Normal or Gaussian Distribution (Continuous)¶
The most common continuous distribution. Symmetric, bell-shaped, and defined by its mean (\(\mu\)) and standard deviation (\(\sigma\)).
- Function: PDF
- Properties: \(\mu = \text{median} = \text{mode}\). Follows the Empirical Rule (e.g., \(\approx 68.3\%\) of data falls within \(\pm 1\sigma\) of the mean).
- Z-Score: Used to convert any Normal Distribution to the Standard Normal Distribution (\(\mu=0, \sigma=1\)): \(Z = \frac{X_i - \mu}{\sigma}\)
Use Case & Example:
| Scenario | Question | Concept Illustrated |
|---|---|---|
| Biology | What percentage of adult male heights are between 5'9" and 6'1"? | Empirical Rule / Area under the PDF curve |
| Finance | How unusual is a stock return of \(-5\%\) given the average return (\(\mu\)) is \(0.5\%\) and the volatility (\(\sigma\)) is \(2\%\)? | Z-Score (finding how many \(\sigma\) away from \(\mu\) the \(-5\%\) return is) |