More on Normal Distributions

Learning Objectives

1. State the mean and standard deviation of the standard normal distribution
2. Use a Z table
3. Use the normal calculator
4. Transform raw data to Z scores

As discussed in the introductory section, normal distributions do not necessarily have the same means and standard deviations. A normal distribution with a mean of 0 and a standard deviation of 1 is called a standard normal distribution.

Areas of the normal distribution are often represented by tables of the standard normal distribution. A portion of a table of the standard normal distribution is shown in Table 1.

Table 1. A portion of a table of the standard normal distribution.

The first column titled "$Z$" contains values of the standard normal distribution; the second column contains the area below $Z$. Since the distribution has a mean of 0 and a standard deviation of 1, the $Z$ column is equal to the number of standard deviations below (or above) the mean. For example, a $Z$ of -2.5 represents a value 2.5 standard deviations below the mean. The area below $Z$ is 0.0062.

The same information can be obtained using the following Java applet. Figure 1 shows how it can be used to compute the area below a value of -2.5 on the standard normal distribution. Note that the mean is set to 0 and the standard deviation is set to 1.

Figure 1. An example from the applet.

Calculate Areas

A value from any normal distribution can be transformed into its corresponding value on a standard normal distribution using the following formula:

$\begin{align*} z=(x-\mu) / \sigma \end{align*}$

where $Z$ is the value on the standard normal distribution, $X$ is the value on the original distribution, $\mu$ is the mean of the original distribution, and $\sigma$ is the standard deviation of the original distribution.

As a simple application, what portion of a normal distribution with a mean of 50 and a standard deviation of 10 is below $26 ?$ Applying the formula, we obtain

$\begin{align*} z=(26-50) / 10=-2.4 \end{align*}$

From Table 1, we can see that $0.0082$ of the distribution is below $-2.4$. There is no need to transform to $Z$ if you use the applet as shown in Figure 2 .

Figure 2. Area below 26 in a normal distribution with a mean of 50 and a standard deviation of 10.

If all the values in a distribution are transformed to $Z$ scores, then the distribution will have a mean of 0 and a standard deviation of 1. This process of transforming a distribution to one with a mean of 0 and a standard deviation of 1 is called standardizing the distribution.