Measures of Central Tendency

Maria Diza C. Febrio BAYUGAN NATIONAL COMPREHENSIVE HIGH SCHOOL Bayugan City mariadizafebrio@gmail.comCreated by:Measures of Central Tendency(Ungrouped Data)

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Read the Instructions First

INTRODUCTIONDISCUSSIONS AND EXAMPLESOBJECTIVESEXERCISES AND CHALLENGES

This material is designed to help you solve the Measures of Central Tendency (Ungrouped): mean, median and mode with each examples and exercises.Just click the ff. icons at the bottom of the slides. - click this icon if you want to go the to the previous slide.- click this icon if you want to go to the next slide.- click this icon if you want to show the home page or the menu.3. Click the following at the homepage or at the menu. - it shows you the background of the topic. - it shows you what will be the outcome after learning this material. - it contains all about the topic with examples to deepen your understanding. - to measure your learning after you read the topics.4. In exercises, its just a multiple choice. Click the letter of your answer. Solutions also were provided if you got the correct answer. 5. Enjoy learning.

INSTRUCTIONS

INTRODUCTIONOBJECTIVESDISCUSSIONS AND EXAMPLESEXERCISES AND CHALLENGES

Measures of Central Tendencyare a key way to discuss and communicate with graphs. The term central tendency refers to the middle, or typical, value of a set of data, which is most commonly measured by using the three m's: mean, median, and mode. The mean, median, and mode are known as the measures of central tendency.

INTRODUCTION

a. mean b. medianc. modeThe learners can develop their understanding of the measures of central tendency ;

OBJECTIVESThis topic is best for Grade 7 and Grade 8 learners.

DISCUSSION

What is mean?

often called the average, of a numerical set of data, is simply the sum of the data values divided by the number of values. This is also referred to as the arithmetic mean. The mean is the balance point of a distribution.

DISCUSSION

What is mean?

Mean ( ) =sum of the values = the number of values

DISCUSSION

What is median? is the number that falls in the middle position once the data has been organized. Organized data means the numbers are arranged from smallest to largest or from largest to smallest.

DISCUSSION

How to compute the median?When the totals of the list are odd, the median is the middle entry in the list after sorting the list into increasing order. When the totals of the list are even, the median is equal to the sum of the two middle (after sorting the list into increasing order) numbers divided by two. Thus, remember to line up your values, the middle number is the median! Be sure to remember the odd and even rule.

DISCUSSION

What is mode? Themodeof a set of data is simply the value that appears most frequently in the set. If two or more values appear with the same frequency, each is a mode. The downside to using the mode as a measure of central tendency is that a set of data may have no mode, or it may have more than one mode. However, the same set of data will have only one mean and only one median.

DISCUSSION

What is mode?The wordmodalis often used when referring to the mode of a data set.If a data set has only one value that occurs most often, the set is calledunimodal.A data set that has two values that occur with the same greatest frequency is referred to asbimodal.When a set of data has more than two values that occur with the same greatest frequency, the set is calledmultimodal.

EXAMPLEA. Find the mean for the following list of values: 13, 18, 13, 14, 13, 16, 14, 21, 13Use the formula in computing the mean:Solution: Mean = sum of the values = the number of valuesMean = ( 13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) 9 = = 15 is the mean

EXAMPLEB. Stephen has been working on programming and updating a Web site for his company for the past 15 months. The following numbers represent the number of hours Stephen has worked on this Web site for each of the past 7 months: 24, 25, 33, 50, 53, 66, 78What is the mean (average) number of hours that Stephen worked on this Web site each month?

EXAMPLESolution:Use again the formula in solving.Mean = (24 + 25 + 33 + 50 + 53 + 66 + 78)7 = 3297 = 47 is the mean number of hours that Stephen worked on the website each month

EXAMPLEC. Look at these numbers and find the mean.

3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29The sum of these numbers is 330There are fifteen numbers.The mean is equal to 330 / 15 = 22

The mean of the above numbers is 22

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EXAMPLED . Clark scores in 4 summative tests are 84, 85, 89, and 90. What must his score be on the fifth test to get an average of 89?

Solution: Let x be his score on the fifth test Sum of the fifth scores = mean x 5 = 89 x 5 = 445

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EXAMPLE

So, 84+ 85+89+90+x = 445 348+x = 445 x = 445 348 = 97 score needs by Clark

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EXAMPLEE. Samuels grades in 6 subjects are shown at the table with corresponding units in different subjects. Find the General-Point Average.

SubjectNumber of UnitsGradeEnglish393Filipino387Math394Science592P.E.186Social Studies388

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EXAMPLESolution:Note that the subjects do not have the same number of units or weight. So, we need to get the weighted average.Grade point average (GPA) = Total Grades Total # of Units GPA = 93(3)+87(3)+94(3)+92(5)+86(1)+88(3)3+3+3+5+1+3 =279+261+282+460+86+264 18 = 1632 = 90.67 18

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F. Find the Median of12, 3, 5.Step 1. Put them in order (increasing order)3, 5, 12Step 2 . Since the number of data values is odd, the median will be found at the middle

The middle number is5, so the median is5.

EXAMPLE

G. Find the median of the following data:7, 9, 3, 4, 11, 1, 8, 6, 1, 4Step 1:Organize the data, or arrange the numbers from smallest to largest. 1, 1, 3, 4, 4, 6, 7, 8, 9, 11Step 2:Since the number of data values is even, the median is equal to the sum of the two middle (after sorting the list into increasing order) numbers divided by two. 1, 1, 3, 4, 4, 6, 7, 8, 9, 11add the two middle values and divide by two: = 4 + 6 = 10 2 2= 5 is the median

EXAMPLE

H. 3, 13, 7, 5, 21, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29. Find the median of this set of data.1. When we put those numbers in order we have:3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 39, 40, 562.There arefifteennumbers. Our middle number will be theeighthnumber:3, 5, 7, 12, 13, 14, 21,23, 23, 23, 23, 29, 39, 40, 56 The median value of this set of numbers is23.

EXAMPLE

EXAMPLEI. Find the mode of the following data:76, 81, 79, 80, 78, 83, 77, 79, 82, 75

There is no need to organize the data, unless you think that it would be easier to locate the mode if the numbers were arranged from least to greatest. In the above data set, the number 79 appears twice, but all the other numbers appear only once. Since 79 appears with the greatest frequency, 79 is the mode of the data values and it is the only value that occurs most often, the set is calledunimodal .

EXAMPLEJ. The ages of 12 randomly selected customers at a local Best Buy are listed below:23, 21, 29, 24, 31, 21, 27, 23, 24, 32, 33, 19What is the mode of the above ages?The above data set has three values that each occur with a frequency of 2. These values are 21, 23, and 24. All other values occur only once. Therefore, this set of data has three modes which are 21, 23 and 24 and the set is called multimodal.

exercise1. What is the Mean of these numbers?6, 11, 7 ?

Multiple Choice: A. Choose the letter of your answer.

8

45

cbad

7

exercise

2. Scott took 7 math tests in one marking period. What is the mean test score? 89, 73, 84, 91, 87, 77, 94

64

9185

abcd

83

exercise

3. Mark operates Technology Titans, a Web site service that employs 8 people. Find the mean age of his workers if the ages of the employees are as follows: 55, 63, 34, 59, 29, 46, 51, 41abc

47.25

31.5050.04

d

43.25

exercise

4. Find the median of this set of data.20, 30, 35, 24, 36, 47, 48

abc

35

4736

d

44

exercise

5.) What is the mode in this given data? 2, 5, 4, 1, 6, 7, 4, 3, 2, 1 abc

1 & 2

1,2 &42 & 4

d

1 & 4

exercise

6.) The scores of 10 students in a 5-item quiz are 2, 2, 3, 3, 4, 4, 4, 4, 4, 5. What is the mode? abc

4

2 3

d

5

exercise

7.) The daily wages of 5 workers are as follows : Php 350, Php 400, Php 400, Php 500, Php 450. Find the mean daily wage.abc

Php 500

Php 350 Php 320

d

Php 420

exercise

8.)Find the median of these grades.90, 84, 79, 80, 77, 89, 95, 93, 90, 86, 85abc

90

9386

d

80

32

exercise

9.) Find Nielzens grade point average.c

SubjectNumber of UnitsGradeEnglish391Filipino389Math393Science588Value Ed.292PE193

exercise

abc

89.16

90.3588.00

d

91.25

exercise

10.)Find the median of these grades : 80, 78, 81, 83, 95, 85.abc

83

9580

d

82

exercise

B. Find the mean, median and mode of the given data. 11. ) 13, 13, 10, 8, 7, 6, 4, 5

meana.b.c.

8

8.257.25

d.

6.50

exercise

12. ) 13, 13, 10, 8, 7, 6, 4, 5

mediana.b.c.

10

8.257.50

d.

13

exercise

a.b.c.mode 13. ) 13, 13, 10, 8, 7, 6, 4, 5

13 10 5

d.

8

exercise

a.b.c.mean

144.17136.25140.15 14.) 130, 140, 135, 125, 160, 175

d.

140.25

exercise

a.b.c.median

140136135 15. ) 130, 140, 135, 125, 160, 175

d.

130

exercise

a.b.c.mean

91 95 88

16. Vics scores in three unit tests are 92, 87, and 88. What must his score be on the fourth test to get an average of 90?

d.

93

exercise

a.17. The number of jeepney passengers for each trip of Mang Dantes jeepney is recorded as follows: 10, 12, 8, 14, 10, 16, 13, 15, 10. Find the mean number of passengers.

b.c.d. 12 14 13 10mean

exercise

Directions: For numbers 18-20, refer to the given data below.The prices of a yellow shirt in 9 stores are surveyed and the results are as follows:

Php 80, Php 100, Php 120, Php 120, Php 130,Php 140, Php 150, Php 675, Php 2200

exercise

a.b.c.median

Php100 Php150 Php120 18.) Find the median. Php 80, Php 100, Php 120, Php 120, Php 130, Php 140, Php 150, Php 675, Php 2200

d.

Php130

exercise

a.b.c.mean

640.40 412.78 500.12 19.) Find the mean.Php 80, Php 100, Php 120, Php 120, Php 130,Php 140, Php 150, Php 675, Php 2200

d.

120.33

exercise

a.b.c.mode

675 150 100 20.) Find the mode. Php 80, Php 100, Php 120, Php 120, Php 130, Php 140, Php 150, Php 675, Php 2200

d.

120

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Youve got the correct answer!Solution: 1. a Mean = 6 + 11 + 7 3 = 24 3 = 8 A.

Youve got the correct answer!Solution: 2. b Mean = 89 + 73 + 84 + 91 + 87 + 77 + 94 7 = 595 7 = 85

Youve got the correct answer!Solution:3. a Mean age of = 55 + 63 + 34 + 59 + 29 + 46 + 51 + 41Marks worker 8 = 378 8 = 47.25

Youve got the correct answer!Solution: 4. aArrange the data (increasing order). 20, 24, 30, 35, 36, 47, 482. Since this is odd, the median is the middle number.20, 24, 30, 35, 36, 47, 48the median is 35.

Youve got the correct answer!Solution: 5. c 2, 5, 4, 1, 6, 7, 4, 3, 2, 1 Since the numbers 1, 2 and 4 appeared with the same frequency therefore these set of numbers is the mode.Modes: 1, 2 and 4

Youve got the correct answer!Solution: 6. a 2, 2, 3, 3, 4, 4, 4, 4, 4, 5 Since the number 4 occurs frequently in the data therefore the mode is 4.

Youve got the correct answer!Solution: 7. d Mean daily wage = 350+400+400+500+4505 = 2100 5= Php 420

Youve got the correct answer!Solution: 8. b Arrange the data (increasing order). 77, 79, 80, 84, 85, 86, 89, 90, 90, 93, 95 2. Since this is odd, the median is the middle number.77, 79, 80, 84, 85, 86, 89, 90, 90, 93, 95 the median is 86

Youve got the correct answer!Solution: 9. c GPA = 91(3)+89(3)+93(3)+88(5)+92(2)+93(1)3+3+3+5+2+1 = 1536 17 = 90.35

Youve got the correct answer!Solution: 10. d 1. Arrange the data (increasing order). 78, 80, 81, 83, 85, 952. Since this is even, identify the two middle number then add and divide it by two. 78, 80, 81, 83, 85, 95= 81 + 83 = 164 = 82 is the median 2 2

Youve got the correct answer!Solution: 11.b Mean = 13 + 13 + 10 + 8 + 7 + 6 + 4 + 5 8 = 66 8 = 8 . 25 B.

Youve got the correct answer!Solution: 12. c Arrange the data (increasing order)4, 5, 6, 7, 8, 10, 13, 132. Since this is even, identify the two middle number then add and divide it by two. 4, 5, 6, 7, 8, 10, 13, 13= 7 + 8 = 15 = 7.50 2 2

Youve got the correct answer!Solution: 13. a Mode:4, 5, 6, 7, 8, 10, 13, 13Which number appeared most often?13, therefore this is the mode.

Youve got the correct answer!Solution:14. a Mean = 130 + 140 + 135 + 125 + 160 + 175 6 = 865 6 = 144.17

Youve got the correct answer!Solution: 15. c Median 1. Arrange the data (increasing order).125, 130, 135, 140, 160, 1752. Since this is odd, the middle value is the median.125, 130, 135, 140, 160, 175

Youve got the correct answer!Solution: 16. d Let x be his score on the fourth test.Sum of the four scores = mean x 4 = 90 x 4 = 360So, 92+ 87+88+x = 360 267+x = 360 x = 360 267 = 93 score needs by Vic

Youve got the correct answer!Solution: 17. b Mean number of passengers = 10 + 12 + 8 + 14 + 10 + 16 + 13 + 15 + 10 9 = 108 9 = 12

Youve got the correct answer!Solution: 18. d 1. Arrange the data (increasing order).Php 80, Php 100, Php 120, Php 120, Php 130,Php 140, Php 150, Php 675, Php 22002. Since this is odd, the middle value is the median.Php 80, Php 100, Php 120, Php 120, Php 130,Php 140, Php 150, Php 675, Php 2200

Youve got the correct answer!Solution: 19. a Mean = 80+100+120+120+130+140+150+675+22009 = 3715 9 = 412.78

Youve got the correct answer!Solution: 20. d Php 80, Php 100, Php 120, Php 120, Php 130,Php 140, Php 150, Php 675, Php 2200Since the number 120 occurs frequently in the data therefore the mode is 120.

You got the correct answer!Solution: 1. a Mean = 6 + 11 + 7 3 = 24 3 = 8

Sorry, you got the Wrong answer.Try Again!!!

Sorry, youve got the Wrong answer.Try Again!!!

a. mean b. medianc. mode

DISCUSSION

You got the correct answer!Solution: 1. a Mean = 6 + 11 + 7 3 = 24 3 = 8

Sorry, you got the Wrong answer.Try Again!!!

THE ENDTHANK YOU AND GOD BLESS!

QUIT

exercise

This a Multiple Choice Test. You should be able to perform at least Proficient level.Advanced-16 - 20 pts.Proficient-12 - 15 pts.Approaching Proficiency- 8 - 11 ptsDeveloping- 4 - 7 pts.Non-Proficient- 0 - 3 pts.If your performance is below Proficient level, read the topic and answer the exercises again. Proceed!

challenge

Problem Solving:Direction: Solve this problem in your own.Two weeks before Mark opened Technology Titans, he launched his company Web site. During those 14 days, Mark had an average of 24.5 hits on his Web site per day. In the first two days that Technology Titans was open for business, the Web site received 42 and 53 hits respectively. Determine the new average for hits on the Web site.

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