L3.3
Special property of Normal distributions
If X has a Normal(mean=
, SD=
) distribution then
σ
μ

X
=
Z
has a Normal(mean=0, SD=1) distribution.
Equivalently, if Z has a Normal(mean=0, SD=1) distribution then
X =
+
Z
has a Normal(mean=
, SD=
) distribution.
Using this property, a probability for any Normal distribution can be
converted into a probability for a Normal (mean=0, SD=1) distribution.
Tables of probabilities for the Normal(mean=0, SD=1) distribution are
available.
This is often called the standard Normal distribution.
Let X = plasma glucose and suppose the distribution of X amongst
obese 4049 year old men is Normal (
= 8,
= 2).
What proportion of obese 4049 year old men have X>10.5?
Since the distribution of X is Normal(
= 8,
= 2)
Pr
(
𝑋
>
10.5
)
=
Pr
𝑋 −
8
2
>
10.5
−
8
2
=
Pr
(
𝑍
>
1.25
)
= (
0.5  0.3944
)
from table for Normal
(
0,1
)
=
0.1056
So, 10.56% of men in this population have plasma glucose
value greater than 10.5.