2. Unit Normal Distribution
This is the simplest of the family of Normal Distributions, also called
the z distribution. It is a distribution of a normal random variable with a
mean equal to zero (μ = 0) and a standard deviation equal to one (μ = 1).
It is represented by a normal curve.
Characteristics:
-It is symmetrical about the vertical line drawn
through z = 0
-the curve is symptotic to the x-axis. This means
that both positive and negative ends approach
the horizontal axis but do not touch it.
-Mean, Median, and Mode coincide with each
other.

3. The Z Score
The number of standard deviations from the mean is called the z-score
and can be found by the formula:
z= x-x
SD
Where:
z= the z score
x= raw score
SD= standard deviation

4. Example
Find the z-score corresponding to a raw score of 132 from a normal
distribution with mean 100 and standard deviation 15.
Solution
We compute
132 - 100
z = ________ = 2.133
15
-2 -1 0 +1 +2.133 z

5. A z-score of 1.7 was found from an observation coming from a
normal distribution with mean 14 and standard deviation
3. Find the raw score.
Solution
We have
x - 14
1.7 = _______
3
To solve this we just multiply both sides by the denominator
3,
(1.7)(3) = x - 14
5.1 = x - 14
x = 19.1

6. Area Under the Unit Normal Curve
The area under the unit normal curve may represent several things like
the probability of an event, the percentile rank of a score, or the
percentage distribution of a whole population. For example, the are
under the curve from z = z1 to z = z2, which is the shaded region in figure
7.6, may represent the probability that z assumes a value between z 1 and
z 2.
Fig. 7.6 The Probability That z1
and z2.
z1 z2

7. Examples
Example no. 1
Find the are between z = 0 and z = +1
Solution:
From the table, we locate z = 1.00 and get the corresponding area
which is equal to o.3413
2nd
0.3413
1 3
s r
t d
1.0 0.3413
0 +1

8. Example no. 2
Find the area between z = -1 and z = 0
Solution
As you can see, there is no negative value of z, so we need the
positive value. Hence, the area is also 0.03413
0.3414
-1 0

9. Example no. 3
Find the area below z = -1
Solution:
Since the whole area under the curve is 1, then the whole area
is divided into two equal parts at z = 0. This means that the area to the
left of z = 0 is 0.5. To get the area below z = -1 means getting the area to
the left of z = -1. The area below z = -1 is then equal to 0.5000 – 0.3414
= 0.1587.
0.1587
-1 0

10. Example no. 4
Find the area between z = -0.70 and z = 1.25
Solution:
The area between z = -0.70 and z = 0 is 0.2580, while that
between z = 0 and z = 1.25 is 0.3944. Therefore, the area between z = -
0.70 and z = 1.25 is 0.2580 + 0.3944 = 0.6524. We add the two areas since
the z values are on both side of the distribution.
0.6524
-0.7 0 1.25

11. Example no. 5
Find the area between z = 0.68 and z = 1.56.
Solution:
The area between z = 0 and z = 0.68 is 0.2518, while the area
between z = 0 and z = 1.56 is 0.4406. Since the two z values are on the
same side of the distribution, we get the difference between the two
areas. Hence, the area between z = 0.68 and z = 1.56 is 0.4406 – 0.2518
= 0.1888.
0.1888
0 0.68 1.56

12. Activity : Plot the following
I. Find the Z SCORE II. Find the area under the unit
normal curve for the following
1. Raw Score = 128 values of z.
Mean = 95
SD = 3 1. Below z = 1.05
2. Above z = 1.52
2. Raw Score = 98 3. Above z = -0.44
Mean = 112 4. Below z = 0.23
SD = 1.5 5. Between z = -0.75 and z = 2.02
6. Between z = -0.51 and z = -2.17
3. Raw Score = 102 7. Between z = -1.55 and z = 0.55
Mean = 87
SD = 1.8