CMH MATH PATH — LESSON 2: PROBABILITY DISTRIBUTIONS

Understanding the Language of “Maybe” — From Bell Curves to Black Swans

PART 1: WHAT IS A PROBABILITY DISTRIBUTION?

The Candy Jar Visualization

Imagine you have a giant, magical candy jar that represents a whole society. The jar has 100 pieces of candy, each representing one person’s “daily happiness score” from 1-10.

If you reach in blindfolded and pull out one piece:

  • What number are you most likely to get?
  • What numbers are rare?
  • What’s the average of all candy numbers?

The probability distribution is the answer to all these questions. It’s a map of possibilities.

The Simple Case: The Perfectly Fair Jar

Let’s say the jar has exactly 10 pieces of each number (1 through 10).

Distribution Map:

Number:   1  2  3  4  5  6  7  8  9  10
Pieces:   10 10 10 10 10 10 10 10 10 10
  • Chance of drawing a “5” = 10/100 = 10%
  • Chance of drawing “8 or higher” = 30/100 = 30%
  • Average = (1+2+3+…+10)/10 = 5.5

This is a uniform distribution — every outcome is equally likely.

CMH Reality Check: Real societies are almost NEVER uniform. Some happiness levels are much more common than others.

PART 2: THE BELL CURVE (NORMAL DISTRIBUTION) — SOCIETY’S DEFAULT SHAPE

The Real Happiness Jar

In reality, our candy jar looks more like this:

Happiness: 1  2  3  4  5  6  7  8  9  10
Pieces:    2  5  10 15 20 20 15 10 5  2

Draw this! Make bar graph with happiness on bottom, pieces on side.

Notice:

  • Most pieces around 5-6 (the “average happy”)
  • Fewer pieces at extremes (very unhappy or very happy)
  • Symmetrical shape

This is the famous BELL CURVE! Also called the Normal Distribution.

Why Life Loves Bell Curves

Most things in nature and society form bell curves because of the Central Limit Theorem (from Lesson 1):

Example 1: Height of Adults

  • Genetics + nutrition + environment = final height
  • Each factor adds randomness
  • Result: Most people near average height, few very short/tall

Example 2: Test Scores

  • Intelligence + studying + luck + health
  • Result: Most near average, few geniuses, few struggling

Example 3: Daily Mood

  • Sleep + events + chemistry + weather
  • Result: Most days “okay,” few terrible, few amazing

The Mathematical Reason:
When something is caused by many small random factors adding together, it forms a bell curve. Most people’s lives have many small random factors every day!

The Three Magic Numbers of Any Bell Curve

Every bell curve can be completely described by just 3 numbers:

  1. The Mean (μ “mu”) = Center point, average
  2. The Standard Deviation (σ “sigma”) = Width, spread
  3. The Total Area = Always equals 1 (100% of population)

Visual Guide:

The 68-95-99.7 Rule (The EMPIRICAL RULE)

This is the most useful rule in all of statistics:

For any bell curve:

  • 68% of data falls within 1 standard deviation of the mean
  • 95% falls within 2 standard deviations
  • 99.7% falls within 3 standard deviations

Example: Adult Male Height in the US

  • Mean (μ) = 5’9″ (69 inches)
  • Standard Deviation (σ) = 3 inches

Then:

  • 68% of men are between 5’6″ and 6’0″ (μ ± σ)
  • 95% are between 5’3″ and 6’3″ (μ ± 2σ)
  • 99.7% are between 5’0″ and 6’6″ (μ ± 3σ)

CMH Application: If we know a society’s average “trust in government” is 6/10 with σ=1, we know immediately that 95% of people rate trust between 4 and 8!

PART 3: SKEWED DISTRIBUTIONS — WHEN LIFE ISN’T FAIR

The Lopsided Candy Jar

Not everything is symmetrical. Some jars are lopsided:

Example: Personal Wealth Jar

Wealth Level:   $0    $10k  $50k  $100k $500k $1M   $10M
People:         20    30    25    15    6     3     1

Draw this! Notice the long tail to the right.

This is a right-skewed distribution. Most people have little, few have lots.

Left-skewed is the opposite (rare but exists):

  • Example: Grades in an easy test (most get A’s, few get C’s)

Why Skewness Matters for CMH

Right-skewed distributions in society:

  • Wealth and income (Gini coefficient measures this!)
  • Social media followers
  • City sizes (many small towns, few megacities)
  • Protest sizes (many small protests, few huge ones)

Left-skewed distributions:

  • Age at death in developed countries
  • Customer satisfaction scores (if you only survey happy customers!)

The Mathematical Measure: Skewness

  • Skewness = 0 → Perfectly symmetrical (bell curve)
  • Skewness > 0 → Right-skewed (tail on right)
  • Skewness < 0 → Left-skewed (tail on left)

PART 4: HEAVY-TAILED DISTRIBUTIONS — WHERE BLACK SWANS LIVE

The Jar with Surprises

Some jars have normal-looking middles but surprisingly fat tails:

Compare:

  • Bell curve: 3σ covers 99.7% (1 in 370 outside)
  • Heavy-tailed: 3σ might cover only 95% (1 in 20 outside!)

Real Examples:

  1. Earthquake magnitudes (most are small, but big ones happen more often than bell curve predicts)
  2. Book sales (most sell few copies, but Harry Potter exists)
  3. War deaths (many small conflicts, occasional catastrophes)
  4. Market crashes (“once in a century” events that happen every decade)

CMH Critical Insight: Social systems often have heavier tails than physical systems because of networks and cascades.

The “Black Swan” Jar

Nassim Taleb’s “Black Swan” = extreme event in the heavy tail that:

  1. Is rare (in the tail)
  2. Has huge impact
  3. We pretend afterwards it was predictable (but wasn’t)

Example: Pandemic Sizes

  • Bell curve prediction: “95% of pandemics infect <10% of population”
  • Reality (heavy-tailed): Spanish Flu infected 27% of world
  • Why wrong? Diseases spread through networks, not independent events

Example: Revolution Sizes

  • Most protests: hundreds of people
  • Occasionally: millions show up (Arab Spring)
  • Why? Social contagion, network effects

PART 5: SPECIAL DISTRIBUTIONS FOR SPECIAL SITUATIONS

The Binary Distribution (Bernoulli)

Simplest possible: Only two outcomes.

Example: Will you vote yes/no in a referendum?

  • p = probability of “yes”
  • 1-p = probability of “no”

CMH Use: Any yes/no question:

  • Will there be a revolution this year? (yes/no)
  • Will the government fall? (yes/no)
  • Will economic growth exceed 3%? (yes/no)

The Poisson Distribution — Counting Rare Events

Models how many times something happens in fixed time/space.

Example: How many protests occur in a city per month?

  • Most months: 0-2 protests
  • Occasionally: 5+ protests
  • Formula: P(k events) = (λ^k × e^{-λ}) / k!
    (λ “lambda” = average rate)

Other Examples:

  • Wars per decade
  • Assassinations per century
  • Technological breakthroughs per year

The Power Law Distribution — When Size Doesn’t Matter

The ultimate heavy-tailed distribution: No “typical” size!

Formula: P(size > x) ∝ 1/x^α

Real Examples (ALL follow power laws!):

  • City populations (Zipf’s law)
  • Wealth distribution (Pareto principle: 80% of wealth owned by 20%)
  • Website visits
  • Earthquake magnitudes (Richter scale)
  • Word frequency in language (the most common word appears 2× as often as second most common, 3× as third, etc.)

CMH Insight: When you see a power law, it usually means network effects or preferential attachment (“rich get richer”).

PART 6: HANDS-ON DISTRIBUTION LAB

Exercise 1: Build a Bell Curve in Spreadsheet

Step 1: Make height data for 1,000 fake people:

Column A (Person 1-1000): =NORM.INV(RAND(), 69, 3)

This gives random heights with mean 69″, standard deviation 3″.

Step 2: Create bins (intervals):

Bins: 60, 63, 66, 69, 72, 75, 78 (increments of 3)

Step 3: Count how many in each bin (use FREQUENCY or COUNTIFS)

Step 4: Make histogram (bar chart)

Step 5: Verify 68-95-99.7 rule:

  • Calculate mean and standard deviation of your data
  • Count % within ±1σ, ±2σ, ±3σ
  • Should be close to 68%, 95%, 99.7%!

Exercise 2: Visualize Wealth Inequality

Step 1: Make wealth data (right-skewed!):

Column A: =LOGNORM.INV(RAND(), 10, 1.5)

This gives log-normal distribution (realistic for wealth)

Step 2: Sort wealth lowest to highest

Step 3: Calculate:

  • Bottom 50% own what % of total wealth?
  • Top 10% own what %?
  • Gini coefficient ≈ (area between line and equality)/(total area)

Step 4: Compare to real world:

  • US: Top 10% own ~70% of wealth
  • Sweden: Top 10% own ~55%
  • Your simulation: __

Exercise 3: Simulate “Black Swans”

Step 1: Generate protest sizes (power law):

Column A: =10^RAND()^3 * 100

This gives numbers from 100 to 1,000,000 with heavy tail

Step 2: Sort and analyze:

  • How many protests < 1,000 people?
  • How many > 100,000?
  • What % of total protesters are in the largest 1% of protests?

Step 3: Compare to bell curve:

  • Generate normal distribution with same mean
  • Notice how different the tails are!

PART 7: CMH APPLICATION GUIDE — WHICH DISTRIBUTION WHEN?

Decision Tree for CMH Practitioners

Step 1: Ask “What am I measuring?”

  • Binary outcome (yes/no) → Bernoulli
  • Count of events (how many?) → Poisson
  • Measurement with many causes → Normal (probably)
  • Size/wealth/population → Log-normal or Power Law
  • Extreme events possible → Heavy-tailed distribution

Step 2: Check your data

  1. Plot histogram
  2. Calculate skewness
  3. Look at tails (log-log plot helps)

Step 3: Choose your model

  • If normal: Use mean ± standard deviation
  • If skewed: Report median, not mean!
  • If heavy-tailed: Prepare for surprises!

Common CMH Mistakes with Distributions

Mistake 1: Assuming normality for everything

  • Wrong: “Average income = $50,000, so most people earn $40,000-$60,000”
  • Right: Income is right-skewed! Median might be $35,000 while mean is $50,000

Mistake 2: Ignoring tails

  • Wrong: “War deaths normally distributed, so 99.7% of wars kill <X”
  • Right: War deaths are heavy-tailed! WW1+WW2 killed more than all other 20th century wars combined

Mistake 3: Using wrong average

  • For normal: Mean = Median = Mode
  • For skewed: Mean ≠ Median ≠ Mode
  • Use median for skewed data (less affected by outliers)

PART 8: YOUR CMH DISTRIBUTION FIELD KIT

Quick Reference Cards

CARD 1: Normal Distribution (Bell Curve)

  • Shape: Symmetrical bell
  • Parameters: Mean (μ), Standard Deviation (σ)
  • CMH Examples: Height, IQ, daily temperature, mood
  • Rule: 68-95-99.7 rule works
  • Danger: Assumes no extreme outliers

CARD 2: Log-normal Distribution

  • Shape: Right-skewed, log-scale makes it normal
  • Parameters: Mean and SD of LOG of data
  • CMH Examples: Income, wealth, city size, company size
  • Rule: “Typical” = median, not mean
  • Danger: Mean much larger than median

CARD 3: Power Law Distribution

  • Shape: Straight line on log-log plot
  • Parameters: Exponent α
  • CMH Examples: Word frequency, website visits, earthquake sizes
  • Rule: Scale-free! No “typical” size
  • Danger: Extreme events not rare!

CARD 4: Poisson Distribution

  • Shape: Discrete, skewed (for small λ)
  • Parameter: λ = average rate
  • CMH Examples: Protests/month, wars/decade, breakthroughs/year
  • Rule: Mean = Variance
  • Danger: Assumes events independent

The CMH Distribution Checklist

Before analyzing any social data, ask:

  1. Is it binary, count, or continuous?
  2. Plot it! Histogram, box plot, QQ-plot
  3. Check symmetry: Calculate skewness
  4. Check tails: Compare to normal distribution
  5. Choose appropriate statistics:
  • Symmetric → mean ± SD
  • Skewed → median + IQR
  • Heavy-tailed → median + tail probabilities

PART 9: REAL-WORLD CMH CASE STUDIES

Case Study 1: Predicting Protest Sizes

Problem: How many people will attend next protest?

Wrong approach: Assume normal distribution

  • Mean from past protests = 5,000
  • SD = 2,000
  • Prediction: 95% chance between 1,000 and 9,000

Right approach: Power law distribution

  • Most protests: 100-1,000 people
  • Occasionally: 100,000+ (Arab Spring)
  • Better prediction: “80% chance <2,000, 1% chance >50,000”

Why it matters: Police/resources needed differ by 100×!

Case Study 2: Economic Inequality Measurement

Problem: How unequal is this society?

Wrong: Just report average income

Right:

  1. Plot income distribution (will be right-skewed)
  2. Report median income (what typical person earns)
  3. Calculate Gini coefficient (0-1 scale)
  4. Report ratio top 10%/bottom 10%

Example:

  • Country A: Mean income $50k, Median $35k, Gini 0.45
  • Country B: Mean $45k, Median $40k, Gini 0.30
  • Country B is more equal despite lower mean!

Case Study 3: Pandemic Preparedness

Problem: How many hospital beds needed?

Wrong: Plan for average pandemic (1918 flu killed 2-3% infected)

Right:

  1. Pandemic sizes follow heavy-tailed distribution
  2. Prepare for reasonable worst case (5-10% mortality)
  3. Have surge capacity for black swan events

Historical evidence:

  • Most pandemics: <1% mortality
  • Occasionally: 1918 flu (2.5%), Black Death (30-50%)
  • Not bell curve! Tail is fat.

YOUR MATH HOMEWORK:

Exercise 1: Distribution Detective

Look at these real datasets (search online):

  1. Heights of NBA players
  2. Populations of US cities
  3. Number of COVID cases per day (2020)
  4. Prices of homes in your city

For each:

  • What distribution family does it belong to?
  • Why does that distribution make sense?
  • What would be wrong with assuming it’s normal?

Exercise 2: Build Your Own Wealth Simulator

In spreadsheet:

  1. Create 100 “people” with random wealth (use log-normal)
  2. Sort poorest to richest
  3. Calculate:
  • What % of wealth do bottom 50% own?
  • What % do top 10% own?
  • What’s the Gini coefficient?
  1. Change parameters: Make society more/less equal
  2. Find parameter values that match real countries

Exercise 3: The “Black Swan” Experiment

  1. Generate 100 years of “disaster sizes”:
  • Normal distribution: mean = 1,000 affected, SD = 300
  • Power law: P(size > x) ∝ 1/x^1.5
  1. For each distribution:
  • How many years have >2,000 affected?
  • What’s the worst year?
  • Which would you rather prepare for?
  1. Insight: Why do insurance companies worry about distributions, not just averages?

Exercise 4: CMH Application Design

You’re advising a government on:

  1. Police staffing for protests
  2. Vaccine production for pandemics
  3. Pension system for aging population

For each:

  • What distribution describes the key variable?
  • What statistics should you report (mean, median, tail probabilities)?
  • What’s the biggest distribution-related risk?

KEY TAKEAWAYS:

  1. The world isn’t always bell-shaped — know your distributions!
  2. Skewness matters — the average can be misleading
  3. Tails matter most — extreme events shape history
  4. Choose statistics wisely — mean for symmetric, median for skewed
  5. Visualize first — always plot your data before analyzing

The CMH Mantra:

“Tell me your distribution, and I’ll tell you your future.
The shape of your society’s variation determines the nature of its risks.”

NEXT LESSON PREVIEW:

Lesson 3: Correlation vs. Causation — The Difference Between “Linked” and “Caused”

Where we’ll learn:

  • Why ice cream sales and drowning deaths are correlated (but one doesn’t cause the other)
  • How to spot spurious correlations in social data
  • The gold standard methods for finding real causes
  • Why this is the most important distinction in all of CMH

Thought to Ponder: If wealth follows a power law distribution (a few have most), but happiness follows a normal distribution (most are moderately happy), what does that say about money and happiness? Can a society be unequal in wealth but equal in happiness? The distributions hold the answer…

Remember: In CMH, we don’t just count things. We understand how they’re distributed. The distribution tells us what’s normal, what’s possible, and what’s catastrophic. It’s the difference between preparing for a rainy day and preparing for a flood that comes once a century but changes everything when it arrives.

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