Gaussian Distribution

Topics: Probability

Definition

The Gaussian distribution, also known as Normal distribution is a continuous probability curved shaped like a bell.

It is fully defined by 2 parameters:

• Mean ($\mu$): the center point of the curve
• Variance ($\sigma^2$) or Standard Deviation ($\sigma$): measures how concentrated or spread out the data is. A small σ makes a tall, narrow bell; a large σ makes a short, wide bell.

The distribution is typically denoted $\mathcal{N}(\mu, \sigma^2)$.

Probability Density Function

$$f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \bigg(-\frac{(x- \mu)^2}{2 \sigma^2} \bigg)$$

The 68-95-99.7 Rule

• About $68\%$ of the data falls within one standard deviation of the mean $\mu \pm \sigma$
• About $95\%$ of the data falls within one standard deviation of the mean $\mu \pm 2\sigma$
• About $99.7\%$ of the data falls within one standard deviation of the mean $\mu \pm 3\sigma$