Polytechnic University of the Philippines
______________________________________________
Module 4: Mathematics in the Modern World
Topic: Statistics
Objectives
Having successfully completed this module, you will be able to:
1.
2.
3.
4.
5.
Identify the uses and importance of Statistics.
Formulate statistical instrument
Organize data in frequency table.
Illustrate and calculate the measures of central tendency
Illustrate and calculate the measures of variability
Essential Questions
With regard to understanding, these are your guide questions.
1.
2.
3.
4.
What are the uses and importance of statistics?
How do you formulate statistical instrument and how do you organize data?
What are the measures of central tendency? How to compute and interpret them?
What are the measures of variability? How to compute and interpret them?
Essential Learning
Section 1.
Introduction to Statistics
A. Importance of Statistics
Activity 1
Measuring the arm span: Stretch out both arms and
measure the length from the tip of a middle finger to
the tip of the other middle finger.
Questions to Ponder:
Do you think students in this class have different arm spans? How many in this class have the
same arm spans? What is the most common measure of arm spans? To answer these questions,
you will to do the following:
1
Instructions:
a. Using a tape measure or a meter stick, measure your individual arm span. Use the centimeter
(cm) unit of length. Round off measures to the nearest cm.
b. On the board, write your measures individually.
Guided Questions:
a. What do these numbers represent?
b. Can we get clear and precise information immediately as we look at these numbers?
c. How can we make these numbers meaningful for anyone who does not know about the
description of these numbers?
d. Based on your activity, what is the meaning of Statistics?
Activity 2
The teacher prepares five pictures related to the topic in which the task of the learner is to describe
it. They can also cite instances that are related with the picture. This activity is individual but the
picture is assigned to each group.
1.
2.
3.
4.
Guided Questions:
a.
What does the picture imply?
b.
How can you relate these pictures to our lesson?
2
Statistics is the science of collection, organization, analysis, and interpretation of data.
Importance of Statistics
a. Business
• Production Planning
• Quality of products
• Location of business
• Advertising of products
• Financial and Capital resources
b. Banking
• Cash flow
• Interest rates
• Lending feasibility
• Estimation on the number of depositors
c. Economics
• Imports and exports
• Inflation rates
• Per capita income
• Law of Supply and Demand
d. Education
• Assessments on student’s performance
• Planning for educational design and curriculum
• Policy planning for school system
e. Mathematics
• Dispersion
• Precision in describing results on measurements
• Estimation of Values
Fill in the blanks. Explain the importance of Statistics by completing the statement below.
a. In weather forecast, statistics help us _____________________________________
__________________________________________________________________________
b. In predicting diseases, statistics helps you ___________________________________
______________________________________________________________________
3
c. In consuming goods, statistics may help you _________________________________
______________________________________________________________________
d. Government agencies use statistics to make _________________________________
______________________________________________________________________
e. In political campaign, statistics help us to ____________________________________
______________________________________________________________________
Match each field of study under Column A to its importance of statistics under column B.
Column A
1. Education
2. Psychology
Column B
a. Monitor status of customers,
employees, orders, and
production.
3. Business and Economics
b. Study of the size, vital
4. Medicine
5. Demographics
characteristics of the population,
and how they might change over
time.
c. Determine attitudinal patterns,
the causes and effects of
misbehavior.
d. Assess students’ performance
and correlate factors affecting
teaching and learning
processes.
e. Collect information about patients
and diseases and to make
decisions about the use of new
drugs or treatment.
4
B. Formulating Simple Statistical Instrument
Data are collected from different sectors such as business, education, medicine, etc.
A leading newspaper conducted a survey on honesty. Below are the five questions asked to the
readers.
HOW HONEST CAN YOU BE?
1. You found someone’s wallet in the canteen, what would you do?
a. Return the wallet to the owner.
b. Return the wallet but keep the money.
c. Keep the wallet and the money.
2. You are mistakenly given 50 pesos extra change when you buy a notebook from the
school’s book store, what would you do?
a. Return the 50 pesos to the cashier
b. Keep the money
3. An extra 10 points mistakenly added to your score in the examination made you pass.
Would you report it to your teacher?
A. Yes
b. No
4. You know that your teacher is not that strict during examination. Would you cheat on
the exam?
a. Yes
b. No
5. Are you honest most of the time?
a. Yes
b. No
1. What is the population for this honest survey? What is the Sample?
2. Suppose 200 students in your school complete the survey and 180 students answer “No” to
Question 4, what percent of the respondents said they would not cheat on the exam?
Situation # 1
Mrs. Rivera wants to find the number of students in her advisory class who have birthdays falling
on each of the 12 months. She collects the data in the following manner.
a. Check the date of birth of each student in the class registry.
b. Ask the students to raise their hands if their birthdays fall in a certain month.
c. Make a form and ask the students to write their birthdates and return the form to her.
5
Situation # 2
A restaurant manager requests her customers to fill out this questionnaire:
BLUE RIBBON RESTAURANT
Customer Satisfaction Survey
1. How would you rate our food?
Very Good
Improvement
Good
Okay
Bad
Needs
Okay
Bad
Needs
2. How would you rate our service?
Very Good
Improvement
Good
Questions:
a. What are the ways/methods of collecting data used in each situation?
b. How would you know which method to use in a given situation?
c. How does Statistics help us in our decision-making?
Data is a collection of facts or information. They may be gathered using the following
methods.
1.
Conducting surveys –
a. interview method is done when a person solicits information from another person (faceto-face)
b. questionnaire method is done through the use of printed questions regarding a certain
matter (pen and paper)
2.
Observing the Outcomes of Events/ Observation- the person who gathers data
is called investigator while the person/object being observed is called the subject.
3.
Experimentation is used by physicists and behavioral scientists in collecting data.
4.
Reading Statistical Publication/ Registration Method- refers to continuous,
permanent, compulsory recording of the occurrence of vital events together with certain
identifying or descriptive characteristics concerning them
6
Match Column A with column B. Write the letters of your answer.
A
B
1. Data collected using face-to-face interviews or
Written questionnaires
a. data
b. survey
2. A small part of a group chosen to represent the
whole group
c. experimentation
d. sample
3. The information collected
e. investigator
4. The person who gathers data using the observation method
5. The method used by the physicist in collecting data
C. Collecting/ Gathering Statistical Data
Collection- is the act or process of getting things from different places and bring them
together.
Data- information, statistics number, facts, figures, and records that usually to calculate,
analyze or plan something.
Data collection- is the process of preparing and collecting data.
Types of Data
a. Primary Data- is the data gathered directly from an original source.
b. Secondary Data- is the data gathered from secondary sources, such as books,
journals, magazines, or thesis of other researchers.
7
Different Methods of Gathering Data
Method
a. Interview
b. Observation
c. Questionnaire
d. Experimentation
e. Registration or Census
Definition
Is a direct method of
gathering data because the
data came directly from the
source.
Makes use of the different
human senses in gathering
information.
Referred as indirect method of
gathering data because this
makes use of written questions
to be answered by
the respondent.
Is usually conducted in
laboratories where specimens
are subjected to some aspects
of control to find out
cause and effect
relationships.
Requires the enactment of law
to take effect because it needs
the participation of a large, if
not the entire
population.
Sources of Data:
 Journals
 Newspapers
 Research papers
 Thesis or dissertations
Data Presentations:
a. Textual form- data are incorporated in the text of the report.
b. Tabular form- data are presented in rows and columns.
Graphical form- data are presented through graphs and
diagrams.
8
Directions: Identify which method is appropriate in the following statistical study.
a. Student’s allowance per week.
b. List of voters in a certain barangay.
c. Heights of Grade 7 students.
d. The most popular television program.
e. Results of entrance examination in a certain school.
Statistics – is a branch of mathematics that deals with the interpretation and analysis of
numerical data.
Primary data – information gathered directly from the source.
Secondary data – information gathered indirectly from the source.
Different methods of gathering data
Interview, Observation, Experimentation, Questionnaire,
Registration or Census
Direction: Tell whether each statement is true or false.
1. We can collect primary data by the use of observation.
2. Secondary data is the information that we can collect from the registration or census method.
3. Interview method makes use of the different human senses in gathering information.
4. Tabular form is a data presentation which the information is presented in rows and columns.
5. Experimentation method is usually conducted in laboratories where specimens are subjected
to find out the cause and effect relationship.
9
D. Organizing Data in Frequency Table
Do you know in what month most babies are born? Let’s do a mini survey in your class. Gather
the birth months of everyone in the class and organize the data in the table.
POPULAR BIRTH MONTHS
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sept
Oct
Nov
Dec
Number of
students
Guide Questions:
1. In which month were most students in the class born?
2. In which month were the least number of students in the class born?
3. What can you infer from data?
Frequency distribution Table is a systematic way of presenting data using a table. The data
are group into different intervals or number of classes assigned by the researcher. Usually, the
ideal number of classes is from 5 to 20 only
Mrs. Angon a Mathematics Teacher in Rosario National High School give a long
quiz to her students. Given the set of scores of 60 students in a 50 item test,
construct a frequency distribution table.
34
15
29
23
33
40
28
42
30
31
26
48
43
33
43
38
24
23
21
21
50
32
43
26
12
18
23
43
45
34
50
17
28
12
30
12
26
46
14
18
18
19
33
10
31
10
11
24
18
13
20
25
23
25
28
38
17
19
21
29
10
Steps in Constructing a Frequency Distribution Table
1. Choose the number of classes of the distribution. In the given set of scores, use 8 as the number
of classes or intervals. (Class interval = 8)
2. Get the range or the difference between the highest and the lowest values. Range = highest
score – lowest score
= 50 – 10
= 40
3. Solve the class width or class size by dividing the range by the number of classes or intervals.
Class size = range = 40 = 5
Class interval 8
4. Use the lowest score as the starting point if the class size is even. If the class size is odd, use
the multiple of the class size, which is less than or equal to the lowest score, as the starting point
or the first lower limit. In the example, the class size is five and the lowest score is 10, which is a
multiple of class size. Therefore, you use the lowest score as the starting point.
5. Determine the next lower limit by adding the class size.
6. The upper limit of the first interval is determined by subtracting one from the second lower value.
Repeat the process to complete the intervals.
7. The lower class limit is the lowest value within the interval, whereas the upper class limit is the
lower class limit and 14 is the upper class limit.
8. Get the tally of each score.
9. Determine the corresponding number in each tally. The number of times the value appears in
the distribution is called the frequency.
Class Interval
(Scores)
10 – 14
15 – 19
20 – 24
25 – 29
30 – 34
35 – 39
40 – 44
45 – 49
50 – 54
Tally
IIII – III
IIII – IIII
IIII – IIII
IIII – IIII
IIII – IIII
II
IIII - I
III
II
Frequency
Number of Students
8
9
10
10
10
2
6
3
2
11
Analysis
a. How did you find the activity?
b. What is the importance of frequency distribution table?
c. Based on the given activity, how can we construct a frequency distribution?
1. A survey was taken on Costa Verde. In each of 20 homes, people were asked how many
cars were registered to their households. The results were recorded as follows:
1, 2, 1, 0, 3, 4, 0, 1, 1, 1, 2, 2, 3, 2, 3, 2, 1, 4, 0, 0
2. One of the Companies in EPZA are producing batteries. Thirty AA batteries were tested to
determine how long they would last. The results, to the nearest minute, were recorded as
follows:
423, 369, 387, 411, 393, 394, 371, 377, 389, 409, 392, 408, 431, 401, 363,
391, 405, 382, 400, 381, 399, 415, 428, 422, 396, 372, 410, 419, 386, 39
1. One of the Grade 7 teachers of Rosario National High School conduct a test. Here are the
results of the 60 – item test of a class. Construct a Frequency distribution.
35, 28, 34, 8, 41, 40, 43, 13, 29, 35, 46, 39, 21, 19, 31
33, 39, 51, 57, 45, 18, 24, 44, 36, 48, 37, 32, 38, 29, 36
2. The following are test scores of Section 1. Construct a suitable
frequency table. Use intervals of width 6.
14
15
30
19
10
18
26
30
10
15
15
28
10
30
34
40
20
43
20
30
10
22
36
36
12
Section 2. Measures of Central Tendency
1. The grades in Mathematics of Grade 7 students are 82, 85, 79, 78, 89, 87, 88, 89,
75, and 77. What is their average grade?
2. The number of books loaned from the library during each day of the week were 36,
31, 24, 45, and 50. What is the median ?
3. The sizes of 15 classes selected at random are: 40, 39, 42, 48, 45, 46, 42, 49, 43,
49, 43, 43, 42, 41, 38, 42, and 47. What is the mode?
Developmental Activity
The set of data shows a score of 35 students in their periodical test.
34
21
19
21
19
35
20
17
20
17
40
19
18
18
29
40
34
15
17
45
48
45
16
10
50
21
21
20
45
48
9
20
28
48
25
Motive Questions:
1. What score is typical to the group of students? Why?
2. What score frequently appears?
3. What score appears to be in the middle? How many students fall below this score?
Mean is commonly referred to as the average of all values.
To compute for the mean, add all the scores and divide the sum by the
number of scores. It is the easiest “average” to compute.
The most frequent score/s in the given set of data is called the mode.
It is also an “average” score. A data set may have two modes (and hence the data
set is called bimodal).
The median is also an “average” score. It is the middle score in the list after the
scores are arranged in decreasing or increasing order.
13
Independent Practice
1. The following sets of data show the height (in centimeters) of two groups of boys playing
basketball at San Pedro town plaza.
Group A: 135
Group B: 167
136 140
136 119
150
136
134
160
129
178
126
126
130
140
a. Compute for the mean
b. What information can you get from these two values?
2. The following sets of data show the weekly income (in peso) of
ten households living in two different barangays in the City of San
Pedro, Laguna.
Brgy. Poblacion A: 150 1500 1700 1800 3000 2100 1700 1500
1750 1200
Brgy. Pacita : 1000 1200 1200 1150 1800 1800 2000
1470
8000
a. Compute for the mean and median for each Barangay.
b. What information can we get from these values?
c. Why do you think the median is more appropriate than the mean?
3. The advance algebra class of Miss. Dela Cruz took an achievement test. The results are
shown below:
84
75
76
83
82
76
88
77
90
86
80
81
82
89
78
79
89
84
80
81
86
84
90
79
85
81
79
78
79
81
82
92
86
82
78
87
90
83
93
84
a. What score is typical to the group of students? Why?
b. What score frequently appears?
c. What score appears to be in the middle? How many students fall below this score?
14
Solve the following problems:
1. Mario took four examinations in a science class. His scores are 48, 65, 78, and 79.
Which measure is more appropriate to use in order to determine how well Mario is
performing in science?
2. A beauty pageant cast their votes using social media, by counting the number of likes on
it . If there are 15 candidates, which of mean, median and mode is more appropriate to
use? Explain your answer.
3. The median for 10, 9, y, 12, and 6 is y. Find the possible values of y, given that the
values are whole numbers.
4. The mean of fifteen numbers is 30 and the mean of ten numbers is 25. What is the mean
of all the twenty-five numbers?
5. Given the set of numbers N = { 7, 9, 10, 14, 8, 16, 13}. When a number x is added to the
set, the new mean is 12. Calculate the value of x.
A. Calculating the measures of Central Tendency of Ungrouped and Grouped Data.
The students will conduct a survey to one another. They shall ask their classmates one by one
of the most preferred social networking site they use now a days. These are the following:
Social Networking Site
Facebook
Instagram
Twitter
Snapchat
Viber
Frequency
After this, they will answer the questions given by the teacher.
1. Arrange the data in order
2. Get the average
3. Get the middle data
4. Which number appeared the most?
15
Follow up questions:
1. How will you arrange the data?
2. Can you formulate an equation to get the average?
3. How will you find the middle data?
4. What is the highest repeating number?
Watch the video clips
www.youtube.com/watch?v=QzcgSCmWcVo
https://.youtube.com/watch?v=abYUPzRMzcQ
Analysis
Answer the following questions:
1. What do you think is the measures of central tendency all about?
2. What are the measurement shown in the video?
3. How will you describe the results shown in the video?
4. What have you noticed in data in first and second video?
5. What is the formula in finding those measures?
Guided Practice
Directions: Ungrouped Data
Calculate the measures of central tendency of the following.
1. A researcher collects data on the ages of recipients of doctoral degree in science and
engineering, and his study yields the following.
37
37
24
28
43
44
36
41
33
27
Find the mean.
Solution: The mean is determined by the sum of the ages and then dividing by the total number
of recipients.
Mean = 37+37+24+28+43+44+36+41+33+27
10
x=
350/10
= 35
therefore, then = 35.
16
2. Seven mothers were selected and given a blood pressure check. Their systolic pressures
were recorded below.
135
121
119
116
130
121
131
121
130
131
135
Find their median.
Solution: Arrange the data in increasing order
116
119
121
Select the middle value. Therefore the median is 121.
3. Eight novels were randomly selected and the numbers of pages were recorded as
follows:.
415
398
402
400
420
415
407
425
407
415
415
420
425
Find the median.
Solution: Arrange the data in order.
398
400
402
Since the middle point falls halfway between 407 and 415, find the median by getting the mean
of these two values.
Median =
407+415
2
= 411, therefore the median is 411
4. The sizes of 15 classes selected at random are:
40
39
42
48
45
46
42
49
43
42
41
44
38
42
47
42
42
42
43
44
45
46
47
48
49
Find the mode.
Solution: Arrange the data in order.
38
39
40
41
42
The mode is 42 because it is the measure that occurs the most number of times.
5. The sizes of 15 families in a chosen barangay chosen at random are as follows:
3
4
4
5
6
6
6
7
7
7
8
8
8
10
12
The modes are 6, 7, and 8. The distribution is multimodal.
17
B. Directions: Grouped Data
Calculate the measures of central tendency of the following.
1. Calculate the mean grade of 50 college students in mathematics in the modern world.
Class Interval
90-94
85-89
80-84
75-79
70-74
f
7
13
16
8
6
X (midpoint)
92
87
82
77
72
n=50
fx
644
1131
1312
616
432
∑ fx = 4135
Solution:
Step 1: Find the midpoint of each grade, and place the values in the third column.
(90+94)/2 = 92,
85+89/2 = 87,
(80+84)/2 = 82
etc.
Step 2: Next, multiply the midpoint by the frequency for each class, and place the results in the
fourth column.
(92)(7) = 644,
(87)(13) = 1131,
(82)(16) = 1312
etc.
Step 3: Find the sum of the fourth column.
∑ fx = 4135
Step 4: Divide the sum by n=number of items in grouped
Mean =
∑ fs
𝑛
=
4135
50
= 82.7 or 83
Therefore the mean grade is 82.7 or 83.
2. Calculate the median of the score of college students in mathematics classes of Mr
Macandog.
Class
boundaries
95-99
Frequency
5
Midpoints
94.5
Cumulative
frequency
100
18
90-94
85-89
80-84
75-79
70-74
65-69
60-64
i=5
11
17
25
20
12
89.5
84.5
79.5
74.5
69.5
64.5
59.5
7
3
95
84
67
42
22
10
3
n=100
Solution:
Step1: Make a table of cumulative frequency.
Class
boundaries
95-99
90-94
85-89
80-84
75-79
70-74
65-69
60-64
i=5
Frequency
Midpoints
5
11
17
25
20
12
7
3
n=100
94.5
89.5
84.5
79.5
74.5
69.5
64.5
59.5
Cumulative
frequency
100
95
84
67
42
22
10
3
Step2: Divide n, number of frequency by 2, to get the halfway Point.
n = 100 /2 = 50 is the halfway point
Step3: Locate the median class in the cumulative frequency Column.
Find the class that contains the 50th value by using the cumulative frequency distribution. Since
50 is less than 67, then the median class is the 4th class.
Step4: Substitute in the formula
𝑛
−𝐹
)𝑖
𝑀𝑑 = 𝐿 + (2
𝑓
100
− 42
)5
𝑀𝑑 = 79.5 + ( 2
25
= 79.5 + (
50−42
25
)5
19
8
= 79.5 + ( ) 5
25
= 79.5 + 1.6
= 81.1
Therefore, the median is 81.1.This means that one-half of the students scored above 81.1 and
the other half scored below 81.1.
3. Calculate the mode of the ages of randomly selected residents of Tagaytay Highlands
28.5 exact lower limit of
the modal class
Ages (years)
79-70
60-69
59-50
49-40
39-30
29-20
19-10
9-0
i=10
Frequency
9
18
22
27
30
44
42
33
225
𝑑1 = 44 -30 = 14
𝑑2 = 44 -42 = 2
Solution: Using the formula:
𝑀𝑜 = 𝐿𝑚𝑜 + (
= 28.5 + (
𝑑1
)𝑖
𝑑1 + 𝑑2
14
) 10
14+2
14
= 28.5 + ( ) 10
16
= 28.5 + (0.875)10
= 𝟑𝟕. 𝟐𝟓
Therefore, the mode is 37.25
A. Directions: UNGROUPED DATA: Arrange in increasing order and calculates the mean,
median and mode/s of each of the following sets of data.
1. 5, 6, 6, 4, 8
2. 3, 5, 6, 7, 4
20
3. 1, 3, 2, 5, 9, 1, 7, 6, 6, 8
4. 5, 6, 0, 6, 8, 2, 9, 4, 6, 8, 2, 3
5. 10, 10, 12, 14, 15, 9, 13, 13, 12
6. 2, 2, 4, 5, 9, 8, 7, 5, 3, 1, 2, 1, 1
7. 11, 5, 5, 5, 6, 2, 12, 9, 3,
8. 8, 7, 9, 10, 8, 6, 5, 4, 3
9. 16, 24, 18, 23, 29, 8, 24, 25, 12
10. 84, 85, 84, 89, 87, 87, 86, 82, 88, 84, 87, 87
B. Directions: GROUPED DATA:
1. Calculate the measures of central tendency of Mid-year test scores of students in
Mathematics.
SCORE
41-45
36-40
31-35
26-30
21-25
16-20
FREQUENCY
1
8
8
14
7
2
2. Calculate the measures of central tendency of the weight of BSEd – 3M students.
Weight in kg
75-79
70-74
65-69
60-64
55-59
50-54
45-69
40-44
Frequency
1
4
10
14
21
15
14
1
3. Calculate the measures of central tendency of the following 50 scores in Mathematics
Test.
31
46
54
40
30
35
74
62
70
73
80
63
81
53
70
63
59
46
46
74
50
41
41
51
37
75
45
50
39
75
71
49
44
52
41
55
36
76
83
79
40
78
68
65
55
60
42
53
71
52
21
Section 3. Measures of Variability
Find the words you think are related in Statistics
The following are the daily wages of 8 factory workers of two garment factories, factory A and
factory B. Find the range of salaries in Peso (Php).
Factory A: 400, 450, 520, 380 482, 495, 575, 450
Factory B: 450, 400, 450, 480, 450, 450, 400, 672
Finding the range of wages:
Range = Highest wage − Lowest wage
Wage range in A = 575 − 380 = 195
Wage range in B = 672 − 400 = 272
Comparing the two wages, which factory has a more scattered wages? Factory B
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Nine students are taking remedial classes. One of their quizzes scores are 12, 17, 13, 18, 18,
15, 14, 17, 11. Their teacher wants to find how the students’ scores scattered. To do that he
used the average deviation instead of range to get a better approximation and he got 2.22.
Based on the given situation, describe how the scores of the students having 2.22 of average
deviation scattered.
The following are the raw scores of students in English quiz: 12, 6, 7, 3, 15, 10, 18, 5. Find the
mean.
Given the variance a2 = 23.75, we can say how far each score is far from
the mean.
Eljohn, Wilmark, and Ebo are comparing their scores in their Mathematics quizzes. They want to
find out whose scores is most consistent. Below are the cores of the three students.
Eljohn: 97, 92, 96, 95, 90
Wilmark:94, 94, 92, 94, 96
Ebo: 95, 94, 93, 96, 92
To do so, they computed for their standard deviation.
Eljohn: 2.6
Wilmark: 1.3
Ebo: 1.4
Interpretation: Since Eljohn’s standard deviation is 2.6 it means that most of his scores are
within 2.6 units from the mean. While 1.36 (Wilmark) and 1.4 (Ebo) standard deviation suggest
that most of their scores are within 1.6 and 1.4 units from the mean.
Conclusion: Wilmark’s scores are clustered closer to the mean. This shows that his scores are
the most consistent among Eljohn and Ebo.
The range of a set of data is the difference between the largest and smallest
values. It is the size of the smallest interval which contains all the data and
provides an indication of statistical dispersion.
Average deviation gives a better approximation than the range. However, it does
not lend itself readily to mathematical treatment for deeper analysis.
Variance measures how far each number in the set is from the mean.
The standard deviation provides some idea about the distribution of scores
around the mean (average). The smaller the standard deviation, the more narrow
the range between the lowest and highest scores or, more generally, that the
scores cluster closely to the average score
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Identify what is being described.
1. It is the size of the smallest interval which contains all the data and provides an indication of
statistical dispersion
2. Provides some idea about the distribution of scores around the mean (average)
3. It gives a better approximation than the range.
A. Calculating Measures of Variability
THUMBS UP OR THUMBS DOWN?
If the statement is correct, do the THUMBS UP. Otherwise,
THUMBS DOWN. Justify your answer.
1. If I have a set of numbers 2, 5,3,10 and 5, therefore the mean is 5.
2. The most common measures of variability are the range, mean, median and mode.
3. Variation is the amount of dispersion, or “spread” of the values in the set of data.
The mean, median and mode do not adequately give the description of the characteristics of a set
of data. To find out how the values are scattered on either side of the central values, variation is
needed. Measures of variability are the spread of values about the mean. Smaller dispersion of
scores arising from the comparison often indicates more consistency and more reliability. These are
the range, average deviation, variance and standard deviation.
a.
To find the range, get the difference between the largest value and smallest value.
R = Highest value – Lowest value
b. To compute the average deviation, first, find the mean for all
the cases, find the absolute difference between each score and the mean, and lastly, find the sum
of the differences and divide it by N.
𝐴. 𝐷. =
∑ │𝑥−𝑥│
𝑁
c. . The variance for ungrouped data is computed as:
∑ │𝑥 − 𝑥│
𝑆
𝑁
2
2
d. The standard deviation is simply the positive square root of the variance of the sample.
𝑆𝐷 = √𝑆 2
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Find the range, average deviation, variance and standard deviation.
Range:
R=H–L
Guyabano:
R = 25 – 5 = 20
Chico:
R = 25 – 5 = 20
Average Deviation:
𝐴. 𝐷. =
∑ │𝑥−𝑥│
𝑁
where:
A.D is the average deviation
x is the individual score
x is the mean; and
N is the number of scores
|x − x¯| is the absolute value of the deviation from the mean

Find the mean x¯.
Guyabano: x¯ =
∑s
=
5+8+11+12+15+17+19+23+25
9
N
Chico: x¯ =
∑s
=
5+7+15+15+15+17+17+19+25
9
• Find the absolute difference between each score and the mean
N
=15
Guyabano
Chico
|𝑥 − 𝑥¯| = |5 − 15|=10
|𝑥 − 𝑥¯| = |5 − 15|=10
= |8 − 15|=7
= |7 − 15|=8
= |11 − 15|=4
= |12 − 15|=3
= |15 − 15|=0
= |17 − 15|=2
= |19 − 15|=4
= |23 − 15|=8
= |25 − 15|=10
=15
= |15 − 15|=0
= |15 − 15|=0
= |15 − 15|=0
= |17 − 15|=2
= |17 − 15|=2
= |19 − 15|=4
= |25 − 15|=10
25

Find the sum of the absolute difference ∑|x − x¯|.
Guyabano: = 48
Chico: 36

Solve for the average deviation by dividing the result in step 3 by N.
Guyabano: = 5.33
Chico: 4
Variance: (a2)
∑ │𝑥 − 𝑥│
𝑆
𝑁
2
2


Compute the mean score
Complete the table below.
Guyabano

Chico
Compute the Variance.
Guyabano: = 39.78
Chico: = 32
Standard Deviation:(SD)
𝑆𝐷 = √𝑆 2

Just get the positive square root of the variance
Guyabano: = 6.31
Chico: 5.66
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If you were to choose from these two sections, which would you prefer? Why?
*The variability in the values of data in section Guyabano is greater than that in section Chico.
Note that the smaller dispersion of scores indicates more consistency and reliability
-Based from the activity, which section has the most consistent scores?
-Why do we need to compute the variation other than the mean?
-How to compute the range? average deviation? variance? standard deviation of a given set of
ungrouped data?
-What is the importance of measuring variability in a given set of data?
Find the range and standard deviation of each set of data.
a. {12, 13, 17, 22, 25, 26}
b. {10, 9, 6, 6, 7, 8, 8, 8, 8, 9}
c. {97, 92, 96, 95, 90}
Follow-up: Answer each of the following.
The scores received by Alden and Maine in ten math quizzes are as follows:
Alden: 4, 5, 3, 2, 2, 5, 5, 3, 5, 0
Maine: 5, 4, 4, 3, 3, 1, 4, 0, 5, 5
a. Compute the standard deviation.
b. Which student has the better grade point average?
c. Which student has the most consistent score?
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