1

Measures of Central Tendency

2

Grouped Data

MEAN

3

Determine the mean of the following.

Determine the mean of the following.

a. 2, 4, 1, 4 and 6

4

a. 2, 4, 1, 4 and 6

answer: 3.4 5

6

b. 77, 80, 90, 65, 77, 89, 80

7

b. 77, 80, 90, 65, 77, 89, 80 answer: 79.71

8

c. 1299,2580, 4098, 9100, 1100

9

c. 1299,2580, 4098, 9100, 1100

answer: 3635.4

10

d. 50, 50, 51, 58, 60

11

answer: 53.8d. 50, 50, 51, 58, 60

12

e. 34, 39, 39, 52, 38, 60

13

answer: 43.67e. 34, 39, 39, 52, 38, 60

14

Measures of Central Tendency of

MEANGROUPED DATA

15

Grouped Data

Grouped data are the data or scores that are arranged in a frequency distribution.

16

Grouped Data

17

Mean

The mean (also known as the arithmetic mean) is the most commonly used measure of central position. It is used to describe a set of data where the measures cluster or concentrate at a point.

18

Mean

Formula:

19

mfXXn

f frequency

mX classmark n sum of frequency

20

X meanmfXX

nWhere

:

21

Illustrative Example:Calculate the mean of the Mid-year Scores of Students in Mathematics.

Score Frequency

41-45 136-40 831-35 826-30 1421-25 716-20 2

Mid-year Test scores of students in Mathematics

22

Solution1. Find the midpoint or class mark ( ) of each class or category.

mX

2mLL ULX

Scores

41-45 136-40 831-35 826-30 1421-25 716-20 2

23

Solution1. Find the midpoint or class mark ( ) of each class or category.

mX

2mLL ULX

Scores

41-45 1 4336-40 831-35 826-30 1421-25 716-20 2

24

Solution1. Find the midpoint or class mark ( ) of each class or category.

mX

2mLL ULX

Scores

41-45 1 4336-40 8 3831-35 826-30 1421-25 716-20 2

25

Solution1. Find the midpoint or class mark ( ) of each class or category.

mX

2mLL ULX

Scores

41-45 1 4336-40 8 3831-35 8 3326-30 1421-25 716-20 2

26

Solution1. Find the midpoint or class mark ( ) of each class or category.

mX

2mLL ULX

Scores

41-45 1 4336-40 8 3831-35 8 3326-30 14 2821-25 716-20 2

27

Solution1. Find the midpoint or class mark ( ) of each class or category.

mX

2mLL ULX

Scores

41-45 1 4336-40 8 3831-35 8 3326-30 14 2821-25 7 2316-20 2

28

Solution1. Find the midpoint or class mark ( ) of each class or category.

mX

2mLL ULX

Scores

41-45 1 4336-40 8 3831-35 8 3326-30 14 2821-25 7 2316-20 2 18

29

Solution2. Multiply the frequency and the corresponding class mark . fXm

Scores Frequency

(f )41-45 1 4336-40 8 3831-35 8 3326-30 14 2821-25 7 2316-20 2 18

30

Solution2. Multiply the frequency and the corresponding class mark . fXm

Scores Frequency

(f )41-45 1 43 4336-40 8 3831-35 8 3326-30 14 2821-25 7 2316-20 2 18

31

Solution2. Multiply the frequency and the corresponding class mark . fXm

Scores Frequency

(f )41-45 1 43 4336-40 8 38 30431-35 8 3326-30 14 2821-25 7 2316-20 2 18

32

Solution2. Multiply the frequency and the corresponding class mark . fXm

Scores Frequency

(f )41-45 1 43 4336-40 8 38 30431-35 8 33 26426-30 14 2821-25 7 2316-20 2 18

33

Solution2. Multiply the frequency and the corresponding class mark . fXm

Scores Frequency

(f )41-45 1 43 4336-40 8 38 30431-35 8 33 26426-30 14 28 39221-25 7 2316-20 2 18

34

Solution2. Multiply the frequency and the corresponding class mark . fXm

Scores Frequency

(f )41-45 1 43 4336-40 8 38 30431-35 8 33 26426-30 14 28 39221-25 7 23 16116-20 2 18

35

Solution2. Multiply the frequency and the corresponding class mark . fXm

Scores Frequency

(f )41-45 1 43 4336-40 8 38 30431-35 8 33 26426-30 14 28 39221-25 7 23 16116-20 2 18 36

36

Solution

Scores Frequency

(f )41-45 1 43 4336-40 8 38 30431-35 8 33 26426-30 14 28 39221-25 7 23 16116-20 2 18 36

3. Find the sum of the results in step 2. fXm

37

Solution3. Find the sum of the results in step 2.

Scores Frequency

(f )41-45 1 43 4336-40 8 38 30431-35 8 33 26426-30 14 28 39221-25 7 23 16116-20 2 18 36

fXm

Solution4. Solve the mean using the formula. Scores Frequenc

y(f )

41-45 1 43 4336-40 8 38 30431-35 8 33 26426-30 14 28 39221-25 7 23 16116-20 2 18 36

n=40

Substitution

Therefore, the mean of Mid-year test is 30.

mfXXn

1, 20040

30X

39

Let’s practice: Find the mean weight of Grade 8 Students.

Weight in kg Frequency75-79 170-74 465-69 1060-64 1455-59 2150-54 1545-49 1440-44 1

Weight in kg

Frequency (f)

75-79 170-74 465-69 1060-64 1455-59 2150-54 1545-49 1440-44 1

40

Weight in kg

Frequency (f)

75-79 1 7770-74 465-69 1060-64 1455-59 2150-54 1545-49 1440-44 1

41

Weight in kg

Frequency (f)

75-79 1 7770-74 4 7265-69 1060-64 1455-59 2150-54 1545-49 1440-44 1

42

Weight in kg

Frequency (f)

75-79 1 7770-74 4 7265-69 10 6760-64 1455-59 2150-54 1545-49 1440-44 1

43

Weight in kg

Frequency (f)

75-79 1 7770-74 4 7265-69 10 6760-64 14 6255-59 2150-54 1545-49 1440-44 1

44

Weight in kg

Frequency (f)

75-79 1 7770-74 4 7265-69 10 6760-64 14 6255-59 21 5750-54 1545-49 1440-44 1

45

Weight in kg

Frequency (f)

75-79 1 7770-74 4 7265-69 10 6760-64 14 6255-59 21 5750-54 15 5245-49 1440-44 1

46

Weight in kg

Frequency (f)

75-79 1 7770-74 4 7265-69 10 6760-64 14 6255-59 21 5750-54 15 5245-49 14 4740-44 1

47

Weight in kg

Frequency (f)

75-79 1 7770-74 4 7265-69 10 6760-64 14 6255-59 21 5750-54 15 5245-49 14 4740-44 1 42

48

Weight in kg

Frequency (f)

75-79 1 77 7770-74 4 7265-69 10 6760-64 14 6255-59 21 5750-54 15 5245-49 14 4740-44 1 42

49

Weight in kg

Frequency (f)

75-79 1 77 7770-74 4 72 28865-69 10 6760-64 14 6255-59 21 5750-54 15 5245-49 14 4740-44 1 42

50

Weight in kg

Frequency (f)

75-79 1 77 7770-74 4 72 28865-69 10 67 67060-64 14 6255-59 21 5750-54 15 5245-49 14 4740-44 1 42

51

Weight in kg

Frequency (f)

75-79 1 77 7770-74 4 72 28865-69 10 67 67060-64 14 62 86855-59 21 5750-54 15 5245-49 14 4740-44 1 42

52

Weight in kg

Frequency (f)

75-79 1 77 7770-74 4 72 28865-69 10 67 67060-64 14 62 86855-59 21 57 119750-54 15 5245-49 14 4740-44 1 42

53

Weight in kg

Frequency (f)

75-79 1 77 7770-74 4 72 28865-69 10 67 67060-64 14 62 86855-59 21 57 119750-54 15 52 78045-49 14 4740-44 1 42

54

Weight in kg

Frequency (f)

75-79 1 77 7770-74 4 72 28865-69 10 67 67060-64 14 62 86855-59 21 57 119750-54 15 52 78045-49 14 47 65840-44 1 42

55

Weight in kg

Frequency (f)

75-79 1 77 7770-74 4 72 28865-69 10 67 67060-64 14 62 86855-59 21 57 119750-54 15 52 78045-49 14 47 65840-44 1 42 42

56

Weight in kg

Frequency (f)

75-79 1 77 7770-74 4 72 28865-69 10 67 67060-64 14 62 86855-59 21 57 119750-54 15 52 78045-49 14 47 65840-44 1 42 42

57

Therefore, the mean weight is 57.25

Weight in kg

Frequency (f)

75-79 1 77 7770-74 4 72 28865-69 10 67 67060-64 14 62 86855-59 21 57 119750-54 15 52 78045-49 14 47 65840-44 1 42 42

mfXXn

458080

57.25X

59

Generalization

60

GeneralizationThe mean (also known as the arithmetic mean) is the most commonly used measure of central position. It is used to describe a set of data where the measures cluster or concentrate at a point.

61

Formula

mfXXn

62

Group Work 

CRITERIA 

5 4 3 2 1

63

Group WorkCRITERIA

 

5 4 3 2 1

ACCURACY    

100% of the steps and solutions have no mathematical errors.

Almost all (90-99%) of the steps and solutions have no mathematical errors.

Almost all (85-89%) of the steps and solutions have no mathematical errors.

Most (75-84%) of the steps and solutions have no mathematical errors.

Less than 75% of the steps and solutions have mathematical errors.

64

Group Work  

CRITERIA 

5 4 3 2 1

ORGANIZATION         

It uses an appropriate and complete strategy for solving the problem. Uses clear and effective diagrams and/or tables.

It uses complete strategy for solving the problem. Uses creative diagrams and/or tables.

It uses strategy for solving the problem. Uses diagrams and/or tables.

It uses an inappropriate strategy or application of strategy unclear. There is limited use or misuse of diagrams and/or tables.

It has no particular strategy for solving the problem. It does not show use of diagrams nor tables.

65

Group Work  

CRITERIA 

5 4 3 2 1

DELIVERY There is a clear and effective explanation of the solution. All steps are included so the audience does not have to infer how the task was completed. Mathematical representation is actively used as a means of communicating ideas, and precise and appropriate mathematical terminology.  

There is a clear explanation and appropriate use of accurate mathematical representation. There is effective use of mathematical terminology.

There is explanation and mathematical representation. There is mathematical terminology

There is an incomplete explanation; it is not clearly represented. There is some use of appropriate mathematical representation and terminology to the task.

There is no explanation of the solutions. The explanation cannot be understood, or is unrelated to the task. There is no use or inappropriate use of mathematical representation and terminology to the task.

TIMER10 minutes

End

67

Assignment1.A telecommunications company is conducting

a study on the average number text messages send per day by high school students in Marikina. A random sample of 50 college students from the said area is taken. Find the mean of the data set. Class Interval Frequency

30-34 825-29 1020-24 1615-19 910-14 7

68

2. Study on Median for Grouped Dataa.Describe Median.b.What is the formula in computing the median for grouped data?Reference: Mathematics Learner’s Module by Emmanuel P. AbunzoPages 564-580

69