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

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.
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.

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:

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.

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

Z = (26 - 50)/10 = -2.4.

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 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.