CMH MATH PATH — LESSON 3: CORRELATION VS. CAUSATION

The Most Important — and Most Dangerous — Idea in Data Science

PART 1: THE CORE CONCEPT — LINKED VS. CAUSED

The Ice Cream & Drowning Mystery

Here’s a strange fact that is 100% statistically true in many countries:

When ice cream sales go UP, drowning deaths go UP.
When ice cream sales go DOWN, drowning deaths go DOWN.

They are strongly correlated. They move together perfectly.

The Dangerous Conclusion: “Ice cream causes drowning! Ban ice cream to save lives!”

The Real Reason: SUMMER.

  • Summer = Hot weather = More people buy ice cream
  • Summer = Hot weather = More people swim = More drowning accidents

Ice cream and drowning are linked by a common cause (summer), but one does not cause the other.

This is the difference in one sentence:

  • Correlation = Two things happen together
  • Causation = One thing makes the other happen

PART 2: THE FOUR TYPES OF “LINKED” RELATIONSHIPS

When two things are correlated, there are only four possible explanations:

Type 1: Direct Causation (A → B)

“Studying more hours → Better test scores”

This is true causation. Changing A directly changes B.

Evidence needed: Controlled experiments (like we’ll discuss later).

Type 2: Reverse Causation (B → A)

“More fire trucks at a fire → Bigger fire”

Does sending more trucks make fires worse? No.
Bigger fires → More trucks sent.
We confused cause and effect!

CMH Example: “Unrest causes police presence” vs. “Police presence causes unrest” — which way does it go?

Type 3: Common Cause (C → A and C → B)

“Ice cream sales ← Summer → Drowning deaths”

The ice cream/drowning example. A third factor causes both.

More CMH Examples:

  • “Shoe size ← Age → Reading ability” (Children: older = bigger feet = read better)
  • “Stork sightings ← Rural areas → Birth rates” (Old myth: storks bring babies!)

Type 4: Pure Coincidence

“Cheese consumption → People dying by becoming tangled in bedsheets”

This is a real correlation found in data! But it’s complete nonsense coincidence.

The lesson: With enough data, you’ll find random patterns that mean nothing.

PART 3: VISUALIZING CORRELATION — THE SCATTER PLOT

The best way to see correlation is with a scatter plot:

Example: Education vs. Income

Income ($)
   ↑
100K |               •  •
     |           •      •
 75K |       •      •
     |     •    •
 50K |   •  •
     | • •
 25K |•
     +--------------------→ Years of Education
       10   12   14   16

What we see: Dots generally go ↗ (up and right)

  • More education → Higher income
  • This is positive correlation

The Three Correlation Patterns:

  1. Positive Correlation (↗):
  • “More X → More Y”
  • Examples: Study hours → Grades; Exercise → Health
  1. Negative Correlation (↘):
  • “More X → Less Y”
  • Examples: Screen time → Sleep hours; Stress → Immune function
  1. No Correlation (• scattered everywhere):
  • “X tells us nothing about Y”
  • Examples: Shoe size → Intelligence; Pizza consumption → Musical talent

Correlation Coefficient (r): A number from -1 to +1:

  • r = +1.0: Perfect positive correlation (all dots on straight line ↗)
  • r = +0.7: Strong positive correlation
  • r = 0: No correlation
  • r = -0.7: Strong negative correlation
  • r = -1.0: Perfect negative correlation (all dots on straight line ↘)

PART 4: THE GOLD STANDARD — HOW WE PROVE CAUSATION

The “Perfect Experiment”: Randomized Controlled Trial (RCT)

The Problem: Does a new teaching method cause better learning?

Wrong approach: Compare School A (uses method) vs. School B (doesn’t)

  • What if School A has richer students? Better teachers? Newer books?
  • Too many differences! Can’t isolate the teaching method.

Right approach (RCT):

  1. Take one school, randomly split students into two groups
  2. Group 1 gets new teaching method (treatment group)
  3. Group 2 gets normal teaching (control group)
  4. Compare results

Why random? It balances EVERYTHING else:

  • Randomness means equal mix of smart/average students in both groups
  • Equal mix of rich/poor, motivated/unmotivated, etc.
  • The only difference is the teaching method

Result: If Group 1 scores higher → Teaching method causes better learning.

Why We Can’t Always Do RCTs in CMH

Ethical/Impossible Experiments:

  • “Randomly assign countries to have revolutions”
  • “Randomly make some people poor to study poverty effects”
  • “Randomly start wars to study conflict”

So CMH uses Natural Experiments & Advanced Methods:

  1. Natural Experiments: Find situations where something “random” happened
  • Example: A policy change in one state but not neighbor state
  • Compare them as if it were an experiment
  1. Longitudinal Studies: Track same people for years
  • Do educated people live longer? Or do healthy people stay in school longer?
  • Track from childhood to old age to see which comes first
  1. Instrumental Variables: Find a “natural randomizer”
  • Example: Does military service cause higher mortality?
  • Use draft lottery numbers (random!) as instrument

PART 5: CMH CASE STUDY — EDUCATION VS. LONGEVITY

The Strong Correlation:

  • People with more education live longer
  • College graduates outlive high school dropouts by ~6 years
  • r ≈ +0.5 (strong positive correlation)

The Million-Dollar Question:

Does education cause longer life? Or are they just linked?

Possible Explanations:

1. Direct Causation (Education → Longevity):

  • Education teaches health knowledge
  • Education leads to better jobs with healthcare
  • Education reduces stress through better coping

2. Reverse Causation (Health → Education):

  • Healthy children miss less school → Learn more
  • Childhood health problems → Less education

3. Common Cause (Wealth/Privilege):

  • Wealthy families → Can afford college + better healthcare
  • It’s not education, it’s the privilege behind it

4. Genetics/Personality:

  • Conscientious people → Stay in school + Take care of health
  • It’s personality, not education

What Research Found (Using Natural Experiments):

  • When states raised compulsory education age, that generation lived longer
  • Twin studies: More educated twin outlives less educated twin
  • Conclusion: Education does cause longer life, but maybe only 2-3 extra years (not 6)

CMH Insight: Even when causation exists, the size matters! And common causes explain much of the correlation.

PART 6: HANDS-ON CORRELATION LAB

Exercise 1: Spot the Spurious Correlation

Go to https://www.tylervigen.com/spurious-correlations (famous site!)

Find 3 hilarious correlations like:

  • “Per capita cheese consumption → People dying by becoming tangled in bedsheets”
  • “Number of people who drowned by falling in pool → Films Nicolas Cage appeared in”

For each:

  1. What’s the correlation?
  2. What’s the likely common cause or coincidence?
  3. Why do people believe it might be causal?

Exercise 2: Build Your Own Correlation

In spreadsheet:

Column A (X): =RAND()*100  (Random numbers 0-100)
Column B (Y): =A1*2 + RAND()*20  (Y depends on X, plus noise)
Column C (Z): =RAND()*100  (Completely unrelated)

Make scatter plots:

  • X vs. Y → Should show clear positive correlation
  • X vs. Z → Should show no pattern

Calculate correlation:

=CORREL(A1:A100, B1:B100)  (Should be high, ~0.9)
=CORREL(A1:A100, C1:C100)  (Should be low, near 0)

Exercise 3: The “Third Variable” Game

Scenario: “Countries with more televisions per person have higher life expectancy.”

  • r = +0.85 (very strong!)

Possible third variables (C) that cause both:

  • Wealth (rich countries buy TVs + have healthcare)
  • Education (educated people buy TVs + understand health)
  • Infrastructure (electricity for TVs + hospitals)

Your task: List 3 more possible C variables for:

  1. “More police → More crime”
  2. “More books at home → Better test scores”
  3. “Facebook usage → Depression”

PART 7: YOUR CMH CORRELATION DECODER KIT

When You See a Correlation, Ask These 5 Questions:

  1. Could it be reversed? (B → A instead of A → B)
  2. What’s the hidden C? (Common cause)
  3. Is it just coincidence? (With enough data, random patterns appear)
  4. How strong is r? (0.1 = weak, 0.5 = moderate, 0.8 = strong)
  5. Is there a mechanism? (How could A possibly cause B?)

Red Flags for Spurious Correlations:

🚩 The “Texas Sharpshooter” Fallacy:

  • Shooting at barn, then drawing target around bullet holes
  • Finding pattern first, then explaining it

🚩 Small Sample Size:

  • “3 of my friends got sick after vaccine” (n=3 means nothing)
  • Need large, representative samples

🚩 P-Hacking/Data Dredging:

  • Testing 1000 correlations, finding 50 “significant” by chance
  • Publishing only the “interesting” ones

🚩 Strange Units/Comparisons:

  • “Per capita” vs. “total” confusions
  • Comparing unrelated metrics

PART 8: ADVANCED CMH APPLICATIONS

Causal Inference in Social Systems

The Fundamental Problem: We can’t run society-sized experiments.

Modern Solutions:

  1. Difference-in-Differences:
  • Compare changes in treatment group vs. control group
  • Example: Minimum wage hike in one state vs. neighbor state
  1. Regression Discontinuity:
  • Exploit sharp cutoffs (like age 18 for voting)
  • Compare people just below vs. just above cutoff
  1. Causal Diagrams (DAGs):
  • Map all possible relationships
  • Identify what to control for

Example: Does unemployment cause crime?

       Education
         ↓    ↘
     Employment → Crime
         ↑    ↖
   Neighborhood

To find if unemployment → crime, we must control for education and neighborhood (common causes).

The Replication Crisis in Social Science

Scary fact: 50-80% of published social science findings don’t replicate!

  • Often due to confusing correlation with causation
  • P-hacking, small samples, publication bias

CMH Solution: Pre-registration, open data, replication studies.

PART 9: YOUR MATH HOMEWORK

Exercise 1: The Chocolate & Nobel Prize Puzzle

Real study found: Countries that eat more chocolate win more Nobel Prizes.

  1. Plot hypothetical data: Chocolate consumption (kg/person) vs. Nobel prizes per 10M people
  2. What are AT LEAST 3 possible explanations (direct, reverse, common cause)?
  3. Design a “natural experiment” to test causality

Exercise 2: Build a Confounder

In spreadsheet, simulate:

Wealth = RAND()*100
Education = Wealth*0.7 + RAND()*30
Health = Wealth*0.5 + Education*0.3 + RAND()*20

Calculate:

  1. Correlation Education-Health (will be high)
  2. Correlation Wealth-Health (will be high)
  3. Now “control for wealth” → Calculate correlation within wealth groups

Insight: The Education-Health correlation shrinks when controlling for wealth!

Exercise 3: CMH Policy Analysis

You’re advising a government with these correlated findings:

  • “Cities with more police have higher crime rates”
  • “Countries with more social media have more protest”
  • “Schools with more funding have lower test scores”

For each:

  1. Why might this correlation be misleading?
  2. What alternative explanations exist?
  3. What evidence would prove causation?
  4. What policy would you recommend?

Exercise 4: The “Before-After” Fallacy

Scenario: “After new mayor took office, unemployment dropped 2%.”

  • Conclusion: “Mayor caused unemployment drop!”

Problems:

  1. Unemployment was already dropping
  2. National economy improved
  3. Seasonal variation

Your task: Design a better analysis using:

  1. Control city comparison
  2. Time series analysis
  3. Multiple data sources

KEY TAKEAWAYS:

  1. Correlation ≠ Causation (most important rule in statistics)
  2. Four possible relationships: Direct, Reverse, Common Cause, Coincidence
  3. Gold standard: Randomized Controlled Trials (when possible)
  4. CMH reality: Use natural experiments, longitudinal studies, advanced methods
  5. Always ask: “What’s the third variable? Could it be reversed?”

The CMH Mantra:

“Just because two things dance together doesn’t mean one leads.
Look for the hidden orchestra conductor (common cause),
check who’s really leading (direction),
and remember sometimes dancers just bump into each other (coincidence).
In social systems, the music is complex, and everyone claims to be the choreographer.”

NEXT LESSON PREVIEW:

Lesson 4: Linear Regression — Finding the Line That Best Predicts the Future

Where we’ll learn:

  • How to draw the “best fit” line through cloud of data points
  • How to make predictions: “If X changes, how much will Y change?”
  • How to measure prediction accuracy
  • Why this is the workhorse of CMH modeling

Thought to Ponder: If we can’t prove causation, is correlation still useful? Yes! Correlation tells us what to expect together, which is valuable for prediction. CMH often needs to predict even when we don’t fully understand causes. But we must never confuse predictive power with true understanding…

Remember: In CMH, we’re trying to predict and improve human societies. Confusing correlation with causation can lead to policies that waste resources or even cause harm. This lesson might save you from your most embarrassing future mistake!

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