The Standard Normal Distribution: Calculation and Interpretation of Probabilities

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 normally distributed where 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*} Calculating Probabilities using z-values under the Standard Normal Distribution Using the standard normal distribution table, we can be able to calculate the that a normally distributed random variable Z, with mean equal to 0 and variance equal to 1, is less than or equal to z, i.e., P(Z ≤ z). However, the table does this when we only have positive values of z. Simply put, if the examiner asks you to find the probability behind a given positive z-value, you’d have to look it up directly on the table. P(Z ≤ z) = θ(z) when z is positive Example: Using the z-score table Using the data from our first example, suppose you were asked to calculate the probability that the return is less than1.

Solution

First, you’d be required to calculate the z-value (2 in this case).

P(Z ≤ 2) can be read off directly from the table.

You just move down and locate the z-value that lies to the right of “2” i.e., 0.9772.

Note: The table above is incomplete.

Negative z-values

If we have a negative z-value and do not have access to the negative values from the table (as shown below), we still can calculate the corresponding probability by noting that:

$$P(Z \le -z) = 1 – P(Z \le z) \text{ or}$$

$$\theta(–z) = 1 – \theta(z)$$

This relationship is true when we consider the following facts:

1. The total area (probability) under the standard normal distribution is 1.
2. The standardized normal distribution is symmetrical about the mean.

Question

Calculate P(Z  ≤ -2.5)

A. 0.9938

B. 0.0062

C. 0.06

Solution

\begin{align*} P(Z \le -2.5) & = 1 – P(Z \le 2.5) \\ & = 1 – 0.9938 \\ & = 0.0062 \\ \end{align*}

Define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution.

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