# 7.5: Standard Normal Distribution

• • Contributed by David Lane
• Associate Professor (Psychology, Statistics, and Management) at Rice University

Learning Objectives

• State the mean and standard deviation of the standard normal distribution
• Use a $$Z$$ table
• Use the normal calculator
• 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 $$\PageIndex{1}$$.

Table $$\PageIndex{1}$$: A portion of a table of the standard normal distribution
Z Area below
-2.5 0.0062
-2.49 0.0064
-2.48 0.0066
-2.47 0.0068
-2.46 0.0069
-2.45 0.0071
-2.44 0.0073
-2.43 0.0075
-2.42 0.0078
-2.41 0.008
-2.4 0.0082
-2.39 0.0084
-2.38 0.0087
-2.37 0.0089
-2.36 0.0091
-2.35 0.0094
-2.34 0.0096
-2.33 0.0099
-2.32 0.0102

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 $$\PageIndex{1}$$ 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 $$\PageIndex{1}$$: 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:

$Z=\frac{X-\mu }{\sigma }$

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.

Example $$\PageIndex{1}$$

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

Solution

Applying the formula, we obtain

$Z = \frac{26 - 50}{10} = -2.4$

From Table $$\PageIndex{1}$$, 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 $$\PageIndex{2}$$. Figure $$\PageIndex{2}$$: 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.

## Contributor

• Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University.