# Using the Standard Normal Distribution to Calculate Probabilities

Using the standard normal distribution table, we can confirm that a normally distributed random variable $$Z$$, with a 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 only when we have positive values of $$z$$. Simply put, if an examiner asks you to find the probability behind a given positive z-value, you will have to look it up directly on the table, knowing that $$P(Z ≤ z) = θ(z)$$ when $$z$$ is positive.

### Positive z-values

#### Example: Using the z-score Table

The returns on ABC stock are normally distributed, where the mean is $0.60 with a standard deviation of$0.20. The probability that the return is less than $1 is closest to: Solution If the return is$0.10, then $$x = 0.1$$ (this is our observed value). Therefore,

\begin{align} z &=\frac{x-\mu}{\sigma}\\&=\frac{1-0.6}{0.2}\\ &=2\ \text{(The return of \1 is two and a half standard deviations below the mean)}\end{align}

First, you’d be required to calculate the z-value (2 in this case). $$P(Z ≤ 1)$$ can be read directly from the table

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

### 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 can still calculate the corresponding probability by noting that:

\begin{align} P(Z \le -z) & = 1 – P(Z \le z) \\ \theta(–z) & = 1 – \theta(z) \\ 0.0228 & = 1 – 0.9772 \end{align}

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*}

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