Correlation vs Causation

Grade 11 · statistics · 100 practice problems · read aloud

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Correlation vs Causation

1. What is it and Why is it Useful?

Correlation measures the strength and direction of a relationship between two variables. Causation means one event directly causes the other. Just because two things are related does not mean one caused the other. This is a critical concept for analyzing data and avoiding false conclusions in science, economics, and everyday life.

2. How to Analyze a Relationship

Identify the Variables: What two things are being compared? (e.g., Ice cream sales and drowning incidents).
Determine Correlation: Is there a relationship? As one goes up, does the other go up (positive) or down (negative)?
Ask "Why?": Could a third, hidden factor (a confounding variable) explain the relationship? For ice cream and drowning, the confounding variable is hot weather.
Conclude: State if the relationship is likely correlational or causal. Causation requires a controlled experiment.

3. Visual Examples

Example 1: Shoe Size & Reading Score

Observation: Data shows a positive correlation: larger shoe sizes are associated with higher reading scores.

Analysis: Does a big foot make you read better? No! The confounding variable is age. As children get older, their feet grow and their reading skills improve.

Conclusion: This is correlation, not causation.

Example 2: Fertilizer & Plant Growth

Observation: A study finds that increased fertilizer use correlates with increased plant height.

Analysis: In a controlled experiment, one group gets fertilizer, another does not. All other factors (water, sunlight) are kept the same.

Conclusion: Because other variables were controlled, we can infer that the fertilizer caused the increased growth. This is causation.

4. Common Mistakes

Mistake: Assuming "correlation equals causation." This is the #1 error.

How to Avoid: Always ask: "What other factor could explain this?" Look for confounding variables.

Mistake: Ignoring the direction of a relationship. A negative correlation (e.g., more study hours, lower failure rate) is still a correlation, not necessarily direct proof of cause.

5. Tips & Tricks

Memory Aid: Remember the phrase: "Correlation does not imply causation."

Strategy: Use the "Ice Cream Test." If you can replace one variable with "ice cream sales" and the other with "drowning incidents" and the logic still seems silly, it's probably just a correlation.

6. How to Practice

Find news headlines. Do they confuse correlation for causation?
Create your own silly correlational examples (e.g., "Number of pirates vs. global temperature").
In class, always identify the variables and potential confounders in data sets.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 ∫(3x² + 2x - 1)dx from 0 to 2 = ?

Hint: Find the antiderivative of each term, then evaluate at the upper and lower limits of integration.

Show the answer

Answer: 10

Find the antiderivative** The antiderivative of 3x² is 3 * (x³/3) = x³. The antiderivative of 2x is 2 * (x²/2) = x². The antiderivative of -1 is -x. So the antiderivative F(x) = x³ + x² - x. --- **
Apply the Fundamental Theorem of Calculus** The definite integral from 0 to 2 is: F(2) - F(0) --- **
Evaluate F(2)** F(2) = (2)³ + (2)² - (2) = 8 + 4 - 2 = 10 --- **
Evaluate F(0)** F(0) = (0)³ + (0)² - (0) = 0 --- **
Subtract** F(2) - F(0) = 10 - 0 = 10 --- **Final Answer:** 10

Let's solve the definite integral step-by-step. We are given: ∫(3x² + 2x - 1) dx from 0 to 2 --- **

2 ∫(6x² - 4x + 3)dx from 0 to 2 = ?

Hint: Find the antiderivative first, then evaluate at the upper and lower limits of integration

Show the answer

Answer: 14

Find the antiderivative of 6x² - 4x + 3 Antiderivative = (6x³/3) - (4x²/2) + 3x = 2x³ - 2x² + 3x
Evaluate at the upper limit (x = 2) F(2) = 2(2)³ - 2(2)² + 3(2) = 2(8) - 2(4) + 6 = 16 - 8 + 6 = 14
Evaluate at the lower limit (x = 0) F(0) = 2(0)³ - 2(0)² + 3(0) = 0 - 0 + 0 = 0
Apply the Fundamental Theorem of Calculus ∫(6x² - 4x + 3)dx from 0 to 2 = F(2) - F(0) = 14 - 0 = 14

The answer is 14.

3 ∫(4x³ - 6x² + 2)dx from 1 to 3 = ?

Hint: Find the antiderivative first, then evaluate at the upper and lower limits

Show the answer

Answer: 32

Find the antiderivative of 4x³ - 6x² + 2 Antiderivative = (4/4)x⁴ - (6/3)x³ + 2x = x⁴ - 2x³ + 2x
Evaluate at the upper limit (x = 3) F(3) = (3)⁴ - 2(3)³ + 2(3) = 81 - 54 + 6 = 33
Evaluate at the lower limit (x = 1) F(1) = (1)⁴ - 2(1)³ + 2(1) = 1 - 2 + 2 = 1
Subtract: F(3) - F(1) = 33 - 1 = 32

The answer is 32.

4 ∫(4x³ - 6x² + 2x)dx from 0 to 2 = ?

Hint: Find the antiderivative of each term using the power rule, then evaluate at the upper and lower limits of integration.

Show the answer

Answer: 4

Find the antiderivative of 4x³ - 6x² + 2x Antiderivative of 4x³ is (4/4)x⁴ = x⁴ Antiderivative of -6x² is (-6/3)x³ = -2x³ Antiderivative of 2x is (2/2)x² = x² So the antiderivative is x⁴ - 2x³ + x²
Evaluate at the upper limit (x = 2) (2)⁴ - 2(2)³ + (2)² = 16 - 2(8) + 4 = 16 - 16 + 4 = 4
Evaluate at the lower limit (x = 0) (0)⁴ - 2(0)³ + (0)² = 0 - 0 + 0 = 0
Subtract the lower limit evaluation from the upper limit evaluation 4 - 0 = 4

The answer is 4.

5 Emma collects data on the number of hours of sunlight per day (x) and the number of ice cream cones sold (y) at her local shop over 7 days. The data yields a Pearson correlation coefficient of r = 0.95. Does this strong correlation prove that more sunlight causes more ice cream sales? Explain why or why not, and identify a possible confounding variable.

Hint: Think about what other factors might change along with sunlight hours. Consider whether there is a third variable that could influence both x and y.

Show the answer

Answer: No, correlation does not imply causation. A possible confounding variable is higher temperature, which increases both sunlight hours and ice cream demand.

The correlation coefficient r = 0.95 indicates a very strong positive linear relationship between sunlight hours and ice cream sales.
However, correlation alone does not prove causation. There could be a lurking or confounding variable.
In this context, temperature is a likely confounding variable: longer sunlight hours often coincide with warmer weather, and warmer weather increases people's desire for ice cream.
Therefore, the observed correlation may be due to the common cause of temperature, not a direct causal link from sunlight to sales.
To establish causation, a controlled experiment (e.g., randomly assigning sunlight exposure) or more advanced statistical methods (e.g., regression with temperature as a control variable) would be needed. Final answer: No, correlation does not imply causation; temperature is a plausible confounding variable.

6 Noah collects data on the number of hours studied (x) and test scores (y) for 6 students, finding a correlation coefficient of r = 0.96. The least-squares regression line is ŷ = 51 + 6x. If a student studies for 11 hours, what is the predicted test score? Then, explain why a high correlation does not imply that studying more hours causes higher test scores.

Hint: First, substitute the given value of x into the regression equation to find the predicted y. Then, think about possible confounding variables that could influence both study time and test scores, such as prior knowledge or motivation. Correlation measures association, not causation.

Show the answer

Answer: 117

Substitute x = 11 into the regression equation ŷ = 51 + 6x.
ŷ = 51 + 6(11) = 51 + 66 = 117.
The predicted test score is 117.
A high correlation (r = 0.96) indicates a strong linear relationship, but it does not prove causation. For example, students who study more may also be more motivated or have better prior knowledge, which could be the actual cause of higher test scores. Without a controlled experiment, we cannot conclude that studying more hours causes higher scores.

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