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Measures of Central Tendency

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Michelle Angela

on 9 July 2015

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Transcript of Measures of Central Tendency

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
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