Normal Distribution

• Definition: Continuous distribution with a bell-shaped curve, fully described by mean $\mu$ and variance ${\sigma }^2$
• Mean: $E[X]=\mu$
• Variance: $Var(X)={\sigma }^2$
• Use case: Models real-valued variables that tend to cluster around an average (heights, test scores, measurement errors).

Mathematical Representation

$$f(x|\mu ,{\sigma }^2)=\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$$

• 68-95-99.7 Rule:
  • ~68% of data within 1 std dev ($\mu \pm \sigma$)
  • ~95% within 2 std devs ($\mu \pm 2\sigma$)
  • ~99.7% within 3 std devs ($\mu \pm 3\sigma$)

Visualization