Normal Distribution: Standard Normal Distribution
m16986
Normal Distribution: Standard Normal Distribution
1.7
2008/06/06 14:18:08 GMT5
2009/02/07 08:34:48.004 US/Central
Susan
Dean
Susan Dean
deansusan@deanza.edu
Barbara
Illowsky
Dr. Barbara Illowsky
illowskybarbara@deanza.edu
Susan
Dean
Susan Dean
deansusan@deanza.edu
Barbara
Illowsky
Dr. Barbara Illowsky
illowskybarbara@deanza.edu
Connexions
Connexions
cnx@cnx.org
Maxfield Foundation
Maxfield Foundation
cnx@cnx.org
elementary
statistics
Mathematics and Statistics
en
The standard normal distribution is a normal distribution of standardized values called
zscores. A zscore is measured in units of the standard deviation. For example, if the
mean of a normal distribution is 5 and the standard deviation is 2, the value 11 is 3 standard
deviations above (or to the right of) the mean. The calculation is:
x
=
μ
+
(
z
)
σ
=
5
+
(
3
)
(
2
)
=
11
The zscore is 3.The mean for the standard normal distribution is 0 and the standard deviation is 1. The transformation
z
=
x

μ
σ
produces the distribution
Z
~
N
(
0
,
1
)
.
The value x comes from a normal distribution with mean μ and standard deviation σ.
Standard Normal Distribution
A continuous random variable (RV)
X~N(0,1). size 12{X "~" N \( 0,1 \) } {}. When X follows the standard normal distribution, it is often noted as
Z~N(0,1) size 12{Z "~" N \( 0,1 \) } {}.
zscore
The linear transformation of the form
z=x−μσ size 12{z= { {xμ} over {σ} } } {}.
If this transformation is applied to any normal distribution
X~N(
μ
,
σ) ,
the result is the standard normal distribution
Z~N(0,1) size 12{Z "~" N \( 0,1 \) } {}. If this transformation is applied to any specific value
x size 12{x} {} of the RV with mean
μ size 12{μ} {} and standard deviation
σ size 12{σ} {} , the result is called the zscore of
x size 12{x} {}. Zscores allow us to compare data that are normally distributed but scaled differently.