We also note the following about the normal distribution.
1
The mean,
μ
, is
independent
of the variance,
σ
2
.
2
A normal variate with mean,
μ
, and variance,
σ
2
is denoted by
X
∼
ND
(
μ, σ
2
)
.
By Arthur Mpazi Yambayamba
The Normal Probability Distribution
November 4, 2015
7 / 32
Areas Under a Normal Distribution Curve
By Arthur Mpazi Yambayamba
The Normal Probability Distribution
November 4, 2015
8 / 32
The Standard Normal Distribution
Since each normally distributed variable has its own mean and
standard deviation, as stated earlier, the shape and location of these
curves will vary.
In practical applications, then, you would have to have a table of
areas under the curve for each variable.
To simplify this situation, statisticians use what is called the
Standard Normal Distribution
.
The standard normal distribution is a normal distribution with a mean of 0
and a standard deviation of 1.
By Arthur Mpazi Yambayamba
The Normal Probability Distribution
November 4, 2015
9 / 32
The Standard Normal Distribution
All normally distributed variables can be transformed into the
standard normally distributed variable by using the formula for the
standard score or
z
score given by:
z
=
value

mean
standard deviation
or z
=
x

μ
σ
The
z
score or value gives the number of standard deviations that a
particular value of
X
,
x
, is away from the mean.
By Arthur Mpazi Yambayamba
The Normal Probability Distribution
November 4, 2015
10 / 32
The Standard Normal Distribution
By Arthur Mpazi Yambayamba
The Normal Probability Distribution
November 4, 2015
11 / 32
Finding Areas Under the Standard Normal Distribution
Curve
The first step is to draw a sketch of the normal distribution curve and
shade the area.
Second step, use the procedure as illustrated in the following figures.
(1)
To the left of any
z
value, look up the
z
value in the table and use
the area given.
By Arthur Mpazi Yambayamba
The Normal Probability Distribution
November 4, 2015
12 / 32
Finding Areas Under the Standard Normal Distribution
Curve
(2)
To the right of any
z
value, look up the
z
value in the table and
subtract the area from 1.
By Arthur Mpazi Yambayamba
The Normal Probability Distribution
November 4, 2015
13 / 32
Finding Areas Under the Standard Normal Distribution
Curve
(3)
Between any two
z
values, look up both
z
values and subtract the
corresponding areas.
By Arthur Mpazi Yambayamba
The Normal Probability Distribution
November 4, 2015
14 / 32
Finding Areas Under the Standard Normal Distribution
Curve
Examples
(1)
Find the area to the left of
z
= 1
.
99
.
(2)
Find the area to the right of
z
=

1
.
16
.
(3)
Find the area between
z
= +1
.
68 and
z
=

1
.
37
.
By Arthur Mpazi Yambayamba
The Normal Probability Distribution
November 4, 2015
15 / 32
Calculating Probabilities Using the Standard Normal
Distribution Curve
A normal distribution curve can be used as a probability distribution
curve for normally distributed variables.
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 F. TAILOKA
 Normal Distribution, normal probability distribution, Arthur Mpazi Yambayamba