Common Distributions
Learn the 3 most important probability distributions
What is a Distribution?
A distribution shows how data is spread out — which values are common, which are rare.
1. Normal Distribution (Bell Curve)
The most important one! Most real-world data follows this.
Shape: Symmetric bell — most values near the middle, fewer at extremes
Examples: Heights, test scores, blood pressure, errors
Key Rule (68-95-99.7):
| Range | % of Data |
|---|---|
| Within 1 SD of mean | 68% |
| Within 2 SD of mean | 95% |
| Within 3 SD of mean | 99.7% |
Example: If mean = 100, SD = 15
- 68% of people score between 85-115
- 95% score between 70-130
Excel: =NORM.DIST(x, mean, sd, TRUE)
2. Binomial Distribution
Use when: Counting successes in repeated Yes/No trials
Examples:
- Heads in 10 coin flips
- Defective items in 100 products
- Customers who click an ad
You need:
- n = number of trials
- p = probability of success each time
Example: Flip coin 10 times, count heads
- n = 10, p = 0.5
- Expected heads = 10 × 0.5 = 5
Excel: =BINOM.DIST(k, n, p, FALSE)
3. Poisson Distribution
Use when: Counting rare events in a fixed time/space
Examples:
- Emails per hour
- Website crashes per month
- Typos per page
You need:
- λ (lambda) = average rate
Example: Average 3 emails/hour
- λ = 3
- Can calculate P(0 emails), P(5 emails), etc.
Excel: =POISSON.DIST(k, lambda, FALSE)
Which One to Use?
| Distribution | Data Type | Example |
|---|---|---|
| Normal | Continuous measurements | Height, weight, scores |
| Binomial | Count of successes (fixed trials) | Heads in 10 flips |
| Poisson | Count of rare events | Calls per hour |
Z-Score: Comparing Apples to Oranges
Problem: How do you compare scores from different scales?
Solution: Convert to Z-score
Z = (Value - Mean) / SD
Example:
- Your exam: 80 (class mean=70, SD=10) → Z = 1.0
- Friend's exam: 150 (class mean=120, SD=20) → Z = 1.5
Friend did relatively better (higher Z)!
Quick Practice
Normal: Mean=100, SD=15
- What % between 85-115? → 68% (1 SD)
- What % above 130? → 2.5% (beyond 2 SD)
Binomial: 10 flips, P(heads)=0.5
- Expected heads? → 5
Tip: Normal distribution is the foundation of most statistical tests!