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 min18 problems
Practice Problems

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.
correlation≠causation\text{correlation} \neq \text{causation}
Even a very strong correlation, such as r=0.95r = 0.95, does not prove that xx causes yy. When two variables are correlated, there are four possible explanations:
  1. 1.xx causes yy** (causation): more hours of practice lead to a better free-throw percentage.
  2. 2.yy causes xx** (reverse causation): the cause runs the other way around.
  3. 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. 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:
rate=number of casesgroup size×100%\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 1515, 1818, 2020, 2222, 2424, 2727, 2929 and 3131 °C. In the same weeks, ice cream sales were 44, 66, 77, 99, 1010, 1313, 1414 and 1616 hundred cones, and there were 55, 88, 99, 1212, 1414, 1818, 1919 and 2323 sunburn cases. The correlation between ice cream sales and sunburn cases is very strong (r≈0.998r \approx 0.998). Does eating ice cream cause sunburn?
1

Describe the correlation

As ice cream sales rise from 44 to 1616 hundred cones, sunburn cases rise from 55 to 2323 → Strong positive correlation
2

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
3

Look for a third variable

Temperature rises every week from 1515 °C to 3131 °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
4

Name the explanation

Temperature →\rightarrow ice cream sales and temperature →\rightarrow sunburn cases → Temperature is a confounding variable

Common Mistakes

Thinking a strong correlation proves causation ("r=0.95r = 0.95, so xx must cause yy")

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≈0.998r \approx 0.998, and neither causes the other.

Correct: A strong rr 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 yy be causing xx? 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 100100 people (percent) or per 10001000 residents.

Interactive Visual

Scatter Plot

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

2.557.51012.51517.5510152025Ice cream sold (hundreds of cones)Sunburn cases15 °C18 °C22 °C24 °C27 °C31 °C

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

15202530352.557.51012.51517.5Average temperature (°C)Ice cream sold (hundreds)

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

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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?

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 500500 patients, randomly split into two groups of 250250. In the medicine group, 2020 patients still have frequent headaches after a month. In the placebo group, 4545 patients do.

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

Step 1: Write the mathematical expression

Compute 45250×100−20250×100\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 6060 health headlines. 4545 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 60−4560×100\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 000400\,000 residents, 2020 libraries and 12001200 traffic accidents per year. City B has 100 000100\,000 residents, 55 libraries and 300300 traffic accidents per year.

How many traffic accidents per 100 000100\,000 residents does City A have?

Step 1: Write the mathematical expression

Compute 1200÷400 000100 0001200 \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 rr close to 11 or −1-1) 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: rate=casesgroup size×100%\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.
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.
Yes. A cause can be hidden by other variables, or the relationship may not be a straight line, so rr 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.
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.
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.

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

rate=number of casesgroup size×100%\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

difference=ratecontrol−ratetreatment\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

expected=number of pairs tested×chance per pair\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

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