Statistics and
Probability
Quarter 3 – Module 3:
The Normal Distribution
Statistics and Probability
Alternative Delivery Mode
Quarter 3 – Module 3: The Normal Distribution
First Edition, 2020
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Statistics and
Probability
Quarter 3 – Module 3:
The Normal Distribution
Introductory Message
This Self-Learning Module (SLM) is prepared so that you, our dear learners,
can continue your studies and learn while at home. Activities, questions, directions,
exercises, and discussions are carefully stated for you to understand each lesson.
Each SLM is composed of different parts. Each part shall guide you step-bystep as you discover and understand the lesson prepared for you.
Pre-tests are provided to measure your prior knowledge on lessons in each
SLM. This will tell you if you need to proceed on completing this module or if you
need to ask your facilitator or your teacher’s assistance for better understanding of
the lesson. At the end of each module, you need to answer the post-test to self-check
your learning. Answer keys are provided for each activity and test. We trust that you
will be honest in using these.
In addition to the material in the main text, Notes to the Teacher are also
provided to our facilitators and parents for strategies and reminders on how they can
best help you on your home-based learning.
Please use this module with care. Do not put unnecessary marks on any part
of this SLM. Use a separate sheet of paper in answering the exercises and tests. And
read the instructions carefully before performing each task.
If you have any questions in using this SLM or any difficulty in answering the
tasks in this module, do not hesitate to consult your teacher or facilitator.
Thank you.
What I Need to Know
This module was designed and written to help you understand the concept of
the normal distribution. The scope of this module helps you to appreciate and
understand learning situations that can be used in a day-to-day basis experience.
The language used in this learning kit recognized the diverse vocabulary level of
students for a higher understanding of the lesson. The lessons were arranged to
follow the standard sequence of the course. However, the manner in which you read
them can be possibly changed to correspond with the textbook that you might be
using now. Thus, this module is divided into two lessons, respectively as follows
•
•
Lesson 1 – The Normal Distribution and Its Properties
Lesson 2 – The Standard Normal Distribution
After going through this module, you are expected to:
1. illustrate a normal random variable and its characteristics (M11/12SP-IIIc1);
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
(M11/12SP-IIIc-d-1).
1
What I Know
Directions: Choose the best answer. Write the chosen letter on a separate
sheet of paper.
1. Which of the following illustrations represents normal distribution?
A.
C.
B.
D.
2. What is another name for normal distribution?
A. Gaussian distribution
B. Poisson distribution
C. Bernoulli’s distribution
D. Probability distribution
3. What is the total area in the distribution under the normal curve?
A. 0
B. 1
C. 2
D. 3
4. Which of the following is a parameter of normal distribution?
A. mean
B. standard deviation
C. mean and standard deviation
D. none of the above
5. The graph of a normal distribution is symmetrical about the ________.
A. mean
B. standard deviation
C. mean and standard deviation
D. none of the above
6. What percent of the area under a normal curve is within 2 standard deviations?
A. 68.3%
B. 95.4%
C. 99.7%
D. 100%
2
7. How many standard deviations are there in each inflection point?
A. 0
B. 1
C. 2
D. 3
8. Which of the following denotes the standard normal distribution?
A. A
B. X
C. Y
D. Z
9. A random variable X has a mean of 12 and a standard deviation of 3.2. What
is the corresponding z-score for x = 8?
A. -1.25
B. -1.50
C. -1.75
D. -2.25
10. What is the area under the normal curve if the z-score given is 2.14?
A. 0.9830
B. 0.9834
C. 0.9838
D. 0.9842
11. What is the z-score if the area under the normal curve is 0.0475?
A. -1.66
B. -1.67
C. 1.66
D. 1.67
12. What is the area under the normal curve between the z-scores -1.99 and 1.56?
A. 0.9173
B. 0.9317
C. 0.9369
D. 0.9639
13. What is the area of the P(Z > 2.58) using the standard normal curve?
A. 0.0049
B. 0.0051
C. 0.9949
D. 0.9951
14. In an achievement test, the mean score of normally distributed values is 70
and the standard deviation is 12. What is the percentage of students who got
a score of 85 and above?
A. 0.1056
B. 0.1075
C. 0.8925
D. 0.8944
15. What is the value of 70th percentile in a standard normal distribution?
A. 0.51
B. 0.52
C. 0.61
D. 0.62
3
Lesson
1
The Normal Distribution and
Its Properties
The normal distribution is just one of the distributions to be discussed in this
course. It is also considered as the most important distribution in Statistics because
it fits many real-life situations. This lesson will bring us a deeper understanding of
the normal distribution and its characteristics.
What’s In
Directions: Anticipation-Reaction Guide. Complete the table by putting check
as to AGREE or DISGAREE on the corresponding columns for the following
statements. Write your answers on a separate sheet of paper.
Statement
Agree
1. The normal curve of the distribution is bell-shaped.
2. In a normal distribution, the mean, median and mode
are of equal values.
3. The normal curve gradually gets closer and closer to
0 on one side.
4. The normal curve is symmetrical about the mean.
5. The distance between the two inflection points of the
normal curve is equal to the value of the mean.
Notes to the Teacher
This contains helpful tips and strategies that will help you in
guiding the learners.
4
Disagree
What’s New
Before proceeding to our discussion, let us consider this activity that will give
us ideas about our lesson.
Consider the random event of tossing four coins once, then follow these steps:
1. List all the possible outcomes using the tree diagram.
2. Determine the sample space.
3. Determine the possible values of the random variables.
4. Assign probability values P(X) to each of the random variable.
5. Construct a probability histogram to describe the P(X).
Answer the following guide questions:
1. How many possible outcomes are there?
2. What composes the sample space?
3. How will you describe the histogram?
What is It
To give us a deeper understanding of the concept of the normal distribution,
let us learn more about its properties.
The following are the properties that can be observed from the graph of a
normal distribution, also known as Gaussian distribution.
1. The graph is a continuous curve and has a domain -∞ < X < ∞.
• This means that X may increase or decrease without bound.
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.
• As the x gets larger and larger in the positive direction, the tail of the
curve approaches but will never touch the horizontal axis. The same
thing when the x gets larger and larger in the negative direction.
3. The highest point on the curve occurs at x = µ (mean).
• The mean (µ) indicates the highest peak of the curve and is found at
the center.
• Take note that the mean is denoted by this symbol µ and the standard
deviation is denoted by this symbol .
5
•
The median and mode of the distribution are also found at the center
of the graph. This indicates that in a
normal distribution, the mean, median
and mode are equal.
4. The curve is symmetrical about the mean.
• This means that the curve will have
balanced proportions when cut in
halves and the area under the curve to
the right of mean (50%) is equal to the
area under the curve to the left of the mean (50%).
5. The total area in the normal distribution under the curve is equal to 1.
• Since the mean divides the curve into halves, 50% of the area is to the
right and 50% to its left having a total of 100% or 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.
• Note that each inflection point of
the normal curve is one standard deviation away from the mean.
7. Every normal curve corresponds to the “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.
6
Consider the following 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 the following:
a. P (39 < X < 51) = 68.3%
c. P (X > 45) = 50%
*Since the area covered is 1 standard
of the deviation to the left and to the right.
b. P (33 < X < 63) = 97.55%
* Since the area covered is half
curve.
d. P (X < 39) = 15.85%
7
What’s More
Directions: Read the following statements carefully. Write ND if the statement
describes a characteristic of a normal distribution, and NND if it does not describe a
characteristic of a normal distribution. Write your answers on a separate sheet of
paper.
1. The curve of the distribution is bell-shaped.
2. In a normal distribution, the mean, median and mode are of equal values.
3. The normal curve gradually gets closer and closer to 0 on one side.
4. The curve is symmetrical about the mean.
5. The distance between the two inflection points of the normal curve is equal to the
value of the mean.
6. A normal distribution has a mean that is also equal to the standard deviation.
7. The two parameters of the normal distribution are the mean and the standard
deviation.
8. The normal curve can be described as asymptotic.
9. Two standard deviations away from the left and right of the mean is equal to
68.3%.
10. The area under the curve bounded by the x-axis is equal to 1.
What I Have Learned
Now that you have learned the concept of normal distribution, you may
proceed to the next activity.
Direction: Complete the given diagram below by filling up the necessary details
about normal distribution.
List 5 properties of normal distribution
NORMAL DISTRIBUTION
State the empirical rule.
Construct a normal curve.
8
What I Can Do
Directions: Make a sketch for each of the 3 areas under the normal curve as
stated in the empirical rule. Using a mosaic art, shade the area that corresponds to
the area under the normal curve. You may use eggshells, old magazines, dried leaves
or any materials available at home.
Assessment
Directions: Read the instructions given and write your answers on your
answer sheets.
A. Multiple Choice. Choose the letter of the best answer.
1. What is another name for normal distribution?
A. Gaussian distribution
B. Poisson distribution
C. Bernoulli’s distribution
D. Probability distribution
2. What is the total area in the distribution under the curve?
A. 0
B. 1
C. 2
D. 3
3. What marks the change in the curve’s concavity?
A. curve
B. inflection points
C. mean
D. standard deviation
4. Which value is found at the center of the normal curve?
A. mean
B. median
C. mode
D. all of the above
5. Which of the following is a parameter of normal distribution?
A. mean
B. standard deviation
C. mean and standard deviation
D. none of the above
9
6. Which of the following symbols is used to denote the mean?
A.
B. µ
C. α
D. ∞
7. Which of the following does not describe a normal curve?
A. asymptotic
B. bell-shaped
C. discrete
D. symmetrical about the mean
8. What percent of the area under a normal curve is within 2 standard
deviations?
A. 68.3%
B. 95.4%
C. 99.7%
D. 100%
9. What percent of the area under a normal curve is within 1 standard deviation?
A. 68.3%
B. 95.4%
C. 99.7%
D. 100%
10. What percent of the area under a normal curve is within 3 standard
deviations?
A. 68.3%
B. 95.4%
C. 99.7%
D. 100%
B. Read, analyze, and answer the given involving normal curve.
1. A continuous random variable X is normally distributed with a mean of 56.3
and standard deviation of 7.2. Illustrate a normal curve and find its
probability.
P (34.7 < X < 63.5)
10
Additional Activities
Distribution of Balls in a Quincunx
In this link, http://www.mathsisfun.com/data/quincunx.html, the webpage
will show a quincunx or "Galton Board" (named after Sir Francis Galton). This has a
triangular array of pegs wherein balls are dropped onto the top peg. Subsequently,
balls bounce their way down to the bottom where they are collected in little bins.
Each time a ball hits one of the pegs, it bounces either to the left or right with equal
probability. As a result, the number of pegs collecting in the bins form a bell-shaped
curve, especially as the number of rows (and bins) as well as the number of balls
increases.
0 1 2 3 4 5 6
(adopted from the Teaching Guide for Senior High School – Statistics and Probability,
pages 172-173)
11
Lesson
2
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.
What’s In
Direction: On your answer sheets, write the area corresponding to the
shaded part of the normal curve.
1.
2.
3.
Notes to the Teacher
This contains helpful tips and strategies that will help you in
guiding the learners.
12
What’s New
Directions: Observe the illustrations then answer the questions that follow.
Write your answers on a separate sheet of paper.
Figure A
Figure B
Guide Questions:
1. What is the mean?
Figure A: _______________ Figure B: _______________
2. What is the standard deviation?
Figure A: _______________ Figure B: _______________
3. What is the area of the shaded region?
Figure A: _______________ Figure B: _______________
4. What did you do to identify the area of the shaded region?
Figure A: _______________________
Figure B: ______________________
5. Did you use the same method? ________________________
13
What is It
As mentioned earlier, normal variable is standardized by setting the mean to
0 and standard deviation to 1. This is for the purpose of simplifying the process in
approximating areas for normal curves. As shown below is the formula used to
manually compute the approximate area.
f(z) =
1
√
𝑧2
𝑒− 2
2𝜋
However, this formula is seldom used because a table was created to
summarize the approximate areas under the standard normal curve and to further
simplify the process. This table of probabilities is known as the z- table.
The Z - Table
Let us get a closer look at the z-table. The outermost column and row
represent the z-values. The first two digits of the z-value are found in the leftmost
column and the last digit (hundredth place) is found on the first row.
Suppose the z-score is equal to 1.85, locate the first two digits 1.8 in the
leftmost column and the last digit, .05, can be located at the first row. Then find their
intersection which gives the corresponding area. Therefore, given z = 1.85, the area
is equal to 0.9678.
z
1.4
1.5
1.6
1.7
1.8
1.9
2.0
0.00
0.9192
0.9332
0.9452
0.9554
0.9641
0.9713
0.9772
0.01
0.9207
0.9345
0.9463
0.9564
0.9649
0.9719
0.9778
0.02
0.9222
0.9357
0.9474
0.9573
0.9656
0.9726
0.9783
0.03
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9788
0.04
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.05
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.06
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.07
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.08
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.09
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
Other examples are as follow:
1. Find the area that corresponds to z = 2.67
z
0.00
0.01
0.02
0.03
0.04
2.4
0.9918 0.9920 0.9922 0.9925 0.9927
2.5
0.9938 0.9940 0.9941 0.9943 0.9945
2.6
0.9953 0.9955 0.9956 0.9957 0.9959
2.7
0.9965 0.9966 0.9967 0.9968 0.9969
2.8
0.9974 0.9975 0.9976 0.9977 0.9977
2. Find the area that corresponds to z = 1.29
3. Find the area that corresponds to z = 3
14
Answer: 0.9962
0.05
0.9929
0.9946
0.9960
0.9970
0.9978
0.06
0.9931
0.9948
0.9961
0.9971
0.9979
0.07
0.08
0.09
0.9932 0.9934 0.9936
0.9949 0.9951 0.9952
0.9962 0.9963 0.9964
0.9972 0.9973 0.9974
0.9979 0.9980 0.9981
Answer: 0.9015
Answer: 0.9987
4. Find the area that corresponds to z = - 0.64
z
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
0.00
0.1841
0.2119
0.2420
0.2743
0.3085
0.3446
0.01
0.1814
0.2090
0.2389
0.2709
0.3050
0.3409
0.02
0.1788
0.2061
0.2358
0.2676
0.3015
0.3372
0.03
0.1762
0.2033
0.2327
0.2643
0.2981
0.3336
0.04
0.1736
0.2005
0.2296
0.2611
0.2946
0.3300
Answer: 0.2611
0.05
0.1711
0.1977
0.2266
0.2578
0.2912
0.3264
0.06
0.1685
0.1949
0.2236
0.2546
0.2877
0.3228
0.07
0.1660
0.1922
0.2206
0.2514
0.2843
0.3192
5. Find the area that corresponds to z = - 2.33
0.08
0.1635
0.1894
0.2177
0.2483
0.2810
0.3156
0.09
0.1611
0.1867
0.2148
0.2451
0.2776
0.3121
Answer: 0.0099
Note: The z-table used is the Cumulative Distribution Function (CDF) of the
Standard Normal Curve. Refer to the attachment on pages 19 and 20.
Now that you already know how to use the z-table to find the corresponding
area for the z-score, let us identify the regions under the nomal curve that
corresponds to different standard normal values. In order to find the regions, a
probability notation is used.
The probability notation P(a < Z < b) indicates that the z-value is between a
and b, P(Z > a) means z-value is above a and P(Z < a) means z-value is below a. It
would not matter whether we are considering P(Z < a) or P(Z ≤ a) or P(Z > a) or P(Z ≥
a). To illustrate, let us consider these examples:
1. 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.
Solution:
STEP 1: Draw a normal curve and locate the
z - scores and shade.
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
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) = 0.8801 or 88.01%
2. Compute the probability using the standard normal curve.
a. P(Z < 1.67) = ______________
Solution:
STEP 1: Draw a normal curve and locate the
z - score and shade.
STEP 2: Locate the corresponding area of the z - score in the z-table.
z = 1.67 has a corresponding area of 0.9525
15
STEP 3: If you are looking for a less than area, the area in the table is the
answer, therefore the P(Z < 1.67) = 0.9525 or 95.25%.
b. P(Z > -0.65) = ______________
Solution:
STEP 1: Draw a normal curve and locate the
z - score and shade.
STEP 2: Locate the corresponding area of the z - score in the z-table.
z = -0.65 has a corresponding area of 0.2578
STEP 3: If you are looking for a greater than area, the area in the table is
subtracted from 1, therefore, 1.0000 – 0.2578 = 0.7422 , and the
P(Z > -0.65) = 0.7422 or 74.22%
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?
Given a normal random variable X with mean (µ) and standrad deviation ( ),
each value of x of the variable can be transformed into z-scores using the formula,
𝑧=
𝑥− 𝜇
𝜎
where z = z- score or standard score
x = observed value
µ = mean
= standard deviation
To illustrate how the value of x can be converted in z-score, here are some
examples.
1. A random variable X has a mean of 6 and a standard deviation of 2. Find the
corresponding z-score for x = 11.
Given:
x = 11
Solution:
𝑥− 𝜇
𝑧= 𝜎
=
11 − 6
2
5
= 2
z = 2.5
µ=6
=2
Step 1: Write the formula.
Step 2: Substitute the given values.
Step 3: Perform the operations.
Step 4: Write the corresponding z-score.
16
2.
Given:
x = 20
Solution:
𝑥− 𝜇
𝑧= 𝜎
=
µ = 12
=3
Step 1: Write the formula.
20 − 12
3
Step 2: Substitute the given values.
8
3.
Given:
= 3 or 2.666666666667
Step 3: Perform the operations.
z = 2.67
Step 4: Write the corresponding z-score.
x = 18
µ = 28
=5
𝑥− 𝜇
𝜎
Step 1: Write the formula.
=
18 − 28
5
Step 2: Substitute the given values.
=
−10
5
Step 3: Perform the operations.
Solution: 𝑧 =
Step 4: Write the corresponding z-score.
z = -2
4. The scores in the summative test of 11- STEM B are normally distributed with
a mean of 65 and a standard deviation of 12. Find the probability that some
students got a score below 40.
Solution:
STEP 1: Convert the normal value in z-score.
Given: x = 40
Solution: 𝑧 =
µ = 65
= 12
𝑥− 𝜇
𝜎
=
40 − 65
12
=
−25
12
z = -2.08
Therefore, the P(X < 40) = P(Z < -2.08)
STEP 2: Draw a normal curve and
locate the z - score and shade.
STEP 3: Locate the corresponding area of the z - score in the z-table.
z = -2.08 has a corresponding area of 0.0188
STEP 4: If you are looking for a less than area, the area in the table is the
answer, therefore, the P(Z < -2.08) = 0.0188 or 1.88%.
5. The height (in meters) of grade 11 students in section A follows a normal
distribution with the mean 1.6 and a standard deviation of 0.3. Find the
pobability that students chosen at random has a height greater than 1.75.
Solution:
STEP 1: Convert the normal value in z - score.
17
Given: x = 1.75
Solution: 𝑧 =
µ = 1.6
= .3
𝑥− 𝜇
𝜎
=
1.75 − 1.6
.3
=
.15
.3
z = 0.5
Therefore, the P(X > 1.75) = P(Z > 0.5)
STEP 2: Draw a normal curve and
locate the z - score and shade.
STEP 3: Locate the corresponding area of the z - score in the z-table.
z = 0.5 has a corresponding area of 0.6915
STEP 4: If you are looking for a greater than area, the area in the table is
subtracted from 1, therefore, 1.0000 – 0.6915 = 0.3085 , and the P(Z >
0.5) = 0.3085 or 30.85%
The Percentile
A percentile is a measure used in statistics indicating the value below which
a given percentage of observations in a group of observations fall.
Imagine you took a standardized test and you scored 91 at the 89th percentile.
This means that 89% of the examiners scored lower than 91 and 11% scored higher
than 91. This explains that 89th percentile is located where 89% of the total
population lies below and 11% lies above that point. To illustrate the 89th percentile
of the normal curve here are the steps:
1. Express the given percentage as probability, remember 89% is the same as
0.8900.
2. Using the z-table (Cumulative Distribution Function (CDF) of the Standard
Normal Curve), locate the area of 0.8900.
3. There is no area corresponding exactly to 0.8900. It is between of 0.8888 with
a corresponding z - score of 1.22 and 0.8907 with a corresponding z - score of
1.23. The nearest value to 0.8900 is 0.8888 and therefore, the distribution
lies below z = 1.22.
4. Construct a normal curve and shade the region to the left of 1.22.
18
Cumulative Distribution Function (CDF) of the Standard Normal Curve
z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
0.00
0.5000
0.5398
0.5793
0.6179
0.6554
0.6915
0.7257
0.7580
0.7881
0.8159
0.8413
0.8643
0.8849
0.9032
0.9192
0.9332
0.9452
0.9554
0.9641
0.9713
0.9772
0.9821
0.9861
0.9893
0.9918
0.9938
0.9953
0.9965
0.9974
0.9981
0.9987
0.9990
0.9993
0.9995
0.9997
0.01
0.5040
0.5438
0.5832
0.6217
0.6591
0.6950
0.7291
0.7611
0.7910
0.8186
0.8438
0.8665
0.8869
0.9049
0.9207
0.9345
0.9463
0.9564
0.9649
0.9719
0.9778
0.9826
0.9864
0.9896
0.9920
0.9940
0.9955
0.9966
0.9975
0.9982
0.9987
0.9991
0.9993
0.9995
0.9997
0.02
0.5080
0.5478
0.5871
0.6255
0.6628
0.6985
0.7324
0.7642
0.7939
0.8212
0.8461
0.8686
0.8888
0.9066
0.9222
0.9357
0.9474
0.9573
0.9656
0.9726
0.9783
0.9830
0.9868
0.9898
0.9922
0.9941
0.9956
0.9967
0.9976
0.9982
0.9987
0.9991
0.9994
0.9995
0.9997
0.03
0.5120
0.5517
0.5910
0.6293
0.6664
0.7019
0.7357
0.7673
0.7967
0.8238
0.8485
0.8708
0.8907
0.9082
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9788
0.9834
0.9871
0.9901
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
0.9988
0.9991
0.9994
0.9996
0.9997
0.04
0.5160
0.5557
0.5948
0.6331
0.6700
0.7054
0.7389
0.7704
0.7995
0.8264
0.8508
0.8729
0.8925
0.9099
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.9838
0.9875
0.9904
0.9927
0.9945
0.9959
0.9969
0.9977
0.9984
0.9988
0.9992
0.9994
0.9996
0.9997
19
0.05
0.5199
0.5596
0.5987
0.6368
0.6736
0.7088
0.7422
0.7734
0.8023
0.8289
0.8531
0.8749
0.8944
0.9115
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.9842
0.9878
0.9906
0.9929
0.9946
0.9960
0.9970
0.9978
0.9984
0.9989
0.9992
0.9994
0.9996
0.9997
0.06
0.5239
0.5636
0.6026
0.6406
0.6772
0.7123
0.7454
0.7764
0.8051
0.8315
0.8554
0.8770
0.8962
0.9131
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.9846
0.9881
0.9909
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.9989
0.9992
0.9994
0.9996
0.9997
0.07
0.5279
0.5675
0.6064
0.6443
0.6808
0.7157
0.7486
0.7794
0.8078
0.8340
0.8577
0.8790
0.8980
0.9147
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.9850
0.9884
0.9911
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.9989
0.9992
0.9995
0.9996
0.9997
0.08
0.5319
0.5714
0.6103
0.6480
0.6844
0.7190
0.7517
0.7823
0.8106
0.8365
0.8599
0.8810
0.8997
0.9162
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.9854
0.9887
0.9913
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.9990
0.9993
0.9995
0.9996
0.9997
0.09
0.5359
0.5753
0.6141
0.6517
0.6879
0.7224
0.7549
0.7852
0.8133
0.8389
0.8621
0.8830
0.9015
0.9177
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
0.9857
0.9890
0.9916
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
0.9990
0.9993
0.9995
0.9997
0.9998
Cumulative Distribution Function (CDF) of the Standard Normal Curve
z
-3.4
-3.3
-3.2
-3.1
-3.0
-2.9
-2.8
-2.7
-2.6
-2.5
-2.4
-2.3
-2.2
-2.1
-2.0
-1.9
-1.8
-1.7
-1.6
-1.5
-1.4
-1.3
-1.2
-1.1
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.00
0.0003
0.0005
0.0007
0.0010
0.0013
0.0019
0.0026
0.0035
0.0047
0.0062
0.0082
0.0107
0.0139
0.0179
0.0228
0.0287
0.0359
0.0446
0.0548
0.0668
0.0808
0.0968
0.1151
0.1357
0.1587
0.1841
0.2119
0.2420
0.2743
0.3085
0.3446
0.3821
0.4207
0.4602
0.5000
0.01
0.0003
0.0005
0.0007
0.0009
0.0013
0.0018
0.0025
0.0034
0.0045
0.0060
0.0080
0.0104
0.0136
0.0174
0.0222
0.0281
0.0351
0.0436
0.0537
0.0655
0.0793
0.0951
0.1131
0.1335
0.1562
0.1814
0.2090
0.2389
0.2709
0.3050
0.3409
0.3783
0.4168
0.4562
0.4960
0.02
0.0003
0.0005
0.0006
0.0009
0.0013
0.0018
0.0024
0.0033
0.0044
0.0059
0.0078
0.0102
0.0132
0.0170
0.0217
0.0274
0.0344
0.0427
0.0526
0.0643
0.0778
0.0934
0.1112
0.1314
0.1539
0.1788
0.2061
0.2358
0.2676
0.3015
0.3372
0.3745
0.4129
0.4522
0.4920
0.03
0.0003
0.0004
0.0006
0.0009
0.0012
0.0017
0.0023
0.0032
0.0043
0.0057
0.0075
0.0099
0.0129
0.0166
0.0212
0.0268
0.0336
0.0418
0.0516
0.0630
0.0764
0.0918
0.1093
0.1292
0.1515
0.1762
0.2033
0.2327
0.2643
0.2981
0.3336
0.3707
0.4090
0.4483
0.4880
0.04
0.0003
0.0004
0.0006
0.0008
0.0012
0.0016
0.0023
0.0031
0.0041
0.0055
0.0073
0.0096
0.0125
0.0162
0.0207
0.0262
0.0329
0.0409
0.0505
0.0618
0.0749
0.0901
0.1075
0.1271
0.1492
0.1736
0.2005
0.2296
0.2611
0.2946
0.3300
0.3669
0.4052
0.4443
0.4840
20
0.05
0.0003
0.0004
0.0006
0.0008
0.0011
0.0016
0.0022
0.0030
0.0040
0.0054
0.0071
0.0094
0.0122
0.0158
0.0202
0.0256
0.0322
0.0401
0.0495
0.0606
0.0735
0.0885
0.1056
0.1251
0.1469
0.1711
0.1977
0.2266
0.2578
0.2912
0.3264
0.3632
0.4013
0.4404
0.4801
0.06
0.0003
0.0004
0.0006
0.0008
0.0011
0.0015
0.0021
0.0029
0.0039
0.0052
0.0069
0.0091
0.0119
0.0154
0.0197
0.0250
0.0314
0.0392
0.0485
0.0594
0.0721
0.0869
0.1038
0.1230
0.1446
0.1685
0.1949
0.2236
0.2546
0.2877
0.3228
0.3594
0.3974
0.4364
0.4761
0.07
0.0003
0.0004
0.0005
0.0008
0.0011
0.0015
0.0021
0.0028
0.0038
0.0051
0.0068
0.0089
0.0116
0.0150
0.0192
0.0244
0.0307
0.0384
0.0475
0.0582
0.0708
0.0853
0.1020
0.1210
0.1423
0.1660
0.1922
0.2206
0.2514
0.2843
0.3192
0.3557
0.3936
0.4325
0.4721
0.08
0.0003
0.0004
0.0005
0.0007
0.0010
0.0014
0.0020
0.0027
0.0037
0.0049
0.0066
0.0087
0.0113
0.0146
0.0188
0.0239
0.0301
0.0375
0.0465
0.0571
0.0694
0.0838
0.1003
0.1190
0.1401
0.1635
0.1894
0.2177
0.2483
0.2810
0.3156
0.3520
0.3897
0.4286
0.4681
0.09
0.0002
0.0003
0.0005
0.0007
0.0010
0.0014
0.0019
0.0026
0.0036
0.0048
0.0064
0.0084
0.0110
0.0143
0.0183
0.0233
0.0294
0.0367
0.0455
0.0559
0.0681
0.0823
0.0985
0.1170
0.1379
0.1611
0.1867
0.2148
0.2451
0.2776
0.3121
0.3483
0.3859
0.4247
0.4641
What’s More
Directions: Read the instructions given and write your answers on a separate
sheet of paper.
A. Complete the table by converting the given values into z-scores. Then find
the corresponding area using the z-table.
Given
Z- score
1. x = 28
µ = 16
=5
2. x = 68
µ = 75
=5
3. x = 1.72
µ = 1.6
= 0.2
4. x = 24
µ = 38
=8
5. x = 50
µ = 45
=6
Approximate area
B. Compute the following probabilities using the standard normal curve.
Construct a curve then shade the region corresponding to the area.
1. P(Z > -1.53)
2. P(Z < 2.89)
3. P(-1.65 < Z < 2.15)
21
What I Have Learned
Now that you have learned the concept of standard normal distribution, you
may proceed to the next activity.
Direction: Complete the given diagram below by filling up the necessary details
about the steps in finding the probability using the standard normal curve and
finding the z-scores.
HOW TO FIND THE PROBABILITY USING
THE STANDARD NORMAL CURVE
HOW TO FIND THE Z-SCORE
STEP 1
STEP 1
STEP 2
STEP 2
STEP 3
STEP 3
STEP 4
What I Can Do
Directions: Analyze the problem below by providing a comprehensive answer
from the given question. Write the complete solution on your answer sheets.
1. The mean score and the standard deviation in the Statistics test are
respectively equal to 80 and 2.5, whereas in the Mathematics test the mean
score is 70 and the standard deviation is 2. If Liza got a score of 85 in Statistics
and a score or 75 in Mathematics in which subject is her standing better
assuming normality in both subjects?
22
Assessment
A. Multiple Choice.
Directions: Choose the best answer and write the chosen letter on your answer
sheets.
1. Which of the following denotes the standard normal distribution?
A. A
B. X
C. Y
D. Z
2. Which of the following describes the standard normal distribution?
A. has a mean of zero (0) and a standard deviation of 1.
B. has a mean of 1 and a variance of zero (0).
C. has an area equal to 0.5.
D. cannot be used to approximate discrete probability distributions.
3. What is the formula in finding the z-score?
𝑥− 𝜇
A. 𝑧 = 𝜎
𝜇−𝜎
𝑥
𝜎− 𝜇
𝑧= 𝑥
𝑥− 𝜎
𝑧= 𝜇
B. 𝑧 =
C.
D.
4. A random variable X has a mean of 4 and a standard deviation of 2. What is
the corresponding z-score for x = 7?
A. 0.5
B. 1.0
C. 1.5
D. 2
5. What is the area if the z - score given is -1.83?
A. 0.0344
B. 0.0336
C. 0.0329
D. 0.0322
6. What is the z-value if the area is 0.9608?
A. -1.76
B. -1.77
C. 1.76
D. 1.77
23
7. Using the standard normal curve, what is the approximate area of P(1.63 < Z
< 2.79)?
A. 0.0049
B. 0.0490
C. 0.9484
D. 0.9974
8. The weights (in kg) of 11-STEM B follow a normal distribution and has a mean
of 50 and a standard deviation of 5. How many students have weights greater
than 57?
A. 1.4
B. 0.0808
C. 0.9192
D. 0.9332
9. What is the value of 97.5th percentile in a standard normal distribution?
A. -1.96
B. -1.97
C. 1.96
D. 1.97
10. What is the value of 67th percentile in a standard normal distribution?
A. -0.43
B. -0.44
C. 0.43
D. 0.44
B. Probability of Standard Normal Distribution
Directions: Compute the following probabilities using the standard normal
curve. On your answer sheets, construct a curve then shade the region
corresponding to the area.
1. P(Z > 1.65)
2. P(Z < 2.36)
Additional Activities
Directions: Analyze the problem below by providing a comprehensive answer
from the given question. Write the complete solution on your answer sheets.
1. The results of a nationwide aptitude test in mathematics are normally
distributed with mean = 63 and standard deviation = 5. What is the 97.5th
percentile?
24
What I Know
1. A
2. A
3. B
4. C
5. A
6. B
7. B
8. D
9. A
10.C
11.B
12.A
13.A
14.A
15.B
Assessment
Lesson 1 - A
1. A
2. B
3. B
4. D
5. C
6. B
7. C
8. B
9. A
10.C
25
What's In
Lesson 1
1. Agree
2. Agree
3. Disagree
4. Agree
5. Disagree
What's New
Lesson 1
1. 16 possible
outcomes
2. {0, 1, 2, 3, 4}
What's More
Lesson 1
1. ND
2. ND
3. NND
4. ND
5. NND
6. NND
7. ND
8. ND
9. NND
10. ND
3. The graph forms
a curve.
What’s In
Lesson 2
1. 68.3%
2. 95.4%
3. 99.7%
What’s New
1. Figure A: 45
Figure B: 0
2. Figure A: 6
Figure B: 1
3. Figure A: 68.3%
Figure B: Answers
may vary
4. Figure A: By
empirical rule
Figure B: By using
the z-table
5. No
B.
1. 84%
Answer Key
26
What's More
Assessment
Lesson 2-A
1. z-value = 2.4
area = 0.9918
2. z-value =-1.4
area = 0.0808
3. z-value = 0.6
area = 0.7257
4. z-value =-1.75
area = 0.0401
5. z-value=0.83
area = 0.7967
Lesson 2 - A
1. D
2. A
3. A
4. C
5. B
6. C
7. B
8. B
9. C
10. D
Lesson 2- B
1. 0.9370
B.
1. 0.0495 or 4.95%
2. 0.9909 or 99.09%
2. 0.9981
3. 0.9347
References
Commission on Higher Education. (2016). Teaching Guide for Senior High School:
Statistics and Probability Core Subject. Quezon City: CHED.
Canlapan, Raymond B. (2016). Statistics and Probability. Diwa Learning Systems
Inc. Makati City, Philippines.
27
For inquiries or feedback, please write or call:
Department of Education – Region III,
Schools Division of Bataan - Curriculum Implementation Division
Learning Resources Management and Development Section (LRMDS)
Provincial Capitol Compound, Balanga City, Bataan
Telefax: (047) 237-2102
Email Address: bataan@deped.gov.ph