|
Standard Normal Distribution
Author(s)
David M. Lane
Prerequisites
Effects
of Linear Transformations, Introduction
to Normal Distributions
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 1.
Table 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 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.
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.
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.
Please answer the questions:
|