Exercises: Distinguish Correlation from Causation
Work through each section in order. For each scenario, separate two
different claims: a CORRELATION claim ("the two variables move together")
from a CAUSATION claim ("one variable produces a change in the other").
When a correlation is not causal, name the most plausible alternative
explanation - reverse causation, a lurking (common) variable, or
coincidence - and identify the lurking variable when one exists. Write
explanations in complete sentences.
Warm-Up: Two Different Claims
These problems review the definitions of correlation and causation.
Which statement best defines a correlation between two variables?
The two variables tend to move together (as one changes, the other tends to change too).
A change in one variable directly produces a change in the other.
The two variables are completely unrelated.
One variable is the hidden cause of the other.
Which statement best defines causation between two variables, and how does it differ from correlation?
Causation means a change in one variable produces a change in the other; correlation only means they move together.
Causation and correlation mean the same thing.
Causation means the variables move together; correlation means one produces the other.
Causation means the correlation is weak, while correlation means it is strong.
A correlation is observed between two variables A and B. Which of the following is not one of the standard explanations for that correlation?
A and B happen to have the same units of measurement.
A causes B (direct causation).
A lurking variable causes both A and B.
The correlation is a coincidence with no real link.
What is a lurking (confounding) variable?
A third variable, not among the two being studied, that influences both of them and can create a correlation between them.
The stronger of the two variables being studied.
A variable that has no effect on anything.
Another name for the correlation coefficient r.
Fluency: Classify the Claim
Decide whether each statement is a correlation claim or a causation claim, and apply the definitions.
A study finds that, among children, shoe size and reading ability have a strong positive correlation. A student concludes, "So having bigger feet makes children read better." What is wrong with this conclusion?
The student jumped from a correlation to a causation claim; a strong correlation does not prove one variable causes the other.
The correlation must be wrong, because bigger feet obviously cannot help reading.
Nothing is wrong; a strong correlation proves causation.
The student should have measured weight instead of shoe size.
Classify the statement: "Students who exercise more tend to have higher fitness levels." Read only as written, is this primarily a correlation claim or a causation claim?
A correlation claim - it says the two variables tend to move together.
A causation claim - it says exercise produces the fitness.
Neither - it is just an opinion.
Both claims are stated explicitly and equally.
A meteorologist notes that ice-cream sales are strongly correlated with the number of drownings at the beach each week. She uses ice-cream sales to help predict busy weeks for lifeguards. Is this a legitimate use of the correlation, even though ice cream does not cause drownings?
Yes - a correlation can be used for prediction even when it is not causal; both track summer heat, so high ice-cream sales flag high-risk weeks.
No - if a correlation is not causal it is worthless and cannot be used for anything.
Yes, because this proves ice-cream sales cause drownings.
No, because correlations can never be used to predict.
Researchers find that stress levels and poor sleep are correlated. A student insists, "This shows stress causes poor sleep." Why can the student not be sure of the direction of causation from the correlation alone?
A correlation is symmetric and carries no direction; poor sleep could just as plausibly cause stress (reverse causation).
Because the variable named first always causes the one named second.
Because stress and sleep cannot possibly be related.
Because the correlation must be a coincidence.
For each correlation, decide whether a causal claim is justified from the description alone. "A randomized experiment found that students given a new study guide scored higher than a randomly assigned control group." Is a causal claim justified here?
Yes - because subjects were randomly assigned to treatment and control, lurking variables are balanced, so the higher scores can be attributed to the study guide.
No - no study can ever justify a causal claim.
No - this is observational, so only correlation is shown.
Yes - but only because the groups were not randomized.
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