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.

Question

The returns on ABC stock are distributed normally. The mean is $0.60 with a standard deviation of $0.20. The z-scores for a return of $0.10 is closest to:

1. -0.5.
2. -2.5.
3. 2.5.

Solution

The correct answer is B.

If the return is $0.10, then x = 0.1 (this is our observed value). Therefore,

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