Republic of the Philippines

Department of Education Regional Office IX, Zamboanga Peninsula

Zest for Progress

Zeal of Partnership

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MATHEMATICS 4th QUARTER – Module 8:

STATISTICS

Name of Learner: ___________________________

Grade & Section: ___________________________

Name of School: ___________________________

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Mathematics – Grade 7 Alternative Delivery Mode Quarter 4 - Module 8: Title: Statistics First Edition, 2020

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office wherein the work is created shall be necessary for exploitation of such work for profit.

Such agency or office may, among other things, impose as a condition the payment of royalty.

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respective copyright owners. The publisher and authors do not represent nor claim ownership

over them.

Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

Printed in the Philippines

Department of Education – Region IX, Zamboanga Peninsula

Office Address: Tiguma, Airport Road, Pagadian City

Telefax: (062) – 215 – 3751; 991 – 5975

E-mail Address: region9@deped.gov.ph

Development Team of the Module

Writer: Zorina F. Piňero

Editors: Haifa D. Ternida

Abdurauf J. Baldomero

Layout Artist: Francis Michael E. Francisco

Reviewers: EPS, Mathematics Vilma A. Brown, Ed. D.

Principal Salvador C. Bucoy, LMD

Management Team: SDS Roy C. Tuballa, EMD, JD, CESO VI

ASDS Jay S. Montealto, CESO VI

ASDS Norma T. Francisco, DM, CESE

EPS Mathematics Vilma A. Brown, Ed. D.

EPS LRMS Aida F. Coyme, Ed. D.

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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-by-step 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 can 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. 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.

This module was written as an aid in the basic statistics lesson of the fourth quarter of

Grade 7-Mathematics. The module follows a step-by-step approach to computational statistics

supported by examples and exercises. It covers the key concepts of statistical measures.

This module was designed to cater the academic needs of diverse learners in

achieving and improving the twin goals of mathematics in basic education levels which are

critical thinking and problem solving. The language used recognizes the vocabulary level of

grade 7 students. The lessons followed a developmentally sequenced teaching and learning

processes to meet the curriculum requirement.

After going through the module, you are expected to:

• Use appropriate statistical measures in analyzing and interpreting statistical data

(M7SP-IVj-1); and

• Draws conclusions from graphic and tabular data and measures of central

tendency and variability. (M7SP-IVj-2)

What I Know

Directions: Choose the letter that corresponds to your answer.

1. Which of the following refers to the branch of mathematics concerned with the techniques

by which information is collected, organized, analyzed and interpreted?

A. Geometry B. Algebra C. Statistics D. Trigonometry

What I Need to Know

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2. It is the average of sets of numbers ___.

A. Median B. Mean C. Mode D. Number

3. It is a graph which uses horizontal or vertical bars to represent information, making a direct

visual comparison of data ___.

A. Line graph B. Bar graph C. Circle graph D. Pictograph

4. Which measure of variability is considered most reliable?

A. Range C. Average deviation

B. Variance D. Standard Deviation

5. What is the median age of a group of employees whose ages are 36, 38, 28, 30, 26 and

25 years?

A. 25 B. 26 C. 29 D. 30

What’s In

Directions: Study the data below and answer the questions that follow.

The data shows the Math quiz scores of 20 grade 7 students. 5, 10, 9, 6, 6, 8, 8, 10, 7, 7, 5, 9, 6, 7, 7, 8, 8,7, 7, 6

1. What measure of central tendency is applicable to know what is the frequent score?

2. What measure of central tendency can be applied to get the general average of a student

quiz scores?

3. What measure of variability is used to find the difference between the highest and the

lowest values in a data set?

4. What is the median score of the data set?

5. What is the standard deviation of the set of data?

What’s New

Directions: Study the data set below and answer the question that follows.

The data below shows the math grades of two groups of students.

GROUP 1

80, 81, 83, 83, 83, 83, 85, 86

GROUP 2

70, 70, 80, 81, 81, 88, 95, 99

QUESTION: Solve for the standard deviation then tell which group performed better?

LESSON 1 Data

What is It

To analyze and interpret data, we often use the following statistical measures:

a. Measures of Central Tendency (Mean, Median, and Mode)

b. Measures of Variability (Range, Variance, and Standard Deviation)

ANALYZING DATA

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STATISTICAL

MEASURES DESCRIPTION

Mean

Mean is the average or the sum of numbers in a data set divided by the

total number of values in the data set. Mean is the balance point of a

distribution.

Median

Median is the number that falls in the middle position or the midpoint value

of a set of data where the values are arranged from smallest to largest or

vice versa. If there are two middle values, find the mean of the two middle

values.

Mode Mode is simply the value that appears most frequently in the set of data.

Range Range is the difference between the largest value and the smallest value.

Variance Variance is the mean squared deviation of the data set, meaning it is the

sum of the squared deviation of the mean divided by the total population.

Standard

Deviation Standard deviation is the square root of the variance.

In this lesson we will be looking for the MEAN, MEDIAN, MODE, RANGE and

STANDARD DEVIATION and we will describe the distribution using numbers.

SITUATION 1

A sales manager in a consumer electronic company is planning to promote one of his

two top salesmen to a supervisory position. To help determine who has a better performance,

the sales manager made a table and compared their sales profile over the last 7 months.

MONTH January February March April May June July TOTAL

Salesman A 9 11 7 7 5 39 6 84

Salesman B 10 12 12 10 11 12 12 79

ANALYSIS

STATISTICAL

MEASURES ILLUSTRATIVE EXAMPLE

Mean

If we compute the mean of both salesmen, we will have,

SALESMAN A: 𝒙 =𝟗+𝟏𝟏+𝟕+𝟕+𝟓+𝟑𝟗+𝟔

𝟕=

𝟖𝟒

𝟕= 𝟏𝟐

SALESMAN B: 𝒙 =𝟏𝟎+𝟏𝟐+𝟏𝟐+𝟏𝟎+𝟏𝟏+𝟏𝟐+𝟏𝟐

𝟕=

𝟕𝟗

𝟕= 𝟏𝟏. 𝟐𝟖 𝒐𝒓 𝟏𝟏

Analysis of data is usually done after all information has been collected,

organized and presented as a frequency distribution and the measures of

central tendency (mean, median and mode) have been computed.

ANALYSIS OF DATA

1. Develop a list of questions to keep you focused on the task.

2. Organize, summarize and manipulate data.

3. Format should maximize clarity and impact.

4. Label, refer to and explain all data in text.

5. Signal analysis and interpretation to the reader.

METHODS / TIPS IN ANALYZING, INTERPRETING AND PRESENTING DATA

MEASURE OF CENTRAL TENDENCY

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As we can see, Salesman A still has an edge with an average sale of

12 computers per month, while salesman B only has an average of 11

computers per month.

Mode

However, based on the given table, Salesman B seems to be a better

salesman in terms of consistent performance because he sold 12

computers more frequently which can be seen in their modes.

SALESMAN A: 𝒙 = 𝟕 SALESMAN B: 𝒙 = 𝟏𝟐

Median

If we find the median of both salesmen, we will have,

SALESMAN A: 5, 6, 7, 7, 9, 11, 39 so the median is 7

SALESMAN B: 10, 10, 11, 12, 12, 12, 12 so the median is 12

Summary

If we summarize the three measures of central tendency, we will have,

Mean Median Mode

Salesman A 12 7 7

Salesman B 11 12 12

Conclusion

We can conclude that Salesman B will be promoted although his mean

is less than that of salesman A. Salesman A has a mean of 12 only

because of his very high output for the month of June which is 39. On

the other hand, salesman B consistently sold 11 or 12 computers.

What’s More

Directions: Represent the following data in a table and answer the following questions.

DATA: List of students’ score/s in the last test.

Amy-90, Bill-95, Jay-100, Lisa-100 and Mary-80

1. What is the Range in the data?

2. How many score/s below 100?

3. What is the most frequent score/s?

4. What is the highest frequency score?

5. What is the less spread in the data?

What I Have Learned

Directions: Read the problem carefully and answer the question that follows.

SITUATION

The editorial department of

the publishing house consists of 8

members whose wages (in pesos)

are given in the table.

MEMBERS WAGES

Editorial-in-Chief 20,000

Supervising Editor 18,500

Assistant to the Editor 17,500

Editorial Assistant II 14,500

Editorial Assistant I 14,000

Artist 12,000

Encoder II 10,000

Encoder I 9,000

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QUESTION

1. Which of the three measures of central tendency is the most appropriate tool to describe

the wages in the publishing house? (Show complete solutions)

What’s In

Directions: Study the graph and answer the questions that follow.

1. How would you describe the shape of the distribution,

skewed left or skewed right?

2. Mark the approximate location of the mean and median.

3. Which inequality symbol (˃, ˂ , or ≈) will make the statement

correct in relation to the graph? Mean _____ Median

4. Which is the “best” measure of central tendency (mean or

median) to be used in describing the graph?

What’s New

Directions: Study the graph and answer the questions that follow.

A classroom of students each tossed a coin 16 times and recorded how many times

their coin landed on heads. The results for the class are shown below.

1. Approximately what is the mean?

2. Approximately what is the median?

3. What would happen to the mean if

several students reported getting 16

heads?

LESSON 2 STATISTICS

What is It

Statistic is a form of mathematical analysis that uses qualified models, representations

and plan for a given set of experimental data.

In this lesson we will be looking for the MEAN, MEDIAN, MODE, RANGE and

STANDARD DEVIATION and we will describe the distribution using graphs.

STATISTICS

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Skewed Right (Positively Skewed) - The mode is in the peak of

distribution (left of center), median is in the center of distribution,

mean is in the right tail of distribution.

Skewed Left (Negatively Skewed) – The mode is in the peak of

distribution (right of center), median is in the center of distribution,

mean is in the left tail of distribution.

Symmetrical Distribution – Mean, Median and Mode are all in

the same location in the distribution.

Outlier – when a data point is detached from the main cluster of

data

Frequency Distribution is a representation, either in a graphical or tabular format that

displays the number of observations within a given interval.

A measure of variability is a summary statistic that represents the amount of dispersion

in a dataset. To find the measure of variability, find the value that will describe the spread of

the set of data or the distance of the data from the mean.

The Variability or dispersion of a variable is often important in practice. For instance,

packaged food producers want to put a consistent amount of product into each package. The

variability in a stock’s price is important in pricing stock options.

There are many measures of variability, let’s start with the simple one the Range.

Consider this sample of 6 observations: 60, 75, 80, 85, 90, 95.

Range = 95 – 60 = 35

The range is pretty easy to calculate and interpret but is not a great measure of

variability the range doesn’t tell us anything about the spread of the values between the

maximum and the minimum.

Another measure of variability is the standard

deviation. What does the Standard Deviation even tell us?

The Standard Deviation tells us how close the values in

the data set are to the Mean. For example, a small

standard deviation indicates a small amount of variability

for a given data set, in other words there will be a lot of

values that closer to the mean which makes the

distribution less spread out.

In contrast, a high standard deviation indicates a high

amount of variability for a given data set, in other words there

will be a lot more values that are farther from the mean which

makes the distribution more spread out.

We can conclude that the variance is the average squared distance from the mean. The

unit of the variance or the squared of the unit of the variable, so if we are taking measurement

in meters then the variance unit of measure is squared, to get back to the same unit of the

variable, we often take the square root of the variance.

MEASURE OF VARIABILITY

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SITUATION 1

In an interview for a job the owner of the company tells you they have 17 employees

and the average salary is Php 50,000.00. The owner gives you a handout with the information

below and you examine it carefully.

Salaries (1000Php): 29, 31, 35,38,38,38,41,42,42,43,51,51,52,61,69,79,110

ANALYSIS

STATISTICAL

MEASURES ILLUSTRATIVE EXAMPLE

Mean If we compute the mean of the salaries, we will have,

𝒙 =𝟖𝟓𝟎,𝟎𝟎𝟎

𝟏𝟕= 𝟓𝟎, 𝟎𝟎𝟎

Median Since the data is already arranged in increasing order and there are odd

number of data, the median is 42, 000

Mode 38, 000 is the mode since it repeated three (3) times in the data set.

Graph The graph is right skewed.

Guide

Questions

1. What did you notice about the salaries?

- Most of the salaries fall around 30, 000 to 40, 000.

2. Is the owner telling the truth?

- Yes

3. What statistic would be more accurate than the mean in describing the

“average” employee’s salaries?

- Median

What I Can Do

Directions: Study the histogram and find indicated statistical measure.

a. Mean

b. Median

c. Mode

d. Range

e. Standard Deviation

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Directions: Choose the letter that corresponds to your answer.

1. If two or more measures appear the same number of times, then each value is ___.

a. Mean b. Median c. Mode d. Range

2. The measure of variation based on measuring how far each data value deviates or is

different from the mean is ___.

a. Range b. Standard Deviation c. Median d. Mean

3. The difference between the highest and lowest values in a set of data is ___

a. Range b. Mean c. Median d. Mode

4. The range, variance and standard deviation are ___

a. Measure of space c. measure of distance

b. Measure of variability d. Measure of central tendency

5. The middle number in an ordered set of data is ___

a. Mean b. Median c. Mode d. Range

6. A value that lies outside the given data set is ___

a. Right Skewed c. Outlier

b. Left Skewed d. Symmetric

7. Which of the following has a smaller spread of values?

a.

b.

c.

d.

8. If the data is roughly symmetric, the best measure of the center is the ___ .

a. Mean b. Median c. Mode d. Range

9. Which distribution occurs when the mean, median, and mode are in the same location?

a. Right Skewed c. Outlier

b. Left Skewed d. Symmetric

10. It helps find the middle or the average of a data set.

a. measure of central tendency c. Measure of variability

b. Measure of spread d. Measure of shape

Assessment

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Answer Key

References: Emmanuael P. Abuzo, Integrated Mathematics Textbook for first year Julieta G. Bernabe, Cecile M. De Leon, Advance Algebra, Trigonometry and Statistics Textbook for fourth year.

What I Know:

1. C 2. B 3. B 4. D 5. C

Lesson 1 What’s In:

1. Mode 3. Range 5. 1.42

2. Mean 4. 7

Lesson 2 What’s In:

1. skewed right

2.

3. >

4. Median

LESSON 1 What’s New: a. group 1 performed better, since its standard deviation is 1.80 which is less than the standard deviation of group 2 which is 10.6. It only shows that the scores of group 1 is closer to the mean than the group 2. LESSON 2 What’s New: 1.Mean is 8 2. Median is 8 3. The Mean would increase

What’s More:

1. 20 2. 3 3. 100 4. 100 5. 80

What I Have Learned:

Mean, since it uses all the data and best represents the average wages.

𝑥ҧ=115,500

8=14,437.50

What I Can Do:

a. 4 b. 4 c. no mode d. 6 e. 2.3

Assessment:

1.c 2.b 3. a 4. b 5. b 6. c 7. b 8.b 9. d 10.a

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I AM A FILIPINO

by Carlos P. Romulo

I am a Filipino – inheritor of a glorious past, hostage to the

uncertain future. As such, I must prove equal to a two-fold

task – the task of meeting my responsibility to the past, and

the task of performing my obligation to the future.

I am sprung from a hardy race – child many generations

removed of ancient Malayan pioneers. Across the centuries,

the memory comes rushing back to me: of brown-skinned

men putting out to sea in ships that were as frail as their hearts

were stout. Over the sea I see them come, borne upon the

billowing wave and the whistling wind, carried upon the

mighty swell of hope – hope in the free abundance of the new

land that was to be their home and their children’s forever.

This is the land they sought and found. Every inch of shore

that their eyes first set upon, every hill and mountain that

beckoned to them with a green and purple invitation, every

mile of rolling plain that their view encompassed, every river

and lake that promised a plentiful living and the fruitfulness

of commerce, is a hollowed spot to me.

By the strength of their hearts and hands, by every right of

law, human and divine, this land and all the appurtenances

thereof – the black and fertile soil, the seas and lakes and

rivers teeming with fish, the forests with their inexhaustible

wealth in wild and timber, the mountains with their bowels

swollen with minerals – the whole of this rich and happy land

has been for centuries without number, the land of my

fathers. This land I received in trust from them, and in trust

will pass it to my children, and so on until the world is no

more.

I am a Filipino. In my blood runs the immortal seed of heroes

– seed that flowered down the centuries in deeds of courage

and defiance. In my veins yet pulses the same hot blood that

sent Lapulapu to battle against the alien foe, that drove Diego

Silang and Dagohoy into rebellion against the foreign

oppressor.

That seed is immortal. It is the self-same seed that flowered

in the heart of Jose Rizal that morning in Bagumbayan when

a volley of shots put an end to all that was mortal of him and

made his spirit deathless forever; the same that flowered in

the hearts of Bonifacio in Balintawak, of Gregorio del Pilar

at Tirad Pass, of Antonio Luna at Calumpit, that bloomed in

flowers of frustration in the sad heart of Emilio Aguinaldo at

Palanan, and yet burst forth royally again in the proud heart

of Manuel L. Quezon when he stood at last on the threshold

of ancient Malacanang Palace, in the symbolic act of

possession and racial vindication. The seed I bear within me

is an immortal seed.

It is the mark of my manhood, the symbol of my dignity as

a human being. Like the seeds that were once buried in the

tomb of Tutankhamen many thousands of years ago, it shall

grow and flower and bear fruit again. It is the insigne of my

race, and my generation is but a stage in the unending

search of my people for freedom and happiness.

I am a Filipino, child of the marriage of the East and the

West. The East, with its languor and mysticism, its passivity

and endurance, was my mother, and my sire was the West

that came thundering across the seas with the Cross and

Sword and the Machine. I am of the East, an eager

participant in its struggles for liberation from the imperialist

yoke. But I know also that the East must awake from its

centuried sleep, shake off the lethargy that has bound its

limbs, and start moving where destiny awaits.

For I, too, am of the West, and the vigorous peoples of the

West have destroyed forever the peace and quiet that once

were ours. I can no longer live, a being apart from those

whose world now trembles to the roar of bomb and cannon

shot. For no man and no nation is an island, but a part of the

main, and there is no longer any East and West – only

individuals and nations making those momentous choices

that are the hinges upon which history revolves. At the

vanguard of progress in this part of the world I stand – a

forlorn figure in the eyes of some, but not one defeated and

lost. For through the thick, interlacing branches of habit and

custom above me I have seen the light of the sun, and I

know that it is good. I have seen the light of justice and

equality and freedom, my heart has been lifted by the vision

of democracy, and I shall not rest until my land and my

people shall have been blessed by these, beyond the power

of any man or nation to subvert or destroy.

I am a Filipino, and this is my inheritance. What pledge

shall I give that I may prove worthy of my inheritance? I

shall give the pledge that has come ringing down the

corridors of the centuries, and it shall be compounded of the

joyous cries of my Malayan forebears when first they saw

the contours of this land loom before their eyes, of the battle

cries that have resounded in every field of combat from

Mactan to Tirad Pass, of the voices of my people when they

sing:

“I am a Filipino born to freedom, and I shall not rest until

freedom shall have been added unto my inheritance—for

myself and my children and my children’s children—

forever.”