Ice Cream and Sunburn

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 $4$ to $16$ hundred cones, sunburn cases rise from $5$ to $23$ = 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.

Police Officers and Crime

A study of 50 city districts finds a positive correlation between the number of police officers in a district and the number of crimes reported there. A politician says: "Police officers cause crime, so we should remove them." What is wrong with this reasoning?

State the correlation: Districts with more police officers have more reported crimes = Positive correlation

Test the claimed direction: Claim: more police $\rightarrow$ more crime. Is there a sensible reason for this? = Very unlikely

Try the reverse direction: More crime $\rightarrow$ the city sends more police officers to that district = Very likely

Name the explanation: The cause runs from crime to police, not from police to crime = Reverse causation

Answer: This is **reverse causation**. High crime is the reason those districts get more officers. Removing the police would not lower crime; it would probably raise it.

Correlations by Pure Chance

A computer compares $200$ pairs of completely unrelated variables (for example, the number of films released each year and the amount of cheese eaten). For any one unrelated pair, there is a $5\%$ chance that the data looks strongly correlated just by luck. How many "strong" correlations should we expect, even though none of them is real?

Write the chance as a decimal: $5\% = \frac{5}{100} = 0.05$ = $0.05$

Multiply by the number of pairs: $200 \times 0.05 = 10$ = $10$ pairs

Interpret the result: About $10$ pairs will look strongly correlated although nothing connects them = Spurious correlations

Answer: We should expect about **$10$** strong-looking correlations that are pure **coincidence**. If you search through enough data, you will always find some "amazing" correlations.

A Randomized Controlled Experiment

To test whether a vitamin prevents colds, $400$ volunteers are split **at random** into two groups of $200$. The treatment group takes the vitamin every day, the control group takes a placebo (a pill with no active ingredient) that looks the same. After three months, $18$ people in the vitamin group and $30$ people in the placebo group have caught a cold. Compare the two groups.

Find the rate in the vitamin group: $\frac{18}{200} \times 100\% = 9\%$ = $9\%$

Find the rate in the placebo group: $\frac{30}{200} \times 100\% = 15\%$ = $15\%$

Find the difference: $15\% - 9\% = 6$ percentage points = $6$ percentage points

Explain why this can show causation: Random assignment makes the groups alike in age, sleep, diet and everything else. The only systematic difference is the vitamin = The vitamin is the likely cause

Answer: The cold rate is $9\%$ with the vitamin and $15\%$ with the placebo, **$6$ percentage points lower**. Because the groups were formed at random and differ only in the vitamin, this experiment is evidence that the vitamin *causes* fewer colds.

Uncovering a Hidden Variable

An observational study of $400$ adults finds that coffee drinkers have more heart disease: $32$ of the $200$ coffee drinkers have heart disease, but only $16$ of the $200$ people who drink no coffee. But many of the coffee drinkers also smoke. Among the smokers, $30$ of $150$ coffee drinkers and $10$ of $50$ non-coffee drinkers have heart disease. Among the non-smokers, $2$ of $50$ coffee drinkers and $6$ of $150$ non-coffee drinkers have it. Does coffee cause heart disease?

Compare the overall rates: Coffee: $\frac{32}{200} \times 100\% = 16\%$. No coffee: $\frac{16}{200} \times 100\% = 8\%$ = Coffee drinkers seem to have twice the rate

Compare the smokers only: Coffee: $\frac{30}{150} \times 100\% = 20\%$. No coffee: $\frac{10}{50} \times 100\% = 20\%$ = Same rate, difference $0$

Compare the non-smokers only: Coffee: $\frac{2}{50} \times 100\% = 4\%$. No coffee: $\frac{6}{150} \times 100\% = 4\%$ = Same rate, difference $0$

Find the real cause: Smokers: $20\%$, non-smokers: $4\%$. Most coffee drinkers ($150$ of $200$) are smokers = Smoking is the confounding variable

Answer: No. Inside each smoking group, coffee drinkers and non-drinkers have **exactly the same** rate of heart disease. The overall difference ($16\%$ vs $8\%$) appears only because most coffee drinkers smoke. **Smoking** is the confounding variable.

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

Why: 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.

Mistake: Assuming the cause runs in the direction you expect

Why: 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.

Mistake: Forgetting about confounding variables that are not in the data

Why: 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.

Mistake: Comparing counts instead of rates

Why: 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.

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.

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.

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.

"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.

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.

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.

Q: If correlation does not prove causation, why do scientists measure it?

A: 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.

Q: Can there be causation without a strong correlation?

A: 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.

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

A: 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.

Q: Can we ever claim causation without an experiment?

A: 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.