Lesson Objectives
At the end of this lesson, you should be able to
 Illustrates a Normal Random Variable
and its Characteristics; and
 convert normal random variable to
standard normal variable and vice versa.

Normal Distribution
NORMAL DISTRIBUTION
• It is a continuous, bell-shaped curve
distribution whose left and right tails
extend
indefinitely but come infinitely and
never touch the x-axis.
• The peak of the curve is the position
of the mean which is found at the
center of the distribution. The
distribution thins out as the values
move away from the
mean

Normal Distribution
PROPERTIES OF NORMAL DISTRIBUTION
• The distribution curve is bell shaped
• The curve is symmetrical about its center
• The mean, median and mode coincide at the corner
• The tails flatten out indefinitely along the horizontal
axis but never touch it ( the curve is asymptotic to the
base line)
• The are under the curve is 1 or 100%

Normal Distribution
2 FACTORS THAT THE GRAPH OF THE NORMAL DISTRIBUTION MAY
DEPEND IN
1. MEAN
• The change of values of the mean shifts the graph of the normal curve to the right
or to the left

Normal Distribution
2 FACTORS THAT THE GRAPH OF THE NORMAL DISTRIBUTION MAY
DEPEND IN
1. Standard deviation
• Large value of standard makes the normal curve
short and wide
• Small value of standard deviation yield a skinnier
and taller normal curve

Normal Distribution
Empirical Rule
• Is also referred to the 68% 95%, and 99.7%
• It tells us that for normally distributed variable the following are
true;

Normal Distribution
Empirical Rule
• Is also referred to the 68% 95%, and 99.7%
• It tells us that for normally distributed variable the following are true;
• Approximately 68% of the data lie within 1
standard deviation of the mean
• Approximately 95% of the data lie within 2
standard deviation of the mean
• Approximately 99.7% of the data lie within 3
standard deviation of the mean

Normal Distribution
EXAMPLE 1: Scores in examination
the scores of the senior high school students in Colegio De Sto Nino in their
Statistics and Probability Examination are normally distributed with a mean 35 and
the standard deviation of 5
A. What percent of the score are between 30-40?
B. What scores fall within 95% of the distribution

Normal Distribution
EXAMPLE 1: Scores in examination
the scores of the senior high school students in Colegio De Sto Nino in their
Statistics and Probability Examination are normally distributed with a mean 35 and
the standard deviation of 5
A. What percent of the score are between 30-40?
B. What scores fall within 95% of the distribution

Normal Distribution
STANDARD NORMAL DISTRIBUTION
1. The total area under the standard normal
curve is equal to 1.
2. The standard normal distribution is
symmetrical about the mean. This means that
the area of the left and right side of the mean
is 0.5.
3. The mean, median and mode are equal.
4. The tails of a standard normal curve are
asymptotic relative to the horizontal line.
5. The probability that the standard random
variable Z will fall into an interval a to b is
equal to the area of the shaded region under
the standard normal curve between two points

Normal Distribution
Understanding the Standard Normal Curve
The standard normal curve is a normal probability distribution that is
most commonly used as a model in inferential statistics. The equation
that describes a normal curve is:
𝛾 =
𝑒 − 1
2
(
𝑋−𝜇
𝜎
)2
𝜎 2𝜋
Where:
Ƴ = height of the curve particular values of X
X = any score in the distribution
σ = standard deviation of the population
µ = mean of the population
π = 3.1416
e = 2.7183

Normal Distribution
Understanding the Standard Normal Curve
• By substituting the mean, µ = 0 and the standard deviation, σ =1 in the
formula, mathematicians are able to find areas under the normal curve.
• The area between -3 and +3 is almost 100% because the curve almost
touches the horizontal line. Thus, there is a small fraction of the area at
the tails of the distribution.

Normal Distribution
Table of Areas under the Normal Curve
• The z-score is a measure of relative standing. It is calculated by subtracting
sample/population mean form the measurement X and then dividing the result by the
corresponding standard deviation. The final result, the z-score, represent the distance between
a given measurement X and the mean, expressed in standard deviations. Either the z-score
locates X within a sample or within a population
𝑧 =
𝑋−𝜇
𝜎
(population data)
𝑧 =
𝑋−X̄
𝑠
(sample data)
Raw scores may be composed of large values, but
large values cannot be accommodated at the base
line of the normal curve. So, they have to be
transformed into scores for convenience without
sacrificing meanings associated with the raw scores.

Normal Distribution
Importance of using z-scores?
• Raw scores may be composed of large values, but large
values cannot be accommodated at the base line of the
normal curve. So, they have to be transformed into scores
for convenience without sacrificing meanings associated
with the raw scores.
• The z-values are matched with specific areas under the
normal curve in a normal distribution table. Therefore, if we
wish to find the percentage associated with X, we must find
its matches z-value using the z-formula.

Normal Distribution
EXAMPLE 1: Reading Scores
Given the mean, µ = 50 and the standard deviation σ = 4 of a
population of reading scores. Find the z-value that corresponds to a
score X=58.
𝑧 =
𝑋−𝜇
𝜎
𝑧 =
58−50
4
𝑧 =
8
4
= 2.00
Thus, the z-value that corresponds to the raw score 58 is 2 in a
probability distribution

Normal Distribution
EXAMPLE 1: Reading Scores
As you can see, score X = 58 corresponds to z=2.00. It is above
the mean. So, we can say that, with respect to the mean, the score
of 58 is above average.

Normal Distribution
EXAMPLE 2: Scores in Science Test
Given X = 20, X̄ = 26 and s = 4. Compute the corresponding z-score.
𝑧 =
𝑋−X̄
𝑠
𝑧 =
20−26
4
𝑧 =
−6
4
=
−3
2
= -1.50
The corresponding z-score is -1.5 to the left of the mean/the
z-value that corresponds to 20 is -1.50 in a sample distribution.
With respect to the sample mean, the score 20 is below the
sample mean

Normal Distribution
EXAMPLE 3:
Suppose the corresponding z-score of the weight of employees in Meycauayan
City Hall is 3.5, what should be the weight of an employee if it resembles a normal
distribution with a mean 58 kg and a standard deviation of 2.

Normal Distribution
Let’s Try!
A. State whether the z-score locates the raw score X
within a sample or within a population
1. X = 50, s = 5, X̄ = 40
2. X = 40, σ = 8, 𝜇 = 52
3. X = 36, s = 6, X̄ = 28
4. X = 74, s = 10, X̄ =60
5. X = 82, σ = 15, 𝜇 =75
B. From exercise A, state whether each raw score lies
below or above the mean.

Normal Distribution
Z-TABLE

Normal Distribution
Z-TABLE

Normal Distribution
Four-step process in Finding the Areas Under the
Normal Curve Given a z-Value
1. Express the given z-value into a three-digit form.
2. Using the z-table, find the first two digits on the left column.
3. Match the third digit with the appropriate column on the right
4. Read the area (or probability) at the intersection of the row and
the column. This is the required area.

Normal Distribution
EXAMPLES
• Find the area that corresponds to z=1
(Finding the area that corresponds to is the same as
finding the area between z=0 and z=1)
• Find the area that corresponds to z=1.36
• Find the area that corresponds to z= - 2.58
(The area that corresponds to z=2.58 is the same as the
area that corresponds to z= - 2.58)

Normal Distribution
Let’s Try!
Find the corresponding area between z = 0 and each of the following:
1. z = 0.96
2. z = - 1.74
3. z = 2.18
4. z = - 2.69
5. z = 3.00