Can there be causation without a strong correlation?

Yes. A cause can be hidden by other variables, or the relationship may not be a straight line, so $r$ can be weak. For example, a medicine might help only some patients, so the overall correlation looks weak even though the medicine really works for them.

What is the difference between an observational study and an experiment?

In an observational study, researchers only record what people already do (who drinks coffee, who smokes). In an experiment, researchers decide who gets the treatment, ideally at random. Only an experiment controls the confounding variables, because random groups are alike in everything except the treatment.

Can we ever claim causation without an experiment?

Sometimes experiments are impossible or unethical (you cannot make people smoke for 20 years). Then scientists collect many kinds of evidence: a strong correlation that appears in many studies, the cause comes before the effect, more of the cause gives more of the effect, and there is a known biological mechanism. This is how smoking was shown to cause lung cancer.

Teacher Resources

Correlation vs Causation

Learn why two variables that rise and fall together do not always cause each other, and how scientists actually prove cause and effect.

Intermediate25 minLesson

Printable worksheet

Definition

Correlation means that two variables tend to change together: when one goes up, the other tends to go up (positive correlation) or down (negative correlation). Causation means that a change in one variable actually *produces* a change in the other.

\text{correlation} \neq \text{causation}

Even a very strong correlation, such as

r = 0.95

, does not prove that

x

causes

y

. When two variables are correlated, there are four possible explanations:

1. $x$ causes $y$ ** (causation): more hours of practice lead to a better free-throw percentage.
2. $y$ causes $x$ ** (reverse causation): the cause runs the other way around.
3.A third variable causes both (a confounding variable, also called a lurking variable): hot weather raises both ice cream sales and sunburn cases.
4.Coincidence (a spurious correlation): the pattern appeared by chance.

A correlation alone cannot tell these four apart. To show cause and effect, scientists use a randomized controlled experiment: they split the subjects into a treatment group and a control group by chance, change only one thing, and compare the results. To compare groups of different sizes, compare their rates:

\text{rate} = \frac{\text{number of cases}}{\text{group size}} \times 100\%

Try it now

In a beach town, ice cream sales and sunburn cases have a very strong positive correlation. Which explanation is most likely?

Worked Examples

A seaside town records ice cream sales and sunburn cases for 8 summer weeks (the numbers are made up for this example). From the coolest week to the hottest, the average temperature was

15

18

20

22

24

27

29

and

31

°C. In the same weeks, ice cream sales were

4

6

7

9

10

13

14

and

16

hundred cones, and there were

5

8

9

12

14

18

19

and

23

sunburn cases. The correlation between ice cream sales and sunburn cases is very strong (

r \approx 0.998

). Does eating ice cream cause sunburn?

Describe the correlation

As ice cream sales rise from

4

16

hundred cones, sunburn cases rise from

5

23

→ Strong positive correlation

Ask whether there is a mechanism

Ice cream cannot burn skin, and a sunburn does not make you buy ice cream → No direct cause in either direction

Look for a third variable

Temperature rises every week from

15

°C to

31

°C. Hot, sunny weeks bring more people to the beach, who buy more ice cream and spend more time in the sun → Temperature drives both variables

Name the explanation

Temperature

\rightarrow

ice cream sales and temperature

\rightarrow

sunburn cases → Temperature is a confounding variable

Answer: No. The correlation is real, but temperature is a confounding variable that raises both ice cream sales and sunburn cases. Banning ice cream would not prevent a single sunburn.

Common Mistakes

Thinking a strong correlation proves causation (" $r = 0.95$ , so $x$ must cause $y$ ")

Why it's wrong: The correlation coefficient only measures how closely two variables move together. In the seaside town example, ice cream sales and sunburn cases have $r \approx 0.998$ , and neither causes the other.

Correct: A strong $r$ tells you the variables are related. To claim causation you need a randomized controlled experiment or very strong additional evidence.

Assuming the cause runs in the direction you expect

Why it's wrong: Two correlated variables can be connected in either direction. People who take more painkillers report more headaches, but the headaches cause the painkillers, not the other way around.

Correct: Always ask: could $y$ be causing $x$ ? Check which variable can change first and which one reacts.

Forgetting about confounding variables that are not in the data

Why it's wrong: A lurking variable is often not measured at all. In the coffee study the overall numbers hide smoking, which is the real cause.

Correct: Ask what else differs between the groups (age, weather, income, population, health). If possible, compare the groups within the same level of that variable.

Comparing counts instead of rates

Why it's wrong: A big city has more accidents, more libraries and more ice cream shops simply because it has more people. Population is a hidden third variable.

Correct: Divide by the group size and compare rates, such as cases per $100$ people (percent) or per $1000$ residents.

Interactive Visual

Scatter Plot

Ice cream sales vs sunburn cases (made-up data, 8 summer weeks, labels = average temperature)

ŷ = 1.47x − 1.01

r = 0.998 · strong positive correlation

Tap a point to see its coordinates.

Scatter Plot

The confounding variable: temperature vs ice cream sales

ŷ = 0.75x − 7.63

r = 0.997 · strong positive correlation

Tap a point to see its coordinates.

Bar Chart

Randomized experiment: share of each group that caught a cold

Vitamin(9%)

Placebo(15%)

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

18 problems

Problem 1 of 18

Easy

In a beach town, ice cream sales and sunburn cases have a very strong positive correlation. Which explanation is most likely?

Practice Problems: Correlation vs Causation

1. In a beach town, ice cream sales and sunburn cases have a very strong positive correlation. Which explanation is most likely?
2. In a randomized experiment, 14 of the 200 people in the treatment group got sick. What percent of the treatment group got sick? (Enter the number only.)
3. To test a new fertilizer, 100 plants are split at random into two groups of 50. In the fertilizer group, 40 plants grow taller than 30 cm; in the control group, 25 plants do. By how many percentage points is the rate higher in the fertilizer group?
4. Which kind of study can show that one variable causes another?
5. In the placebo group of a trial, 40 of 250 patients recovered without medicine. What percent of the placebo group is that? (Enter the number only.)
6. Town A has many bike lanes, 50 000 residents and 100 bike accidents per year. Town B has few bike lanes, 20 000 residents and 60 bike accidents per year. A headline says: "Bike lanes cause accidents!" How many fewer bike accidents per 1000 residents does Town A have?
7. People who take painkillers more often report more headaches. Which explanation is most likely?
8. A researcher tests 300 pairs of unrelated variables. Each pair has a 5% chance of looking strongly correlated just by luck. How many strong-looking correlations should she expect?
9. To test whether music helps studying, 24 students are split at random into two equal groups. One group studies with music, the other in silence. On the test, the music group scores 936 points in total and the silence group 864 points in total. How many points higher is the average score of the music group?
10. More firefighters are sent to bigger fires, and bigger fires cause more damage. So the number of firefighters at a fire is positively correlated with the damage. What is the confounding variable?
11. In a randomized experiment, 12% of the treatment group and 20% of the control group got the flu. By how many percentage points is the rate lower in the treatment group?
12. An online shop shows version A of its website to 1500 randomly chosen visitors and version B to another 1500. With version A, 45 visitors buy something; with version B, 75 do. By how many percentage points is the buying rate higher with version B?
13. 400 volunteers are split at random into 4 equal groups: one placebo group and three groups with different doses of a cold medicine. In the placebo group, 30 people catch a cold; in the highest-dose group, 12 people do. By how many percentage points is the cold rate lower in the highest-dose group?
14. Which statement is correct?
15. 600 volunteers are split at random into two equal groups. In the vaccine group, 27 people get the flu; in the placebo group, 45 people do. By how many percentage points is the flu rate lower in the vaccine group?
16. Hospital A has an overall death rate of 3.4% and Hospital B of 2.6%, so B looks better. But Hospital A treats many more severely ill patients. Hospital A had 800 severe patients, of whom 32 died, and 200 mild patients, of whom 2 died. Hospital B had 200 severe patients, of whom 10 died, and 800 mild patients, of whom 16 died. For severe patients, by how many percentage points is Hospital A's death rate lower than Hospital B's?
17. A lifeguard records 100 summer days. On the 50 days with high ice cream sales there were 7 swimmer rescues, and on the 50 days with low sales there were 4. Split by weather: there were 30 hot days with high sales (6 rescues) and 10 hot days with low sales (2 rescues), 20 cool days with high sales (1 rescue) and 40 cool days with low sales (2 rescues). On hot days only, what is the difference in rescues per day between high-sales days and low-sales days?
18. A researcher tests 20 foods that have no real link to a disease. For each food there is a 5% chance that the test shows a false "link" just by luck, and the tests are independent. What is the probability that at least one food shows a false link? Give it in percent, rounded to a whole number.

Why It Matters

Every day you meet claims built on correlations:

News headlines: "People who drink coffee live longer" or "Kids who eat breakfast get better grades". Is it the coffee and the breakfast, or something else about these people?
Health: A medicine is only approved after randomized trials show that it, and not chance, makes patients better.
Business: Companies test two versions of a website on randomly chosen visitors before deciding which one sells more.
Public policy: Cities decide where to spend money on schools, police or bike lanes. A wrong cause means wasted money.
Your own decisions: Knowing the difference protects you from ads and fake science that use a real correlation to sell a false cause.

If you can ask "Could it be the other way around? Is there a third variable? Could it be chance?", you think like a scientist.

Real World Applications

Medical Trials

New medicines are tested in randomized controlled trials. Patients are assigned to the medicine or to a placebo by chance, so that the medicine is the only systematic difference between the groups.

Example:

In a famous type of trial, neither the patients nor the doctors know who gets the real medicine (a double-blind trial). This stops expectations from changing the results.

1Try It Yourself

A headache medicine is tested on $500$ patients, randomly split into two groups of $250$ . In the medicine group, $20$ patients still have frequent headaches after a month. In the placebo group, $45$ patients do.

By how many percentage points is the headache rate lower in the medicine group?

Step 1: Write the mathematical expression

Compute $\frac{45}{250} \times 100 - \frac{20}{250} \times 100$ :

Judging News Headlines

Many health and lifestyle headlines are based on observational studies, where researchers only watch what people already do. Such studies can find correlations but cannot prove causes.

Example:

"Students who play a musical instrument get better grades." Families that pay for music lessons may also have more time and money for school support. That is a possible confounding variable.

2Try It Yourself

A journalist checks $60$ health headlines. $45$ of them are based only on observational studies. The others are based on randomized experiments.

What percent of the headlines are based on randomized experiments?

Step 1: Write the mathematical expression

Compute $\frac{60 - 45}{60} \times 100$ :

City Statistics

City data is full of correlations caused by population: bigger cities have more of almost everything. Planners compare rates per $100\,000$ residents to remove this confounding variable.

Example:

Across many cities, the number of libraries is strongly correlated with the number of traffic accidents. Libraries do not cause accidents: both grow with the number of people living in the city.

3Try It Yourself

City A has $400\,000$ residents, $20$ libraries and $1200$ traffic accidents per year. City B has $100\,000$ residents, $5$ libraries and $300$ traffic accidents per year.

How many traffic accidents per $100\,000$ residents does City A have?

Step 1: Write the mathematical expression

Compute $1200 \div \frac{400\,000}{100\,000}$ :

Key Takeaways

1Correlation means two variables change together; causation means one actually produces a change in the other
2A strong correlation (even $r$ close to $1$ or $-1$ ) never proves causation on its own
3A correlation can come from causation, reverse causation, a confounding (lurking) variable or coincidence
4A confounding variable affects both variables, like temperature for ice cream sales and sunburn cases
5Compare rates, not counts: $\text{rate} = \frac{\text{cases}}{\text{group size}} \times 100\%$
6A randomized controlled experiment, with random assignment to a treatment and a control group, is the best way to show cause and effect

Frequently Asked Questions

Correlation is still very useful. It tells us which variables are worth studying and it lets us make predictions: a strong correlation between temperature and ice cream sales helps a shop plan its stock, even though the shop does not need to know the cause. Correlation is the first clue, an experiment is the proof.

Glossary

Causation: A relationship in which a change in one variable actually produces a change in another variable
Confounding variable (lurking variable): A third variable that affects both variables being studied and can create a correlation between them
Reverse causation: When the variable thought to be the effect is really the cause
Spurious correlation: A correlation that appears by coincidence or because of a hidden variable, not because of a real cause
Randomized controlled experiment: A study in which subjects are assigned to a treatment group or a control group by chance, so the groups differ only in the treatment
Placebo: A pill or treatment with no active ingredient, given to the control group so that both groups have the same experience

Formula Card

Rate in a group

\text{rate} = \frac{\text{number of cases}}{\text{group size}} \times 100\%

Turns counts into percents so groups of different sizes can be compared

Difference of two rates

\text{difference} = \text{rate}_{\text{control}} - \text{rate}_{\text{treatment}}

Measured in percentage points; in a randomized experiment it estimates the effect of the treatment

Expected chance correlations

\text{expected} = \text{number of pairs tested} \times \text{chance per pair}

How many strong-looking correlations appear by pure luck when many unrelated pairs are tested

Today's challengeOne problem, three tries

Correlation vs Causation

Definition

Worked Examples

Common Mistakes

Interactive Visual

Scatter Plot

Ice cream sales vs sunburn cases (made-up data, 8 summer weeks, labels = average temperature)

Scatter Plot

The confounding variable: temperature vs ice cream sales

Bar Chart

Randomized experiment: share of each group that caught a cold

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Medical Trials

Judging News Headlines

City Statistics

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Rate in a group

Difference of two rates

Expected chance correlations

More in This Topic

Introduction to Correlation

Correlation Coefficient

Interpreting Correlation

Correlation Word Problems