Standard Normal Distribution

Prerequisites
Effects of Linear Transformations, Introduction to 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

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.
 Z Area below Z -2.50 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 0.0080 -2.40 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.

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

Z = (X - μ)/σ

where Z is the value on the standard normal distribution, X is the value on the original distribution, μ is the mean of the original distribution and σ is the standard deviation of the original distribution.

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