**Measures of Central Tendency**

**Median**

**Mode**

Central Tendency

refers to the numerical value in the central region of a distribution of scores.

Mean (Commonly called Average)

The sum of all numbers divided by

n.

Example:

Given the set of numbers: 2, 7, 5, 10, 4, find the mean.

(2+4+5+7+10)

5

= 5.6

Median

The middle value when the data is arranged in numerical order.

Example:

Given the set of numbers: 2, 7, 5, 10, 4, find the median.

2 4 5 7 10

Mode

The value that appears the most.

A set of data can have more than one mode.

If all the numbers appear the same number of times, there is no mode for that data.

Example:

Given the set of numbers: 2, 7, 5, 10, 4, find the mode.

2 4 5 7 10 = No mode.

Example:

1, 3, 3, 3, 4, 4, 6, 6, 6, 9

The mode here is 3 and 6.

**Mean**

Ungrouped Data

Mean

Sum of the scores

Total frequency

What will be the average of a student if his grades are : 93, 95, 89, 90, 96?

93 + 95 + 89 + 90 + 96

5

= 92.6

Weighted Mean

Mean

Sum of the product of frequency and score

Total frequency

A class of 25 students took a science test. 10 students got a score of 80; 5 students got a score of 87; 4 students scored 76 and 6 students scored 64. Calculate the mean.

80(10) + 87(5) + 76(4) + 64(6)

25

=

1923

25

= 76.92

Grouped Data

Mean

Total frequency

Sum of the product of frequencies and class marks

Calculate the average working hours of 100 college students.

f

24

14

39

18

5

CI

0-9

10-19

20-29

30-39

40-49

X

m

4.5

14.5

24.5

34.5

44.5

f

X

m

108

203

955.5

621

222.5

=

2110

100

=

21.1 or 21

Grouped Data

Mean

Assumed mean

Total frequency

Size of the class interval

Class mark

The average of lower interval and upper interval.

Coded

value

X

m

- X

o

i

**Ungrouped Data**

Grouped Data

Take the cumulative frequency distribution of the working hours of 100 students and calculate for its median.

CI

40-49

30-39

20-29

10-19

0-9

f

5

18

39

14

24

<cf

100

95

77

38

24

N

2

=

100

2

=

50th score

The class interval that contains the 50th score is 20-29.

X

= 19.5

cf

= 38

f

= 39

i

= 10

LB

b

m

19.5 + (

50-38

39

) 10

=22.58

Calculate the mean.

83, 105, 46, 79, 66, 58, 99

Calculate the mean.

Calculate the average money donated by 20 parents.

Number of parents

5

7

4

3

1

Amount of Money ($)

500

780

400

350

100

Find the mean time that students spend reviewing for their exams.

Calculate the mean score of 40 students in a Math quiz.

Class Interval

98 - 100

95 - 97

92 - 94

89 - 91

86 - 88

83 - 85

80 - 82

77 - 79

74 - 76

71 -73

f

2

1

1

6

6

5

9

2

3

5

X

m

f

X

m

Mode of Ungrouped data

Time

2 hrs

3 hrs

2 1/2 hrs

5 hrs

4 hrs

Number of Students

5

15

10

7

13

Class Interval

96 - 110

81 - 95

66 - 80

51 - 65

36 - 50

21 - 35

Frequency

20

8

10

45

5

12

N = 100

Calculate for the median.

Class Interval

96 - 110

81 - 95

66 - 80

51 - 65

36 - 50

21 - 35

Frequency

20

8

10

45

5

12

<cf

Calculate for the mean.

Median

Number of values

Median

Lower boundary of

the median class

Size of the class interval

Frequency of the median class

Cumulative frequency before the median class

Median class

Given the assumed mean of 17, find the mean for the number of hobbies of the students in the school.

CI

0-4

5-9

10-14

15-19

20-24

25-29

30-34

35-39

f

45

58

27

30

19

11

8

2

X

m

2

7

12

17

22

27

32

37

X

c

-3

-2

-1

0

1

2

3

4

f

X

c

-135

-116

-27

0

19

22

24

8

= - 205

N = 200

X

o

= 17

i

= 5

X = 17 + ( ) 5

- 205

200

= 17 - 5.125

= 11.875

Calculate the mean working hours of 100 college students if the assumed mean is 24.5.

Hours worked per week

0 - 9

10 - 19

20 - 29

30 - 39

40 - 49

f

24

14

39

18

5

X

m

4.5

14.5

24.5

34.5

44.5

X

c

-2

-1

0

1

2

f

X

c

-48

-14

0

18

10

Calculate for the mean if X

o

is 10.

Class Interval

15 -17

12 -14

9 - 11

6 - 8

3 - 5

f

7

9

9

12

3

X

m

16

13

10

7

4

Find the median set

12 15 11 13 19 16 20

11

12

13

15

19

16

20

= ( ) = 4th score

7+1

2

Find the median set

25 28 22 20 18 23 30 24

18

20

22

23

24

25

28

30

=( ) =4.5th score

8 + 1

2

23 + 24

2

= 23.5

Find the mode in the given set of data.

100 115 110 108 100 125 110 120 90 110 90 105

The mode is

110

since it occurs three times in the distribution.

Find the mode in the given set of data.

Size of shoe

4

4 1/2

5

5 1/2

6

6 1/2

7

Number of Pairs sold

4

6

10

10

5

3

1

The highest number of shoes sold is

10

. Therefore, there are two modes:

5

and

5 1/2

. The data is

bimodal

.

Calculate for the median.

Class Interval

15 - 17

12 - 14

9 - 11

6 - 8

3 - 5

f

7

9

9

12

3

<cf

N

2

=

X =

fm

=

cf

=

LB

B

i

=

X =

~

Mode of a Grouped data

Mode

Lower boundary of the modal class

Difference between the frequency of the modal class and the frequency of the class interval preceding it

Difference between the frequency of the modal class and the frequency of the class interval succeeding it

Size of the class interval

Find the mode.

C. I.

96 - 110

81 - 95

66 - 80

51 - 65

36 -50

21 - 35

f

20

8

10

45

5

12

Modal class

= 50.5

= 45 - 5 = 40

= 45 - 10 = 35

i

= 15

50.5 + ( )15

40

75

= 58.5

Find the mode.

C. I.

15 - 17

12 - 14

9 - 11

6 - 8

3 -5

f

10

6

10

9

5

Modal class

Modal class

The numerical value in the central region of a distribution of numbers.

What are the three measures of central tendency?

Calculate for the mean, median and mode.

x

15.5

15.7

12.0

12.1

12.2

12.8

f

22

4

29

2

12

6

4 11 8 2