Objectives

At the end of this lesson, the learner should be able to

● accurately illustrate a normal random variable;
● accurately locate a given score that lies in a normally distributed set
of scores in terms of its standard deviation ;
● correctly determine the probability density function of a normal
distribution given the mean and standard deviation; and
● correctly solve real-life problems involving normal random variable.
 Essential Questions

● How can we say that a set of data is normally distributed?

● How can the standard deviation help in interpreting the distance of a

score from the mean?
 Warm Up!

This lesson will tackle about normal distribution. Before we start with
the discussion, let us use an online calculator to give us an introduction
about the graph of a normal distribution.

Open the Desmos Calculator.

Please go to https://www.desmos.com/calculator
 Warm Up!

•
Guide Questions
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3
1
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5
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5
Try It!
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Let’s Practice!
Let’s Practice!
Key Points

Normal Distribution
1 a continuous probability distribution where most of the scores tend to be closer to the mean

Normal Random Variable

12 a continuous random variable of a normal distribution
Key Points

3
1
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Standard Normal Distribution

4 the most common example of a normal distribution with a mean of 0 and standard deviation of
1

5
 Synthesis

● How do you approximate the location of a score in terms of

standard deviation?

● How can you apply the concept of normal random variable in

evaluating your performance in an exam?

● How would you describe data that is normally distributed?

Lesson 1

The Normal Random

Variable
 Learn about It!

Normal Distribution
1 a continuous probability distribution where most of the scores tend to be closer to the mean

Note:
The term normal refers to the fact that this kind of distribution occurs
in many different kinds of common measurements.
Synthesis
 Learn about It!

Normal Distribution
1 a continuous probability distribution where most of the scores tend to be closer to the mean

Example:
The height of people in a population follows a normal distribution.
 Learn about It!

Normal Random Variable

12 a continuous random variable of a normal distribution
Lesson 2

Characteristics of a
Normal Random
Variable
 Objectives

At the end of this lesson, the learner should be able to

● accurately determine the characteristics of a normal random variable;

● correctly identify if a given score is an outlier; and

● correctly solve real-life problems involving characteristics of a

normal random variable.
 Essential Questions

● What are the advantages of a normal distribution over other data

distributions?

● What is the importance of knowing the empirical rule on normal

curves in getting the number of scores within a specific range?
 Warm Up!

This lesson will tackle about the characteristics of a normal

distribution. An interactive graph is available online where you can
change the value of the mean and the standard deviation and observe
the behaviour of the graph.

(Click the link to access the interactive graph.)

Bourne, Murray. “Normal Probability Distribution Graph Interactive.”

Interactive Mathematics. Retrieved 15 July 2019 from
https://www.intmath.com/counting-probability/normal-distribution-gra
ph-interactive.php
 Warm Up!

Uncheck “Show standard normal curve.”

Check “Show probability calculation.”

Drag the red colored dots representing the mean and the standard
deviation and observe the behavior of the graph.

You can also drag the green colored dots so that the two are of equal
multiple of standard deviations from the mean.
 Guide Questions

● Where in the curve does the mean located?

● What is the probability that a score is within one standard deviation

away from the mean?

● What is the probability that a score is within two standard

deviations away from the mean?
 Learn about It!

Characteristics of a Normally Distributed Set of Data

1 a. A normal distribution is symmetric about its mean.

In a normally distributed set of data, the mean will be placed at the

center if the data are arranged in ascending order.
 Learn about It!

Characteristics of a Normally Distributed Set of Data

1 a. A normal distribution is symmetric about its mean.

Moreover, approximately 50% of the data is less than the mean and
approximately 50% of the data is greater than the mean.
 Learn about It!

Characteristics of a Normally Distributed Set of Data

1 b. The mean, median, and mode of a normal distribution are all equal.

Since the mean, media, and more are all equal, the median and the
mode are also located at the middle of the distribution.
 Learn about It!

Characteristics of a Normally Distributed Set of Data

1 c. A normal distribution is thicker at the center and less thick at the tails.

”Thicker” at the center means that there are more scores located
near the center of the distribution and there are fewer scores found
near both ends of the distribution.
 Learn about It!

Standard Normal Distribution

4 the most common example of a normal distribution with a mean of 0 and standard deviation of
1
 Learn about It!

Note that the area of a normal distribution refers to the

graphical representation of the percentage, proportion, or
probability of a normal distribution.
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1
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1
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1
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12
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Outliers
13 scores that are more than two standard deviations away from the mean.
 Learn about It!

Extreme Outliers
14 scores that are more than three standard deviations away from the mean.
Try It!
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Solution:
1. Make a plan in solving the problem.

A score is considered an outlier if it is more than two standard

deviations away from the mean.
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Solution:
Thus, we should first solve for the scores two standard deviations
away from the mean and determine whether the given score lies
beyond those points.
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Solution:
2. Solve for the scores two standard deviations away from the mean.
Try It!
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Let’s Practice!
Let’s Practice!
Key Points

1
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12

Outliers
13 scores that are more than two standard deviations away from the mean.

Extreme Outliers
14 scores that are more than three standard deviations away from the mean.
 Synthesis

● What are the different characteristics of a set of data that is

normally distributed?

● Why do you think that test scores tend to be closer to the mean?

● How do we illustrate a normal distribution using a graph?

Lesson 3

The Normal Curve

 Objectives

At the end of this lesson, the learner should be able to

∙ properly construct a normal curve;

∙ correctly find the area of a region under the normal curve;

and

∙ accurately solve real-life problems involving the normal

curve.
 Essential Questions

∙ Why is a normal curve said to be normal?

∙ What is the importance of the empirical rule in finding the area of

the region under the normal curve?
 Warm Up!

Before we learn about The Normal Curve, let us recall the characteristics
of a normal distribution by watching a video.

(Click on the link to access the video.)

Steve Mays. “Properties of a Normal Distribution.” YouTube video, 2:58.

Posted 17 November 2011. Retrieved 30 June 2019 from
https://www.youtube.com/watch?v=4rdgdDmyCpg
 Guide Questions

● What are the characteristics of a normal distribution?

● Why is the total area of a normal distribution equal to one?

● How do we get the inflection points of a normal distribution?

 Learn about It!

Normal Curve (or Bell Curve)

1 a graph that represents the probability density function of a normal probability distribution. It is
also called a Gaussian curve named after the mathematician, Carl Friedrich Gauss.
 Learn about It!

A. Area of the Region between One Standard Deviation away from

the Mean
 Learn about It!

B. Area of the Region between Two Standard Deviations away

from the Mean
 Learn about It!

C. Area of the Region between Three Standard Deviations away

from the Mean
 Try It!

Example 1: What is the area of the scores in a normally distributed

data that is more than one standard deviation above the mean?
 Try It!

Example 1: What is the area of the scores in a normally distributed

data that is more than one standard deviation above the mean?

Solution:

1. Shade the region in the normal curve that is one standard deviation
above the mean.
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2. Construct the normal curve using the set of scores obtained from the
previous step.

Using the mean as the center and the set of scores, we can construct the
normal curve as follows:
Let’s Practice!
Let’s Practice!
Key Points

Normal Curve (or Bell Curve)

1 a graph that represents the probability density function of a normal probability distribution. It is
also called a Gaussian curve named after the mathematician, Carl Friedrich Gauss.

2
 Synthesis

● How did you get the area of a region under the normal curve?

● What other field of study you can use the normal curve?

● How will you solve word problems involving a normal distribution?

Lesson 4

Solving Problems
Involving the Normal
Random Variable
 Objectives

At the end of this lesson, the learner should be able to

∙ correctly recall how to locate the given scores on a normal

curve;
∙ correctly recall how to find the area of a region under the
normal curve; and
∙ solve problems involving the normal random variable.
 Essential Questions

∙ How can we easily solve problems involving the normal random

variable?

∙ How can the empirical rule help you in solving problems involving
the normal random variable?
 Warm Up!

Before we learn about Solving Problems Involving the Normal Random

Variable, le us construct a normal curve by using an interactive normal
distribution.

Show the following areas:

1. 0.5
2. less than one standard deviation below the mean
3. more than two standard deviations above the mean
 Warm Up!

(Click on the link to access the

website.)

Bognar, Matt. “Normal Distribution.”

Homepage Stat. Retrieved 1 July 2019
from http://bit.ly/2XTQZdF
 Guide Questions

∙ How are you able to assign a mean and standard deviation to show
the given area in the normal curve?

∙ What rules do you follow in showing the given areas in the normal
curve?
 Learn about It!

Normal Curve (or Bell Curve)

1 a graph that represents the probability density function of a normal probability distribution. It is
also called a Gaussian curve named after the Mathematician Karl Friedrich Gauss.
 Learn about It!

Normal Curve Areas under the Empirical Rule

To solve problems involving the normal random variable, we can use

the following areas in the normal curve based from the empirical rule.
Learn about It!
Learn about It!
 Learn about It!

Normal Curve Areas under the Empirical Rule

A. Areas between the mean and one standard deviation above or

below the mean
 Learn about It!

Normal Curve Areas Based on the Empirical Rule

B. Area between the mean and two standard deviations above or

below the mean
 Learn about It!

Normal Curve Areas Based on the Empirical Rule

C. Areas between the mean and three standard deviations above or

below the mean
Try It!
 Try It!

Solution:

1. Construct the normal curve of the distribution and locate the given
weights.
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Example 2: The average price of carrots was ₱60.5 per kilogram with
a standard deviation of 2.4 for the last three months. If the price is
normally distributed for 90 days, what percent of 90 days when the
price was above ₱62.9?
Solution:

1. Construct the normal curve of the distribution and locate the given
price.
Try It!
Try It!
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Let’s Practice!
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Individual Practice:

2. A mechanic can finish repairing a car in an average of 2.3 hours with a

standard deviation of 0.12. Given that the number of hours to repair a
car is normally distributed, what is the percentage of the cars repaired in
less than 2.54 hours?
Let’s Practice!
Key Points

Normal Curve (or Bell Curve)

1 a graph that represents the probability density function of a normal probability distribution. It is
also called a Gaussian curve named after the Mathematician Karl Friedrich Gauss.
 Key Points

Normal Curve Areas Based on the Empirical Rule

A. Area between the mean and one standard deviation above or

below the mean
 Key Points

Normal Curve Areas Based on the Empirical Rule

B. Area between the mean and two standard deviations above or

below the mean
 Key Points

Normal Curve Areas Based on the Empirical Rule

C. Area between the mean and three standard deviations above or

below the mean
 Synthesis

∙ What strategies did you use in solving each problem involving the
normal random variable?

∙ Does planning play a vital role in solving problems? Why or why

not?

∙ How can we convert a random variable to a standard normal variable

and vice versa?