Page 48 - Basic Statistics

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3.3 STANDARD NORMAL DISTRIBUTION Z

Not efficient if we always tried to organize a separate table for each of

the normal curve for the possible each pair of values  and  . But we have to
use the table if we want to avoid having to use the integral calculus.

Fortunately, we can transform any observation from any normal random

variable X be the value of the standard normal random variable Z with a mean

2
value 0 (  =0 ) and the variance 1 ( =1 ).

X - 
This can be done through the transformatio Z = ( Z2 )


The mean value of Z is zero, because

1 1
E(Z) = E(X-  ) = (  -  ) = 0
 

While the variance is

1  2
2
2
 =  ( 2 X  / )  =  2 / =  = = 1.
−
X
z
 2 x  2

Definition: Standard Normal Distribution. Distribution of the normal random
variable with mean value 0 and a standard deviation 1 is called the standard

normal distribution.

If X is between x = x 1 and x = x , the random variable Z will be among the
2
values of equivalent

x −  x − 
z = 1  and z = 2  ( Z3 )
1
2

~~* CHAPTER 3 NORMAL PROBABILITY DISTRIBUTION *~~