📊 Common Statistical Distributions

 

📊 Foundational Thought Card: Common Statistical Distributions

The distinct shapes data takes—each with its own meaning, use, and personality

1. Background Context

Not all data behaves the same. Some events:

  • Are binary (yes/no)
  • Cluster around a center
  • Spread evenly
  • Happen rarely but with outsized effects

To work with data wisely, we must know what shape it follows—this is the role of distributions.

A distribution is a map of how often values occur.
It tells us what’s likely, what’s rare, and what’s normal—for this kind of data.

2. Core Concept

A probability distribution describes how values in a dataset are expected to be distributed.
Each type of distribution has its own rules, applications, and implications.

3. Common Distributions & Their Traits

Distribution

Shape & Behavior

Common Uses

🧭 Normal (Gaussian)

Bell-shaped, symmetric, most data near mean

Human height, test scores, measurement errors

⚖️ Uniform

All values equally likely (flat)

Dice rolls, random number generators

✅ Binomial

Discrete outcomes of repeated yes/no trials

Coin tosses, success/failure counts

📉 Exponential

High values rare, time until event

Time between radioactive decays, failure of devices

🧨 Poisson

Counts of rare events in fixed space/time

Number of emails per hour, mutation counts

💥 Power Law

Many small events, few very large

Earthquakes, city sizes, wealth distributions

📊 Skewed

Long tail on one side

Income, wait times, biological processes

4. Current Relevance

  • Finance: Risk often misunderstood by assuming normality (when returns may follow fat-tailed distributions)
  • Epidemiology: Disease outbreaks modeled with exponential or Poisson distributions
  • Machine Learning: Models assume data follows certain distributions; wrong assumption = bad predictions
  • Inequality Analysis: Wealth often follows power-law distribution—not symmetric or fair

5. Metaphoric Forms (Short Preview)

  • Normal = a hill
  • Uniform = a plateau
  • Binomial = a stepwise staircase
  • Poisson = a bumpy sidewalk
  • Power law = a steep cliff with a long flat plain
  • Exponential = a sudden drop
    (See full Metaphor Card next: "The Landscape of Distributions")

6. Reflective Prompts

  • What kind of distribution does my data assume?
  • Am I modeling using normal assumptions when the data is not normal?
  • Where do the rare but impactful events live in this curve?

7. Fractal & Thematic Links

  • 🎯 Probability – every distribution reflects an underlying event logic
  • 📈 Spread & Shape – distributions are the "personality" of variation
  • ⚠️ Outliers – tail behavior differs dramatically by distribution
  • 🔄 Simulation & Inference – most models simulate or fit to distributions

Use This Card To:

  • Identify the right distribution for your data
  • Understand risk, rarity, and regularity
  • Improve modeling, forecasting, and interpretation
  • See the story your data’s shape is telling you

 

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