72 − 64 = 8 beats per minute.
- No one was assigned to exercise; people chose. This is an observational study.
- Daily exercisers may also differ in diet, age or health, which are possible lurking variables.
- So the survey supports an association between daily exercise and lower heart rate, not proof that exercise caused it.
Answer: The difference is 8 beats per minute; the survey shows an association, not causation.
Common mistakes
When two counts both grow with size, they correlate even if neither affects the other.
The correlation coefficient measures how closely points follow a line. It says nothing about which variable, if either, is the cause.
Check for a third variable before drawing a causal arrow between the two you measured.
Not to be confused with
Correlation: two variables move together, as with ice cream sales and sunburns. Causation: one actually makes the other change, as with sunlight and plant growth.
An association is any pattern linking two variables in data, linear or not. Causation is a claim about why the pattern exists, and it needs stronger evidence.
Quick check: 3 questions
Which study can best show that a new fertilizer causes taller corn?
Answer: C. Random assignment spreads soil, sunlight and water differences evenly, so a height difference can be credited to the fertilizer. A farmer survey is observational, two years differ in weather, and sales versus prices measures a different relationship entirely.Show answer and explanation
Across many cities, the number of firefighters at a fire is positively correlated with the amount of damage. What best explains this?
Answer: B. Bigger fires bring more firefighters and do more damage. Fire size is the lurking variable. Choice A reads the correlation as a cause, C gets the direction wrong, and D ignores that real data can correlate for a reason.Show answer and explanation
In an experiment, 18 of 60 randomly assigned students using flashcards failed a quiz, compared with 30 of 60 who did not. How many percentage points lower was the failure rate with flashcards?
Answer: D. 1860 = 30% and 3060 = 50%, so the difference is 20 percentage points. 12 subtracted the counts, and 30 and 50 are the two separate rates, not their difference.Show answer and explanation
Causation: frequently asked questions
What does correlation does not imply causation mean?
It means a pattern in data is not enough to prove that one thing causes another. Two variables can rise together because a third variable drives both, because the cause runs the other way, or by chance. To show causation, researchers run an experiment that randomly assigns the treatment, so the groups differ only in the variable being tested.
What is a lurking variable?
A lurking variable is a variable that was not measured or considered but affects both variables in a study, making them look related. Age links shoe size and reading level in children. Hot weather links ice cream sales and sunburns. Lurking variables are the main reason an observational study cannot prove cause and effect.
Can an observational study ever suggest causation?
It can suggest it, but not prove it alone. When many different observational studies agree, the association is strong, it follows the expected time order, and there is a sensible mechanism, scientists may treat causation as likely. Smoking and lung cancer is a classic case. For a single study on a test, only a randomized experiment supports a causal conclusion.
Practice and tools
Sources
- Common Core State Standards, High School Statistics: Interpreting Categorical and Quantitative Data (HSS-ID.C.9)
- OpenStax, Introductory Statistics 2e, 1.4 Experimental Design and Ethics
- OpenStax, Introductory Statistics 2e, 12.3 The Regression Equation
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSS-ID.C.9
Cite this entry
Effortless Math Team. “Causation.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/causation/
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