MEASURE OF CENTRAL TENDENCY TYPES OF AVERAGES Arithmetic mean Median Mode Geometric mean Harmonic mean

1. 1
MEASURE OF CENTRAL TENDENCY
TYPES OF AVERAGES
1. Arithmetic mean
2. Median
3. Mode
4. Geometric mean
5. Harmonic mean

2. 2
MEASURE OF CENTRAL TENDENCY
The diagrammatic representation of a set of data can give
us some impression about its distribution. Even then there
remains a need for a single quantitative measure which
could be used to indicate the center of the distribution . An
average is a single value which represent a set of data
or a distribution as a whole. It is more or less central
value round which the observations in the set of data or
distribution tend to cluster. Such a central value is also
called a measure of central tendency or measure of
location.

3. 3
MEASURE OF CENTRAL TENDENCY
Arithmetic Mean
It is obtained by dividing the sum of all the observations
by the total number of observations. It is denoted by
" X ".
For ungrouped data
n
X
=
n
x...+x+x+x
=X n321 
For grouped data
f
fx
=
f...f+f+f
xkf...xf+xf+xf
=X
k321
k332211



4. 4
Example
The following data is the final plant height (cm) of thirty plants of wheat.
Construct a frequency distribution
87 91 89 88 89 91 87
92 90 98 95 97 96 100
101 96 98 99 98 100 102
99 101 105 103 107 105 106
107 112

5. 5
MEASURE OF CENTRAL TENDENCY
63.97
30
2929
==X
For ungrouped data
n
X
=
n
x...+x+x+x
=X n321 

6. Class
Limit
s
Class
Bounda
ries
Mi
d
Poi
nts
X
C.
F
Frequ
ency
(f)
fx
86---
90
85.5---
90.5
88
93
6
1
6
4
52
8
FREQUENCY DISTRIBUTION TABLE

7. 7
MEASURE OF CENTRAL TENDENCY
f
fx
=
f...f+f+f
xkf...xf+xf+xf
=X
k321
k332211


For grouped data
83.97
30
2935



f
fx
=X

8. 8
Median

9. 9
Median
The median is a value that divides a set of data in to
two equal parts after arranging the values in ascending
or descending order of magnitude.
For ungrouped data
(i) When "n" is odd then the median is
[(n+1)/2]th observation.
(ii)When "n" is even then the median is
1/2[(n/2) + (n/2+1)]th observation.

10. 10
Median
Example
Given below are the marks obtained by 20 students.
53 74 82 42 39 28 20
81 68 58 67 54 93 70
30 61 55 36 37 29.
Find Median
Solution:
First arrange the data in ascending order of magnitude
20 28 29 30 36 37 39
42 53 54 55 58 61 67
68 70 74 81 82 93
Median = 1/2[(n/2) + (n/2+1)]th observation.
Median = 1/2[ 10 + 11]th observation = 1/2[ 54 +55]
Median = 54.5

11. 11
Median
Example
Given below are the marks obtained by 9 students.
45 32 37
41 48 36
46 39 36
Find Median
Solution:
First arrange the data in ascending order of magnitude
32 36 36
37 39 41
45 46 48
Median = [n + 1]/2th observation.
Median = [9 + 1]/ 2th observation = 5th observation
Median = 39

12. 12
Median
For grouped data
l = Lower class boundary of the class containing median
h = Class interval of the class containing median*
f = Frequency of the class containing median
n = Total number of observations
c = Cumulative frequency of the class preceding the class
containing median.
c)-
2
n
(
f
h
+l=Median

13. 13
Median
Example: Estimate the Median
Daily
Income Rs
No. of
Work
ers( f )
c. f class
boundaries
5-----24 4 4 4.5---24.5
25-----44 6 10 24.5---44.5
45-----64 14 24 44.5---64.5
65-----84 22 46 64.5---84.5
85----104 14 60 84.5---104.5
105---124 5 65 104.5---124.5
125---144 7 72 124.5---144.5
145---164 3 75 144.5---164.5

14. 14
Median
Step -1 : Calculate n/2, its helps which class
containing the median.
n/2 = 75 / 2 = 37.5 see this observation in
cumulative frequency column
Step- 2: The class containing the median is 64.5 ---- 84.5
Step- 3: l = lower class boundaries containing the median
= 64.5
Step- 4: h = class interval of the class containing the median
= 20
Step- 5: f = frequency of the class containing median = 22
Step- 6: c = cumulative frequency of the preceding class = 24

15. 15
Median
77= 76.
- 24)
2
75
(
22
20
64.5 +Median =
- c)
2
n
(
f
h
l +Median =