Normal Distribution

A normal distribution is a distribution whereby the mean, mode and median are all the same. We represent a normal distribution with mean $\mu$ and standard deviation $\sigma$ as: $$ \begin{equation}\begin{aligned} X\sim N(\mu,\sigma^2)\\ \end{aligned}\end{equation} $$

For example, if the masses of cows is normally distributed, with a mean of $55kg$, and a standard deviation of $3kg$, the distribution of the masses is represented as: $$ \begin{equation}\begin{aligned} X\sim N(55kg,9kg^2)\\ \end{aligned}\end{equation} $$

Notice that the variance is stated here with a unit ($kg^2$) which is the square of the unit for the quantity being modelled (mass of the cows). This is because variance ($\sigma^2$) is the square of standard deviation ($\sigma$) which has units $kg$.

The standard normal distribution

This is an important variation of the normal distribution whereby the mean is $0$ and the standard deviation is $1$: $$ \begin{equation}\begin{aligned} Z\sim N(0,1)\\ \end{aligned}\end{equation} $$

The relationship between any normal distribution and the standard normal distribution is that any value $X$ in the original distribution is: $$ \begin{equation}\begin{aligned} Z=\frac{X-\mu}{\sigma}\\ \end{aligned}\end{equation} $$

The $Z$ score corresponding to an $X$ value tells us how many standard deviations from the mean ($\mu$) that $X$ value is. For example, a $Z$ value of $1$ means that the $X$ is exactly $1$ standard deviation to the right of the mean.

The Empirical Rule

This rule states that for a normal distribution:

• 68% of values fall within $1$ standard deviation of the mean
• 95% of values fall within $2$ standard deviations of the mean
• 99.8% of values fall within $3$ standard deviations of the mean