Standard Normal Distribution
David M. Lane
of Linear Transformations, Introduction
to Normal Distributions
- 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
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
Figure 1. An example from the applet.
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 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
Please answer the questions: