Normal Distribution

1. The Normal
Distribution
Statistics and
Probability
Quarter 3 – Week 3
Maricar P. Dimas
Teacher

2. Most Essential
Learning
Competencies
After going through this module, you are expected to:
1. illustrate a normal random variable and its
characteristics (M11/12SP-IIIc-1);
2. identify regions under the normal curve that
correspond to different standard
normal values (M11/12SP-IIc-3);
3. convert a normal random variable to a standard
normal variable and vice
versa (M11/12SP-IIIc-4); and
4. compute probabilities and percentiles using the
standard normal distribution

3. The Normal
Distribution
and Its
Properties
 1. The graph is a
continuous curve and has
a domain -∞ < X < ∞.
 2. The graph is asymptotic
to the x-axis. The value of
the variable gets closer
and closer but will never
be equal to 0.
 3. The highest point on
the curve occurs at x = µ
(mean).
 4. The curve is
symmetrical about the
mean.
 5. The total area in the
normal distribution under
the curve is equal to 1.
 6. In general, the graph of a
normal distribution is a bell-
shaped curve with two
inflection points, one on the
left and another on the right.
Inflection points are the
points that mark the change
in the curve’s concavity.
• Inflection point is the point at
which a change in the direction
of curve at mean minus
standard deviation and mean
plus standard deviation

4. Empirical Rule
(also called the
68 - 95 -99.7%
rule):
 about 68.3% of
the area under the
curve falls within
1 standard
deviation of the
mean
 about 95.4% of the area
under the curve falls
within 2 standard
deviations of the mean
about 99.7% of the area
under the curve falls within
3 standard deviations of the
mean.

5. Examples
1. Suppose the mean is 60
and the standard deviation
is 5, sketch a normal curve
for the distribution. This is
how it would look like.
2. A continuous random
variable X is normally
distributed with a mean of 45
and standard deviation of 6.
Illustrate a normal curve and
find the probability of
thefollowing:
a. P (39 < X < 51) = 68.3%
c. P (X > 45) = 50%
b. P (33 < X < 63) = 97.55%
a. P (39 < X < 51) = 68.3%
b. P (33 < X < 63) = 97.55%
d. P (X < 39) = 15.85%

6. The
Standard
Normal
Distribution
The standard normal
distribution, which is denoted
by Z, is also a normal
distribution having a mean of
0 and a standard deviation of
1. Since the normal
distribution can have different
values for its mean and
standard deviation, it can be
standardized by setting the µ
= 0 and the = 1.

7. The Z -
Table

8. Identifying
Regions under
a Normal Curve
1
2
STEP 1: Draw a normal
curve and locate thez -
scores and shade.
3
STEP 3: If you are looking
for the area between two z
- scores, simply subtract
the corresponding areas to
arrive at the answer.
Therefore, 0.9857 - 0.1056 =
0.8801 and the
P(-1.25 < Z < 2.19) =
88.01%
STEP 2. Locate the
corresponding area of the
z - scores in the z-table
z = -1.25 has a
corresponding area of
0.1056
z = 2.19 has a corresponding
area of 0.9857
Find the proportion of the
area between z = -1.25 and
2.19, this can be expressed
as P(-1.25 < Z < 2.19), read
as the probability that Z is
greater than -1.25 but less
than 2.19.

9. The Z-
Score
The z-score is an essential component in standard
normal distribution. This allows us to describe a given
set of data by finding the z-scores. This leads us to a
question of how z-scores are identified?

10. Example

11. Example

12. Example

13. The
Percentile

14. Thank You!
Quarter 3 Week 3