Demonstration of a Z Transformation of a

Normal Distribution

We begin with a normal distribution. In this example the distribution has a mean of 10 and a standard deviation of 2

Normally distributed Random Variable

= 10

= 2

We would like to translate it to a standard normal distribution with a mean of 0 and a standard deviation of 1

Normally distributed Random Variable

= 10

= 2

Standard Normal Distribution

= 1

= 0

We use the formula for Z transformation:

Normally distributed Random Variable

= 10

= 2

Standard Normal Distribution

= 1

= 0

X

Z

The mean of the original distribution is 10 and it translates to:

Normally distributed Random Variable

= 10

= 2

Standard Normal Distribution

= 1

= 0

02

1010

Z

On the original distribution the point 8 is one standard deviation below the mean of 10 and it translates to:

Normally distributed Random Variable

= 2

Standard Normal Distribution

= 1

12

108

Z

On the original distribution the point 12 is one standard deviation above the mean of 10 and it translates to:

Normally distributed Random Variable

= 2

Standard Normal Distribution

= 1

12

1012

Z

In sum, the Z formula transforms the original normal distribution in two ways:1. The numerator of the formula, (X-shifted the distribution so it is centered on 0. 2. By dividing by the standard deviation of the original distribution, it compressed the width of the distribution so it has a standard deviation equal to 1.

Normally distributed Random Variable

8 10 12

= 2

Standard Normal Distribution

= 1

X

Z

This transformation allows us to use the standard normal distribution and the tables of probabilities for the standard normal table to answer questions about the original distribution.

Normally distributed Random Variable

8 10 12

= 2

For example, if we have a normally distributed random variable with a mean of 10 and a standard deviation of 2, we may want to know what is the probability of getting a value of 8 or less. This is equal to the highlighted area under the probability distribution. For this particular distribution , it is a complicated mathematical operation to calculate the area under the curve. So, it is easier to use the Z transformation.

Normally distributed Random Variable

8 10 12

= 2

This transformation allows us to use the standard normal distribution and the tables of probabilities for the standard normal table to find out the appropriate probability. The Z transformation tells us the 8 on the original distribution is equivalent to -1 on the standard normal distribution. So, the area under the standard normal distribution to the left of -1 represents the same probability as the area under the original distribution to the left of 8.

Normally distributed Random Variable

8 10 12

= 2

Standard Normal Distribution

= 1

12

108

Z

Look up probability on a standard normal probabilities table

Z Prob.

-2.0 0.02275

-1.9 0.02872

-1.8 0.03593

-1.7 0.04456

-1.6 0.05480

-1.5 0.06681

-1.4 0.08076

-1.3 0.09680

-1.2 0.11507

-1.1 0.13566

-1.0 0.15865

-0.9 0.18406

-0.8 0.21185

-0.7 0.24196

-0.6 0.27425

-0.5 0.30853

-0.4 0.34457

-0.3 0.38209

-0.2 0.42074

-0.1 0.46017

-0.0 0.50000

I have reproduced a portion of standard normal table

Standard Normal Probabilities:(The table is based on the area P under the standard normal probability curve, below the respective z-statistic.)

So, probability of getting -1 or less in a random event with a standard normal distribution is 15.9%

Because the point 8 on our original distribution corresponds to -1 on the standard normal distribution, the area to the left of 8 equals 15.9%. (Please excuse my poor drawing skills)

Normally distributed Random Variable

8 10 12

= 2

15.9%