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# The Standard Normal Distribution

The standard normal distribution refers to a normal distribution that has been standardized such that it has a mean of 0 and a standard deviation of 1. The shorthand notation used is:

$$N \sim (0, 1)$$

In the context of statistics and mathematics, standardization is the process of converting an observed value for a random variable into a z-value where:

\begin{align*} Z & = \cfrac {(\text{observed value} – \text{population mean})} {\text{standard deviation}} \\ & =\cfrac {(x – \mu)}{\sigma} \\ \end{align*}

The z-value, also referred to as the z-score in some books, represents the number of standard deviations a given observed value is from the population mean.

#### Example: Standard Normal Distribution

The returns on ABC stock are distributed normally. The mean is $0.60 with a standard deviation of$0.20. Calculate the z-scores for a return of $0.10. Solution If the return is$0.10, then x = 0.1 (this is our observed value). Therefore,

\begin{align*} z & =\cfrac {(x – \mu)}{\sigma} \\ & =\cfrac {(0.1 – 0.6)}{0.2} \\ & = -2.5 \quad (\text{The return of }0.1 \text{ is two and a half standard deviations below the mean.}) \end{align*}

We could get the z-score for any other observed value following a similar approach. For instance, the z-score for a return of \$1 will be:

\begin{align*} z & =\cfrac {(1 – 0.6)}{0.2} \\ & = 2 \quad (\text{The return of } 1 \text{ is two standard deviations above the mean.}) \\ \end{align*}

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