What is Normal Distribution?
The normal distribution, also called the Gaussian distribution or "bell curve," is the most important probability distribution in statistics. It describes how data values are distributed around a central mean value.
The Classic Bell Curve
The normal distribution is fully defined by just two parameters: the mean (μ) which determines the center, and the standard deviation (σ) which determines the spread.
Key Properties
Symmetry
Mean = Median = Mode
Asymptotic
Total Area = 1
How Standard Deviation Affects the Shape
Standard deviation controls the "spread" of the normal distribution. A smaller σ creates a tall, narrow curve; a larger σ creates a short, wide curve.
Visual Comparison
Low SD (σ = 0.5)
Data clustered tightly around the mean
High SD (σ = 2)
Data spread widely from the mean
Z-Scores and Standardization
A z-score tells you how many standard deviations a value is from the mean. This allows you to compare values from different normal distributions.
Z-Score Formula
| Z-Score | Meaning | Percentile |
|---|---|---|
| -2 | 2 SDs below mean | ~2.3% |
| -1 | 1 SD below mean | ~15.9% |
| 0 | At the mean | 50% |
| +1 | 1 SD above mean | ~84.1% |
| +2 | 2 SDs above mean | ~97.7% |
Real-World Examples
Many natural phenomena follow a normal distribution:
- Human heights:Most people are near average height, with fewer very tall or very short individuals
- IQ scores:Designed to follow a normal distribution with mean 100 and SD 15
- Measurement errors:Random errors in scientific measurements
- Blood pressure:Population blood pressure readings
When Data Isn't Normal
Not all data follows a normal distribution. Be cautious with:
Non-Normal Distributions