Causation Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Causation.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
Causation exists when one variable directly produces or influences a change in another variable — distinct from mere correlation or association.
causes means changing will change . Not just 'they move together.'
Read the full concept explanation →How to Use These Examples
- Read the first worked example with the solution open so the structure is clear.
- Try the practice problems before revealing each solution.
- Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea: Causation means altering one variable directly produces a change in another, beyond just moving together.
Common stuck point: The procedure for causation is the easy part; the trap is reading correlation as causation. Asking "Would deliberately changing X reliably change Y, not just co-occur with it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint: Ask: Would deliberately changing X reliably change Y, not just co-occur with it?
Worked Examples
Example 1
mediumIce cream sales and drowning deaths are positively correlated. Explain why correlation does not imply causation here, identifying the confounding variable.
Answer
Confounding variable is temperature. Ice cream and drowning are both caused by hot weather, not by each other.
First step
1
Observed correlation: more ice cream sold → more drowning deaths
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Example 2
hardA study finds students who eat breakfast score higher on tests. Design an argument for why this might not be causal, and describe how you would establish causation.
Example 3
mediumCite-and-explain Bradford Hill criterion: dose-response. Why does it strengthen causal claims?
Example 4
hardIn a DAG , , , can we estimate 's causal effect on from observational data?
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easyCountries with more TV sets per capita have higher life expectancy. Does this mean buying TVs causes longer lives? Identify the likely confounding variable.
Example 2
hardA pharmaceutical company's observational study finds Drug X correlates with recovery. Explain three criteria needed to claim Drug X causes recovery, and why a randomized trial is required.
Example 3
easyIce cream sales and drownings both rise in summer. What is the likely confounder making them correlate?
Example 4
easyDoes a strong correlation between X and Y prove that X causes Y?
Example 5
easyCausation means changing X will change Y. If forcing X higher leaves Y unchanged, is X a cause of Y?
Example 6
easyTo establish causation reliably, which study design is strongest: an observational survey or a randomized controlled experiment?
Example 7
easyRoosters crow before sunrise every day. Do roosters cause the sun to rise?
Example 8
easyA study finds students who sleep more get higher grades. Could the causal direction be reversed?
Example 9
easyTrue or false: a confounding variable can create a correlation between two things that have no direct causal link.
Example 10
easyCities with more firefighters have more fire damage. Does adding firefighters cause more damage?
Example 11
mediumCountries with more Nobel laureates also consume more chocolate. Name a plausible confounder and explain why this is not causation.
Example 12
mediumA drug trial randomly assigns patients to drug or placebo; the drug group improves more. Why does randomization let us infer causation?
Example 13
mediumAn observational study shows coffee drinkers have higher heart-disease rates, but coffee drinkers also smoke more. How does smoking threaten the causal claim?
Example 14
mediumTwo variables are statistically dependent. List the three causal explanations besides 'X causes Y'.
Example 15
mediumA town adds bike lanes and accidents drop the next year. Why might this not prove the lanes caused the drop?
Example 16
mediumSunscreen use correlates with higher skin-cancer rates in some data. Give a confounder that explains this without sunscreen causing cancer.
Example 17
mediumWhy is 'X precedes Y in time' necessary but not sufficient for 'X causes Y'?
Example 18
mediumA city notices that months with more police on patrol also have more reported crimes. Why might this not mean police cause crime?
Example 19
mediumResearchers want to know if a tutoring program raises test scores. Why is comparing volunteers to non-volunteers weaker than randomly assigning students?
Example 20
challengeA study reports that people who take vitamins are healthier. Identify the most likely confounder and explain the 'healthy-user' bias.
Example 21
challengeData show that hospitals with the best surgeons have higher patient death rates. Explain how confounding by case severity resolves this paradox.
Example 22
challengeA headline claims 'students who own more books score higher, so buy your child books.' Critique the causal leap and propose how to test causation properly.
Example 23
easyTrue or false: a correlation of between two variables proves one causes the other.
Example 24
easyShoe size and reading ability are strongly correlated in elementary students. What is the obvious confounder?
Example 25
easyChildren who attend preschool tend to do better in college. Why is this not direct evidence of causation?
Example 26
easyDoes experimental manipulation of X (forcing a value), all else random, allow causal inference?
Example 27
easyA study shows people who exercise are happier. Could the causal arrow go the other way?
Example 28
easyName three alternative explanations to 'X causes Y' for an observed correlation.
Example 29
mediumA pharmaceutical RCT shows the drug group recovers faster than placebo. Is causal inference justified?
Example 30
mediumA city's parks correlate with high property values. Explain confounding and reverse-causation candidates.
Example 31
mediumWhat is a placebo effect, and why does it require blinding for fair causal comparison?
Example 32
mediumWhy does observational data from 'natural experiments' (e.g., policy changes) sometimes support causal claims?
Example 33
mediumA study tracks smokers vs non-smokers over 30 years. Smokers die younger. Is this causal?
Example 34
mediumHouseholds with two cars have higher incomes. Does owning two cars cause higher income?
Example 35
mediumWhy is blinding ineffective for surgical interventions, and how do researchers handle it?
Example 36
hardSimpson's paradox: an effect reverses when groups are combined. Why does this matter for causation?
Example 37
hardAn instrumental variable Z affects Y only through X. Why does this allow causal inference?
Example 38
hardTwo papers report opposite causal conclusions on the same exposure. What three explanations should you consider before believing either?
Example 39
hardWhat is a directed acyclic graph (DAG) used for in causal inference?
Example 40
hardWhy is it wrong to control for a variable that lies on the causal path from X to Y?
Example 41
challengeBerkson's paradox: a hospital-based study finds two diseases negatively correlated, despite being unrelated in the population. Explain.
Example 42
challengeCritique the headline: 'Eating chocolate makes you smarter — Nobel laureates eat more chocolate per capita.'
Example 43
challengeDesign an experiment to test whether music improves test scores, listing randomization, controls, and the outcome measure.
Related Concepts
Background Knowledge
These ideas may be useful before you work through the harder examples.
correlationdependence