This document outlines measures of central tendency, focusing on mean, median, and mode, as well as their calculations and examples. It discusses various types of data and provides definitions and methods for calculating arithmetic mean, geometric mean, and harmonic mean. Moreover, it highlights the merits, demerits, and uses of these central tendency measures, emphasizing their importance in statistical analysis.

1. Module
2 Measures of Central Tendency
OUTLINE ( Teaching Hours - 5)
1. Introduction
1
1. Introduction
2. Mean
3. Median
4. Mode
By Tushar Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot. 2/28/2020

2. Types of Data
Here we are study mainly 3-types of data (observations) :
2By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020

3. Measure of Central Tendency
A single expression, representing the whole group , is selected which may
convey a fairly enough idea about the whole group . This single expression in
statistics is known as the average .
The average are generally the central part of the distribution and therefore
they are also called the measure of central tendency.
3
Now there are five types of measure of central tendency which are
commonly used . These are ,
1. ARITHMATIC MEAN
2. GEOMETRIC MEAN
3. HARMONIC MEAN
4. MEDIAN
5. MODE
By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020

4. Measure of Central Tendency
4By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020

5. Arithmetic Mean
There are three ways to obtain A.M for the given data :
1. A.M for Individual Observations :
Let
5By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
2. A.M for Discrete Observations :
Let
; , -
f di iX A A Assumed Mean d X Ai ifi
∑
= + = =
∑

6. Arithmetic Mean
There are three ways to obtain A.M for the given data :
3. A.M for Continuous Observations :
In this type of observations we have class and their
corresponding frequency are given then the A.M of
the given data is defined as :
6By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
the given data is defined as :

7. Arithmetic Mean
Ex-1 : Find the A.M of the marks obtained by 10 students of class X in
mathematics in a certain examination. The marks obtained are :
25,30,21,55,47,10,15,17,45,35.
Solu : Here the given observations are individual then
A.M =
7By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
A.M =
Therefore A.M is

8. Arithmetic Mean
Ex-2 : Find the A.M from the following frequency table:
Solu : Here the given observations are discrete then
Marks 52 58 60 65 68 70 75
No. Of
students
7 5 4 6 3 3 2
8By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
Solu : Here the given observations are discrete then
A.M = ; , -
f di iX A A Assumed Mean d X Ai ifi
∑
= + = =
∑

9. Arithmetic Mean
Marks(x) f d = x-A fd
52 7 -13 -91
58 5 -7 -35
60 4 -5 -20
65=A 6 0 0
9By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
65=A 6 0 0
68 3 3 9
70 3 5 15
75 2 10 20
Total 30 -7 -102

10. Arithmetic Mean
102
65 3.4 61.6
30
65
f d
i iX A
f
i
= − =
∑
= + = −
∑
10By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020

11. Arithmetic Mean
Ex-3 : Find the A.M from the following data :
Class 0-30 30-60 60-90 90-120 120-150 150-180
Frequency 8 13 22 27 18 7
Solu : Here given data is continuous therefore,
11By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
Solu : Here given data is continuous therefore,
Where,

12. Arithmetic Mean
Class f Mid Value(X) d = X- A/ i fd
0-30 8 15 -2 -16
30-60 13 45 -1 -13
60-90 22 75 = A 0 0
12By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
60-90 22 75 = A 0 0
90-120 27 105 1 27
120-150 18 135 2 36
150-180 7 165 3 21
Total 95 55

13. Arithmetic Mean
55
75 30
95
75 17.36
= + ×
= +
13By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
75 17.36
92.36
= +
=

14. Arithmetic Mean
Ex-4 : Find the A.M from the following data :
Class < 10 < 20 < 30 < 40 < 50
Frequency 2 18 30 17 3
Solu : Here given data is continuous therefore,
14By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
Where,

15. Arithmetic Mean
Class f Mid
Value(X)
d = X- A/ i fd
0-10 2 5 -2 -4
10-20 18 15 -1 -18
20-30 30 25=A 0 0
15By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
30-40 17 35 1 17
40-50 3 45 2 6
Total 70 1
1
25 10 25 0.14 25.14
70
X
 
= + × = + = 
 

16. Geometric Mean
16By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
Ex – 1 : Find the Geometric mean of the numbers
50,100,200.
Solu : 3
. 50 100 200 100G M = × × =

17. Harmonic Mean
17By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
Ex – 1 : Find the Harmonic mean of 3 –observations
2,4 and 8.
Solu : 3 3
. 3.429
1 1 1 0.5 0.25 0.125
2 4 8
H M = = =
+ ++ +

18. Median
18By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020

19. Median
19By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020

20. Median
20By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
• M is near to the those cumulative frequency then their
corresponding value of observation is required median.

21. Median
21By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020

22. Median
Ex- 1 : Find the median of the data
10, 18, 23, 40, 58, 65, 92,38
Solu : Arranging the data in ascending order, we get
10, 18, 23, 38, 40, 58, 65, 92
22By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
.
1
2 2
2
4 5
2
38 40
39
2
th th
th th
Here no of observations are even
n n
Value of observation Value of observation
Median
value of observation value of observation
   
+ +   
   ∴ =
+
=
+
= =

23. Median
Ex- 2 : Find the median of the data
6,20,43,50,19,53,0,37,78,1,15
Solu : Arranging the data in ascending order, we get
0,1,6,15,19,20,37,43,50,53,78
23By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
.
11 1
2
6
20
th
th
Here no of observations are odd
Median Value of observation
Value of observation
+ 
∴ =  
 
=
=

24. Median
Ex- 3 : Find the median of the following data
Solu : Here given data is discrete therefore we are using
the formula :
X 20 9 25 50 40 80
f 6 4 16 7 8 2
24By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
the formula :

25. Median
First arrange the data into ascending order :
X f Cumulative frequency
9 4 4
20 6 10
25By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
25 16 26
40 8 34
50 7 41
80 2 43

26. Median
1
; 43
2
22 ; 26
25, . 26
th
n
M observation n total of frequencies
whichis near tothecumulative frequency
whichis a corressponding observation of c f
+ 
= = = 
 
=
=
26By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020

27. Median
Ex- 4: Find the median from the following data :
Class 0-30 30-60 60-90 90-120 120-150 150-180
Frequency 8 13 22 27 18 7
Solu : Here given observations are continuous therefore
27By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
Solu : Here given observations are continuous therefore
we will use the formula :
2
n
c
M L i
f
 
− 
= + × 
 
 

28. Median
Class f c.f
0-30 8 8
30-60 13 21
60-90 22 43
90-120 27 70
120-150 18 88
95
, 47.5 islies in the class 90 120
2 2
The median class 90 120
90
43
27
n
now
L
c
f
= = −
∴ = −
=
=
=
28By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
150-180 7 95
Total 95
27
30
f
i
=
=
47.5 43 4.5 30
90 30 90 90 5 95
27 27
M
− ×   
∴ = + × = + = + =     

29. 29By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020

30. Mode
30By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020

31. Mode
31By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020

32. Mode
32By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020

33. Mode
Solu -1 : (i) Here the number 45 is repeated therefore
Mode = 45.
(ii) Here no number is repeat therefore the
given series has no mode.
(iii) Here two numbers 10 and 18 are repeated
33By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
(iii) Here two numbers 10 and 18 are repeated
therefore the given series has two mode
10 and 18.

34. Mode
2:Solu -2:
Class Frequency
0-10 10
10-20 14
34By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
20-30 19 Max. Frequency
30-40 17
40-50 13

35. Mode
Solu -2:
1
Modelclass is 20 30 becauseit has max.frequency
L= 20
f 14
= −
=
35By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
2 17
19
10
19 14 50
20 10 20 20 7.14 27.14
2(19) 14 17 7
m
f
f
i
Mode
=
=
=
 −
= + × = + = + = 
− − 

36. Mode
Solu -:3
Here 25 20
Mode 3 2 3(20) 2(25) 10
X and M
Z M X
= =
∴ = − = − =
36By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020

37. Merits, Demerits and uses of Mean
Merits :
It can be easily calculated
Its calculations are based on all the observations
It is easy to understand
it is the average obtained by calculations and it does
37By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
it is the average obtained by calculations and it does
not depend upon any position.
It is rigidly defined by the mathematical formula
Demerits :
It may not be represented in actual data and so it is
theoretical.
The extreme values have greater effect on mean.

38. Merits, Demerits and uses of Mean
Demerits :
It can not be calculated if all the values are not
known.
It can not be determined for the qualitative data like
beauty, honesty etc.,.
38By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
beauty, honesty etc.,.
Uses of Mean :
It is extensively used in practical statistics
Estimates are always obtained by mean

39. Merits, Demerits and uses of Median
Merits :
It is easily understood.
It is not affected by extreme values
It can be located graphically
It is the best measure for qualitative data like beauty,
39By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
It is the best measure for qualitative data like beauty,
honesty etc.,.
Demerits :
It is not subject to algebraic treatments
It can not represent the irregular distribution series
It is a positional average and is based on the middle
item.

40. Merits, Demerits and uses of Median
Uses of Median :
It is useful in those cases where numerical
measurements are not possible.
It is generally used in studying phenomena like skill,
honesty, intelligence, etc.
40By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
honesty, intelligence, etc.

41. Thank
You
41By Tushar Bhatt, Assistant Professor in Mathematics,
Atmiya University, Rajkot.
2/28/2020
You