Measures of central tendency

1. NADEEM UDDIN
ASSOCIATE PROFESSOR
OF STATISTICS

2. Introduction:
This chapter extended our discussion of
descriptive statistics, which deals with
methods of summarizing and presenting
the essential information contained in a
set of data.
After constructing a frequency distribution
to obtain a general idea about the
distribution of a data set, we can use
numerical measures to describe the central
location of the data.

3. Average:
“An average is a central value which can
represent the whole data.”
The average tends to lie in the centre of a
distribution they are called measure of
central tendency or measure of central
location.

4. Types of Averages:
The following averages are usually used.
(a) Mathematical Average:
(i) Arithmetic Mean
(ii) Geometric Mean
(iii) Harmonic Mean
(b) Average of Location:
(i) Mode
(ii) Median

5. ARITHMETIC MEAN:
The arithmetic mean of a set of n observation is defined
as the sum of all the observation divided by the number
of observation.
The arithmetic mean is the most commonly used
average. In view of its common use, it is usually referred
to as the average or simply the mean.
Calculating the Mean:
a) In case of ungroup data, the mean can be found by
the following formula:
b) In case of group data, the mean can be found by the
following formula:
n
x
x



f
fx
x

6. Example-1:
Find arithmetic mean of the values 3, 7, 9,
10 and 5.
Solution:
𝑥 =
𝑥
𝑛
𝑥 =
3 + 7 + 9 + 10 + 5
5
𝑥 =
34
5
𝑥 = 6.8

7. 1-Find the arithmetic mean of the set of numbers
94, 101, 82, 78, 97 and 88. (90)
2-Find the arithmetic mean of the following
measurements of the diameter of a cylinder
recorded by a scientist 4.59, 4.15, 4.42, 4.47, 4.52,
4.35, 4.53, 3.98, 4.06 and 3.92 inches. (4.29)
3-The per day incomes of ten families in rupees in
a certain locality are given below, Find mean.(438.5)
Family A B C D E F G H I J
Incomes “Rs” 500 250 730 150 200 1000 670 485 100 300

8. C.I 0 – 4 5 – 9 10 – 14 15 – 19 20 – 24 25 – 29
F 2 4 6 8 3 1
Example-4:
Calculate arithmetic mean for the following data:

9. C.I f x fx
0 – 4
5 – 9
10 – 14
15 – 19
20 – 24
25 – 29
2
4
6
8
3
1
2
7
12
17
22
27
4
28
72
136
66
27
Sum 24 333
Solution:


f
fx
x
24
333

88.13x 

10. Find A.M for the following
ANSWERS(11.2)(75.04)(65.31)
C.I 4 – 6 7 – 9 10 – 12 13 – 15 16 – 18
f 2 8 10 6 4
Groups 0 – 10 10 – 40 40 – 90 90 – 100 100 – 105 105 – 120 120 - 140
f 50 100 140 210 100 50 20
Height
“Inches”
60 61 62 63 64 65 66 67 68 69 70
No. of
Students
2 4 9 10 12 20 18 12 9 6 5

11. Definition of 'Geometric Mean:
The average of a set of products, the calculation
of which is commonly used to determine the
performance results of an investment or portfolio.
Technically defined as "the 'n'th root product of 'n'
numbers",
the formula for calculating geometric mean is
most easily written as:
G.M = ( x1 . x2 . x3 ……. xn)1/n
Or
G.M = 



n
x
anti
log
log

12. Example-21(a):
Find G.M of 2 and 8.
Solution:
Example-21(b):
The rate of inflation in four years was
7%,11%,15% and19%
find the average rate of inflation per year.
Solution:
   
1 1
2 2G.M= x y 2 8 16 4    
   
1
0.25
4. 7 11 15 19 21945 12.17%G M      

13. Example-22:
A data set contains the observations.
2, 4, 6, 8, 10 find G.M .
Solution:
x logx
2 0.3010
4 0.6020
6 0.7781
8 0.9030
10 1
𝑙𝑜𝑔𝑥 = 3.5841

14. 𝐺. 𝑀 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔
𝑙𝑜𝑔𝑥
𝑛
𝐺. 𝑀 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔
3.5841
5
𝐺. 𝑀 =
𝑎𝑛𝑡𝑖𝑙𝑜𝑔 0.7168
𝐺. 𝑀 = 5.21

15. 1-The monthly income of ten families in rupees in a certain
city are 85, 70, 10, 75, 500,8, 42, 250, 40, 36 $. Calculate G.M
(Ans:55.35)
2-Find the geometric mean of 1, 4 and 128 (Ans:8)
3-Find the geometric mean of 7% , 10% ,12% , 15% and 17% .
(Ans:11.65%)
4-The rate of inflation in three successive year in a country
was 13% , 17% ,and 21%. Find the average rate of inflation
per year. (Ans:16.68%)

16. Example-23: Calculate Geometric mean.
C.I f x Log x flog x
65 – 84
85 – 104
105 – 124
125 – 144
145 – 164
165 – 184
185 – 204
9
10
17
10
5
4
5
74.5
94.5
114.5
134.5
154.5
174.5
194.5
1.8722
1.9754
2.0589
2.1287
2.1889
2.2418
2.2889
16.8498
19.7540
35.0013
21.2870
10.9445
8.9672
11.4445
60 f 2483.124log  xf
𝐺. 𝑀 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔
𝑓𝑙𝑜𝑔𝑥
𝑓
𝐺. 𝑀 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔
124.2483
60
𝐺. 𝑀 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔 2.0708
𝐺. 𝑀 = 117.7

17. Maximum
Load
(Tons)
8 – 10 11 – 13 14 – 16 17 – 19 20 – 22 23 – 25
No. of
cables
2 4 6 4 3 1
Find the geometric mean of the following table.
(Ans:15.22)

18. Definition of 'Harmonic Mean:
The reciprocal of the arithmetic mean of
the reciprocal is called as Harmonic Mean.
It is calculated by dividing the number of
observations by the sum of reciprocal of the
observation.
The formula to find the harmonic mean is
given by:
Harmonic Mean H =
n
1
x1
+
1
x2
+
1
x3
+ … . . +
1
xn
H.M =
x
n
1


19. Example-24:
A data set contains the observations.
2, 4, 6, 8, 10 find Harmonic mean.
Solution:
X
2 0.5
4 0.25
6 0.1666
8 0.125
10 0.1
x
1
1
1.1416
x
 
𝑯. 𝑴 =
𝒏
𝟏
𝒙
𝑯. 𝑴 =
𝟓
𝟏.𝟏𝟒𝟏𝟔
𝑯. 𝑴 = 𝟒. 𝟑𝟖

20. 1-Find the Harmonic Mean of
40, 33, 10, 12, 16, 20, 25, 22. (Ans:18.32)
2-Find the Harmonic mean of 1, 4 and 128.
(Ans:2.39)
3-IF an investor buys shares of Rs.9000/- at a
price of Rs.45/- per share of Rs.9000/-
at Rs.36/- per share. Calculate the average
price per share. (Ans:40.81)

21. Example-25: Calculate H.M.
Solution:
C.I f X
65 – 84
85 – 104
105 – 124
125 – 144
145 – 164
165 – 184
185 – 204
9
10
17
10
5
4
5
74.5
94.5
114.5
134.5
154.5
174.5
194.5
0.12081
0.10582
0.14847
0.07435
0.03236
0.02292
0.02571
=
0.53044
x
f
60 f
f
x

𝐻. 𝑀 =
𝑓
𝑓
𝑥
𝐻. 𝑀 =
60
0.53044
𝐻. 𝑀 = 113.11

22. Maximu
m Load
(Tons)
8 – 10 11 – 13 14 – 16 17 – 19 20 – 22 23 – 25
No. of
cables
2 4 6 4 3 1
•Find the Harmonic mean of the following table .
(Ans:14.68)

23. Example-26:A data set contains the observations.
2, 4, 6, 8, 10 find A.M, G.M and H.M.
Also show that A.M > G.M > H.M
Solution:
X Logx
2 0.3010 0.5
4 0.6020 0.25
6 0.7781 0.1666
8 0.9030 0.125
10 1 0.1
x
1
30x log 3.5841x 
1
1.1416
x
 

24. A.M=
𝒙
𝒏
=
𝟑𝟎
𝟓
= 𝟔
𝑮. 𝑴 = 𝒂𝒏𝒕𝒊𝒍𝒐𝒈
𝒍𝒐𝒈𝒙
𝒏
𝑮. 𝑴 = 𝒂𝒏𝒕𝒊𝒍𝒐𝒈
𝟑.𝟓𝟖𝟒𝟏
𝟓
𝑮. 𝑴 = 𝒂𝒏𝒕𝒊𝒍𝒐𝒈 𝟎. 𝟕𝟏𝟔𝟖
𝑮. 𝑴 = 𝟓. 𝟐𝟏
𝑯. 𝑴 =
𝒏
𝟏
𝒙
𝑯. 𝑴 =
𝟓
𝟏.𝟏𝟒𝟏𝟔
𝑯. 𝑴 = 𝟒. 𝟑𝟖
A.M > G.M > H.M

25. 1-Find Arithmetic, Geometric and Harmonic
Means of 40, 33, 10, 12, 16, 20, 25, 22, Also
verify the relationship between these
measures. (Ans:22.25, 20.20, 18.32)
2-Given the following index numbers for 10
years. 101, 99, 95, 105, 102, 87, 85, 112,
110, 112. Find Arithmetic, Geometric and
Harmonic Means. (Ans:100.8, 100.37, 100)

26. Example-27: Calculate A.M, G.M and H.M
C.I f x fx Logx flogx
65 – 84
85 – 104
105 – 124
125 – 144
145 – 164
165 – 184
185 – 204
9
10
17
10
5
4
5
74.5
94.5
114.5
134.5
154.5
174.5
194.5
670.5
945.0
1946.5
1345
772.5
698
972.5
1.8722
1.9754
2.0589
2.1287
2.1889
2.2418
2.2889
16.8498
19.7540
35.0013
21.2870
10.9445
8.9672
11.4445
0.12081
0.10582
0.14847
0.07435
0.03236
0.02292
0.02571
7350 =0.53044
x
f
60 f log 124.2483f x 
f
x

𝑥 =
𝑓𝑥
𝑓
=
7350
60
= 128.5

27. 𝑮. 𝑴 = 𝒂𝒏𝒕𝒊𝒍𝒐𝒈
𝒇𝒍𝒐𝒈𝒙
𝒇
𝑮. 𝑴 = 𝒂𝒏𝒕𝒊𝒍𝒐𝒈
𝟏𝟐𝟒. 𝟐𝟒𝟖𝟑
𝟔𝟎
𝑮. 𝑴 = 𝒂𝒏𝒕𝒊𝒍𝒐𝒈 𝟐. 𝟎𝟕𝟎𝟖
𝑮. 𝑴 = 𝟏𝟏𝟕. 𝟕
𝑯. 𝑴 =
𝒇
𝒇
𝒙
𝑯. 𝑴 =
𝟔𝟎
𝟎.𝟓𝟑𝟎𝟒𝟒
𝑯. 𝑴 = 𝟏𝟏𝟑. 𝟏𝟏
A.M > G.M > H.M

28. X 0 1 2 3 4 5
f 2 2 4 6 8 3
C.B F
10 – 12 14
12 – 14 26
14 – 16 42
16 – 18 30
18 – 20 8
Calculate AM, GM, HM for the given data
Verify that A.M>G.M> H.M
(ANSWERS 3, G.M and H.M not possible)
(ANSWERS 14.87, 14.70, 14.53)