![Page 1: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/1.jpg)
Measures of Central Tendency
CPT Section D Quantitative Aptitude Chapter 11 Dr. R. B. Tiwari
![Page 2: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/2.jpg)
Introduction
Central Behavior of the data.
Off Central Behavior of the data.
![Page 3: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/3.jpg)
1. Measures of Central Tendency.
The tendency of a given set of observations to cluster around a single central or middle value and the single value that represents the given set of observations is described as a measure of central tendency or location or average.
e.g. a company is recognized by its high average profit and an educational institution is judged on the basis of the average marks obtained by the students.
![Page 4: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/4.jpg)
Arithmetic Mean
![Page 5: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/5.jpg)
Arithmetic Mean
Case (i) Arithmetic Mean (AM)
1 2 ...(i) nX X X
Xn
Xor X
n
+ + +=
=∑
![Page 6: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/6.jpg)
Case (ii) For Discrete Series
i.e. For x1-f1, x2-f2,…xn-fn type data
![Page 7: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/7.jpg)
If values of X are large enough
(ii)
1, 2,3....
.
ii
fdX A X C
NX A
where d i nC
A Assumed Mean
or C Common divisor for ungrouped data
= +
−= ∀ =
=
=
∑
![Page 8: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/8.jpg)
Case (iii) For Continuous Series
i. e. If frequencies are given to corresponding class intervals
![Page 9: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/9.jpg)
Example 1. Following are the daily wages in rupees of a sample of 9 workers: 58,62,48,53,70,52,60,84,75.Compute the mean wage.
Solution: Given that no. of workers, n = 9 ,and
58 62 48 53 70 52 60 84 75 562
562 62.449
Sum of the wages X
XTherefore the mean wage is givenby AM X
n
= + + + + + + + + =
= = = =
∑∑
![Page 10: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/10.jpg)
Solution:
Example 2: Find the AM for the following data: X : 10 20 30 40 50 f : 2 5 6 3 4
![Page 11: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/11.jpg)
Example 3.
Compute the mean weight of a group of BBA students of St. Xavier’s College from the following data:
• Weights in Kgs: 44-48 49-53 54-58 59-63 64-68 69-73
• No. of students. 3 4 5 7 9 8
![Page 12: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/12.jpg)
Solution Weights in Kgs. No. of
Students (f)
Mid –Values (X) fX
44-48 3 46 138
49-53 4 51 204
54-58 5 56 280
59-63 7 61 427
64-68 9 66 594
69-73 8 71 568
Totals N=36 2211
![Page 13: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/13.jpg)
Answer
2211 61.4236
fXX kgs
N= = =∑
![Page 14: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/14.jpg)
Example 4.
Find the AM for the following frequency distribution.
• Class Intervals: 350-369 370-389 390-409 410-429 430-449 450-469 470-489
• Frequency : 23 38 58 82 65 31 11
![Page 15: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/15.jpg)
Class intervals
Frequency (f)
Mid-Values
(X)
fd
350-369 23 359.50 -3 -69
370-389 38 379.50 -2 -76
390-409 58 399.50 -1 -58
410-429 82 419.50=A
0 0
430-449 65 439.50 1 65
450-469 31 459.50 2 62
470-489 11 479.50 3 33
Totals N=308 -43
419.5020
i ii
X A Xd
C− −
= =
Solution
![Page 16: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/16.jpg)
Solution
( )43419.50 20
308416.71
fdX A X C
N
X
= +
−= +
=
∑
Note that here A is taken as guessed mid – value and C is a common divisor. Then AM is given as
![Page 17: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/17.jpg)
Properties of AM
If all observations are equal to a constant, k say, Then AM is also equal to k.
![Page 18: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/18.jpg)
The algebraic sum of deviations taken about AM is always zero. i.e. e.g. the AM of 3,5 and 7 is 5 and (3-5)+(5-5)+(7-5)=0
( )( )
0 .
0 .i
i i
X X for ungrouped data and
f X X for grouped frequancy distribution
− =
− =
∑∑
Properties of AM
![Page 19: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/19.jpg)
AM is affected by change of origin and/or scale. i.e. if
.
fdX Ad and dC N
then X A C d
−= =
= +
∑
Properties of AM
![Page 20: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/20.jpg)
If there are two groups having n1 and n2 observations and
as their respective AMs , then the AM of combined group is given as
1 2X and X
1 1 2 2
1 2
n X n XX
n n+
=+
Properties of AM
![Page 21: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/21.jpg)
Example 5.
The mean salary for a group of 40 female workers is Rs. 5200 per month and that for a group of 60 male workers is Rs. 6800 per month. What is the combined salary?
![Page 22: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/22.jpg)
Solution.
1 1
2 2
1 1 2 2
1 2
40, 520060, 6800
40 5200 60 680040 60
.6160
n Xn XThen combined mean salary is givenby
n X n XX
n nX X
Rs
= =
= =
+=
++
=+
=
![Page 23: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/23.jpg)
Example 6.
Find the AM of the following data.
• Marks: less than 10 <20 <30 <40 <50
No. of students: 5 13 23 27 30
![Page 24: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/24.jpg)
Solution
Marks No. of students
(f)
Mid-value (X)
fx
0-10 5 5 25
10-20 13-5=8 15 120
20-30 23-13=10 25 250
30-40 27-23=4 35 140
40-50 30-27=3 45 135
N=30 670
![Page 25: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/25.jpg)
Solution: AM
670 22.3330
fXX
N= = =∑
![Page 26: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/26.jpg)
Example.7
Find the missing value Y, if the following frequency distribution has AM as 5.
• X: 3 5 Y 6 • f: 1 3 2 4
![Page 27: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/27.jpg)
Solution. AM is given by
1 3 3 5 2 6 4 42 21 3 2 4 10
42 2. . 510
50 42 24
fX Y YXN
Yi e
or Ytherefore Y
× + × + + × += = =
+ + ++
=
− ==
∑
![Page 28: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/28.jpg)
MCQ’s on Arithmetic Mean Question Time
![Page 29: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/29.jpg)
Question 1.
a) 2.5
b) 3.5
c) 4.5
d) 5.5
Answer: (b) 3.5
AM of 1,2,3,4,5,6 is
![Page 30: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/30.jpg)
Question 2.
(a) 3
(b) 5
(c) 7
(d) 9
Answer: (b) 5
AM of 5,5,5,5,5, is
![Page 31: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/31.jpg)
Question.3.
(a) 0
(b) 5
(c) -5
(d) None of these
Answer: (a) 0
The algebraic sum of deviations taken about the AM of 15, 20 and 25 is
![Page 32: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/32.jpg)
Question 4.
(a) 11/3
(b) 5
(c) 4
(d) 4.5
Answer: (a) 11/3
If a variable assumes the values 1,2,3,4,5 with frequencies as 1,2,3,4,5 respectively, then AM is
![Page 33: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/33.jpg)
(a) 40%
(b)50%
(c) 60%
(d) None of these
Answer: (a) 40%
The average salary of a group of unskilled workers is Rs. 10000 and that of a group of skilled workers is Rs. 15000.If
the combined salary is Rs. 12000, then what is the percentage of skilled workers?
Question.5
![Page 34: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/34.jpg)
Question 6 Find the AM for the following frequency distribution
Marks : 5-14 15-24 25-34 35-44 45-54 55-64 No. Of students: 10 18 32 26 14 10
(a) 30
(b) 29
(c) 33.68
(d) 34.21
Answer: (c) 33.68
![Page 35: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/35.jpg)
Question7
Find the AM of the following data.
• Marks: less than 20 <40 <60 <80 <100
• No. of students: 5 17 28 42 50
![Page 36: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/36.jpg)
Question 7
(a) 51.2
(b) 52.2
(c) 53.2
(d) 54.2
Answer: (c) 53.2
Question 7
(a) 51.2
(b) 52.2
(c) 53.2
(d) 54.2
Answer: (c) 53.2
![Page 37: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/37.jpg)
Question.8
Find the missing value Y, if the following frequency distribution has AM as 5.
• X: 6 4 Y 2 • f: 4 3 2 1
![Page 38: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/38.jpg)
Question 8
(a) 5
(b) 6
(c) 7
(d) 8
Answer: (b) 6
![Page 39: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/39.jpg)
Question.9
Find the AM of the following frequency distribution.
• X: 0-10 10-20 20-30 30-40 40-50 • f: 3 5 7 4 1
![Page 40: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/40.jpg)
Question 9
(a) 20.5
(b) 21.5
(c) 22.5
(d) 23.5
Answer: (c) 22.5
![Page 41: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/41.jpg)
Question.10
Find the AM of the following frequency distribution.
• X: 0-50 50-100 100-150 150-200 200-250 • f: 15 25 30 20 10
![Page 42: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/42.jpg)
Question 10
(a) 115.5
(b) 116.5
(c) 117.5
(d) 118.5
Answer: (c) 117.5
![Page 43: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/43.jpg)
Question.11
If the AM of the following frequency distribution is 14.7 for age in years of 20
individuals. Find the missing values. • X: 0-6 6-12 12-18 18-24 24-30 • f: 3 5 - - 4
![Page 44: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/44.jpg)
Question 11
(a) 4, 4
(b) 6, 2
(c) 5, 3
(d) 7, 1
Answer: (b) 6, 2
![Page 45: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/45.jpg)
Question.12 If the AM of the series 1,2,3,…n is 6. Find the value of n.
![Page 46: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/46.jpg)
Question 12
(a) 7
(b) 9
(c) 11
(d) 12
Answer: (c) 11
![Page 47: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/47.jpg)
Median
![Page 48: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/48.jpg)
Median (Md) and other Partition Values
Median is a positional average, which divides the data into two equal parts.
![Page 49: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/49.jpg)
If Y= a +b X then Md(Y) = a + b Md(X)
Properties of Median
![Page 50: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/50.jpg)
The sum of absolute deviations is minimum when it is measured about Md. i.e.
minimum if AX A is Md− =∑
Properties of Median
![Page 51: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/51.jpg)
Determination of Median Case (i): When number of observations is odd, i.e. n is odd.
First arrange the observations in ascending or descending order . If no. of observations is n which is an odd number, then
12
nThen Md th term+ =
![Page 52: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/52.jpg)
Example 1 : Find median of the following data 40, 29, 15, 58, 70
1 5 1 32 2
. . 40
rdnMd th term term
i e Md
+ + = = =
=
Solution: Let us arrange the observations in ascending order as 15, 29, 40, 58, 70 Here n= 5,Therefore
![Page 53: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/53.jpg)
Case(ii)
1 ( ) ( 1)2 2 2
n nThen Md th term th term = + +
When number of observations is even, i.e. n is even. First arrange the observations in ascending or descending order.
![Page 54: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/54.jpg)
Example 2. Find the Median of the following observations. 5, 25, 11, 14, 26, 30.
( )
1 ( ) ( 1)2 2 21 6 6 1( ) ( 1) 14 25 19.52 2 2 2
n nMd th term th term
Then Md th term th term
= + + = + + = + =
Solution: Let us arrange these observations into ascending order as 5, 11, 14, 25, 26, 30. Here n = 6 i.e. n is even, hence
![Page 55: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/55.jpg)
Case (iii)
For x1-f1, x2-f2,…xn-fn type data, Form a column of less than type cumulative
frequencies.
Then Md = Value of the variable X corresponding to the cumulative frequency
just greater than N/2
![Page 56: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/56.jpg)
Example 3.
Find the median of the following frequency distribution.
• X: 60 65 70 75 80 85 90 • f : 7 12 20 22 18 14 9
![Page 57: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/57.jpg)
Solution:
X f Less than type c.f.
60 7 7
65 12 19
70 20 39
75 22 61
80 18 79
85 14 83
90 9 92
Total N=92
![Page 58: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/58.jpg)
Solution
Here the c.f. just greater than N/2 i.e. 46, is 61. Therefore the corresponding value of X is Median, i.e. Md = 75
![Page 59: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/59.jpg)
Case (iv)
For grouped data i.e. if frequencies are given corresponding to the class
intervals.
First find the Md class corresponding to the cumulative
frequency just greater than N/2.
![Page 60: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/60.jpg)
2
limit
(besure that theclass intervals are , . . )
N cThen Md l h
fwhere l lower of the Md class
f frequency of the Md classc cumulative frequency preceeding the Md classh class width exclussivetype i e continuous classes
− = + ×
====
Median Formula for Continuous Series
![Page 61: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/61.jpg)
Example. 4.
Find the Median of the following data.
• Marks: 0- 10 10-20 20-30 30-40 40-50 • No. of students: 5 8 10 7 6
![Page 62: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/62.jpg)
Solution Marks
Class Intervals No. Of students
(f) c.f.
0-10 5 5
10-20 8 13=c
20-30 10=f 23
30-40 7 30
40-50 6 36
Totals N=36
![Page 63: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/63.jpg)
Solution
2
36 132. . 20 10 25
10
N cThen Md l h
f
i e Md
− = + ×
− = + × =
The c.f. just greater than N/2 i.e. 18, is 36, which is the Md class. Hence, Note that if class intervals are not continuous then they are first made continuous by adding d/2 to the all upper limits and subtracting d/2 from all the lower limits, where d is the difference between lower limit of any class and upper limit of the previous class.
![Page 64: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/64.jpg)
Quartiles There are three quartiles viz. Q1, Q2 and Q3,which divide the data into four equal parts.
![Page 65: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/65.jpg)
Case (i) For x1-f1, x2-f2,…xn-fn type data
First form a column of cumulative frequencies then the value of the variable corresponding to the c.f. just greater than
iN/4 gives the ith quartile.
![Page 66: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/66.jpg)
Then ith (i=1,2,3) Quartile (Qi) is given by
4
limit
(besure that theclass intervals are , . . )
i
i
i
i
iN cThen Q l h
fwhere l lower of the Q class
f frequency of the Q classc cumulative frequency preceeding the Q classh class width exclussivetype i e continuous classes
− = + ×
====
Quartile
![Page 67: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/67.jpg)
Example 1 Find the third quartile for the following
frequency distribution.
X: 10 20 30 40 50 60 70 80 90 100 f: 3 5 10 20 30 40 45 35 7 5
![Page 68: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/68.jpg)
Solution X f c.f. 10 3 3 20 5 8 30 10 18 40 20 38 50 30 68 60 40 108 70 45 153 80 35 188 90 7 195
100 5 200 Total N= 200
![Page 69: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/69.jpg)
Here to calculate third quartile we see the value of X
corresponding to the
cummulative frequency just greater than
3N/4=3x200/4=150, which is 153. Hence third quartile
is given by
Q3 = 70
Solution
![Page 70: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/70.jpg)
Case (ii)
If frequencies are given to corresponding class intervals.
![Page 71: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/71.jpg)
Then ith (i=1,2,3) Quartile (Qi) is given by
4
limit
(besure that theclass intervals are , . . )
i
i
i
i
iN cThen Q l h
fwhere l lower of the Q class
f frequency of the Q classc cumulative frequency preceeding the Q classh class width exclussivetype i e continuous classes
− = + ×
====
Quartile
![Page 72: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/72.jpg)
Example 2
• Marks: 0- 10 10-20 20-30 30-40 40-50 50-60 • No. of students: 5 8 10 7 6 4
Find the First and Third Quartiles for the following frequency distribution.
![Page 73: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/73.jpg)
Solution Marks
Class Intervals No. Of students
(f) c.f.
0-10 5 5
10-20 8 13
20-30 10 23
30-40 7 30
40-50 6 36
50-60 4 40
Totals N=40
![Page 74: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/74.jpg)
1
1
4
40 54. . 10 10 16.25
8
N cThen Q l h
f
i e Q
− = + ×
− = + × =
And
3
3
34
3 40 304. . 40 10 40
6
N cThen Q l h
f
i e Q
− = + ×
× − = + × =
Solution
![Page 75: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/75.jpg)
Deciles
There are nine deciles viz. D1, D2,…D9, which divide
the data into 10 equal parts.
![Page 76: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/76.jpg)
Case (i) For x1-f1, x2-f2,…xn-fn type data, Form a column of less than type cumulative frequencies.
Then ith decile is the value of the variable X corresponding to cumulative frequency just
greater than iN/10.
![Page 77: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/77.jpg)
Example 1
Find the 5th decile for the following frequency distribution.
X: 10 20 30 40 50 60 70 80 90 100 f: 3 5 10 20 30 40 45 35 7 5
![Page 78: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/78.jpg)
Solution X f c.f. 10 3 3 20 5 8 30 10 18 40 20 38 50 30 68 60 40 108 70 45 153 80 35 188 90 7 195
100 5 200 Total N= 200
![Page 79: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/79.jpg)
Solution
Here we have formed the column of cumulative frequencies, now see the
c.f. just greater than 5N/10=5x200/10=100, which is 108,
hence the corresponding value of X is fifth decile, i.e.
D5 = 60
![Page 80: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/80.jpg)
Case (ii) For grouped data, i.e. frequencies against class intervals are given.
Then ith decile class is obtained corresponding to
the cumulative frequency just greater than iN/10.
![Page 81: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/81.jpg)
Then ith (i-1,2,…9) decile (Di) is given by
10
limit
(besure that theclass intervals are , . . )
i
i
i
i
iN cD l h
fwhere l lower of the D class
f frequency of the D classc cumulative frequency preceeding the D classh class width exclussivetype i e continuous classes
− = + ×
====
![Page 82: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/82.jpg)
Example 2
Find the seventh decile for the following frequency distribution.
Marks : 0- 10 10-20 20-30 30-40 40-50 50-60 No. of students: 5 8 10 7 6 4
![Page 83: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/83.jpg)
Solution
Marks Class
Intervals
No. of students (f)
c.f.
0-10 5 5
10-20 8 13
20-30 10 23
30-40 7 30
40-50 6 36
50-60 4 40
Totals N=40
![Page 84: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/84.jpg)
First we find the seventh decile class
corresponding to the c.f. just greater than 7N/10=28, i.e. 30. Hence
7
7
7
710
7 40 2310. . 30 10
7i. . 30 7.14 37.14
N cThen D l h
f
i e D
e D
− = + ×
× − = + ×
= + =
![Page 85: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/85.jpg)
Percentiles:
Percentiles divide the whole data into 100 equal
parts. They are 99 in number and denoted by
P1,P2,…P99
![Page 86: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/86.jpg)
Case (i) For x1-f1, x2-f2,…xn-fn type data.
First, form a column of less than type cumulative frequencies. Identify the ith
percentile class for which the cumulative frequency is just greater than iN/100.
![Page 87: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/87.jpg)
Example 1
Find the 65th percentile for the following frequency distribution.
X: 10 20 30 40 50 60 70 80 90 100 f: 3 5 10 20 30 40 45 35 7 5
![Page 88: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/88.jpg)
Solution X f c.f. 10 3 3 20 5 8 30 10 18 40 20 38 50 30 68 60 40 108 70 45 153 80 35 188 90 7 195
100 5 200 Total N= 200
![Page 89: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/89.jpg)
The 65th percentile is the value of X for which corresponding cumulative frequency is just greater
than 65N/100=65x200/100=130, which is 153 .
Hence P65 = 70
The 65th Percentile
![Page 90: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/90.jpg)
Case (ii) For grouped data, i.e. frequencies are given corresponding to class intervals.
First obtain the ith percentile class which is corresponding to the cumulative frequency just
greater than iN/100.
![Page 91: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/91.jpg)
Then ith (i=1,2,…99) percentile (Pi) is given by
100
limit
(besure that theclass intervals are , . . )
i
i
i
i
iN cP l h
fwhere l lower of the P class
f frequency of the P classc cumulative frequency preceeding the P classh class width exclussivetype i e continuous classes
− = + ×
=
=
=
=
Percentile
![Page 92: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/92.jpg)
Example 2.
• Marks: 0- 10 10-20 20-30 30-40 40-50 50-60 • No. of students: 5 8 10 7 6 4
Find the 52nd percentile for the following frequency distribution.
![Page 93: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/93.jpg)
Solution Marks
Class Intervals No. Of students
(f) c.f.
0-10 5 5
10-20 8 13
20-30 10 23
30-40 7 30
40-50 6 36
50-60 4 40
Totals N=40
![Page 94: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/94.jpg)
The 52nd percentile
52
52
52
52100
52 40 1310020 10
1020 7.8 27.8
N cP l h
f
or P
P
× − = + ×
× − = + ×
= + =
First we obtain the 52nd Percentile class which is corresponding to the c.f. just greater than 52N/100 i.e. 52x40/100=20.8, which is 23, shown as red.
![Page 95: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/95.jpg)
Partition Values for Ungrouped Data:
For unclassified data the pth Quartile is given by (n+1)pth value.
Take p=1/4 for first quartile p=2/4 for second quartile
p=3/4 for third quartile
![Page 96: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/96.jpg)
Example
Find Q3, for the wages of labourers: • Rs. 82, 56, 90, 50, 120, 75,
79, 80, 130, and 65.
![Page 97: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/97.jpg)
Solution
( )
( )
33143i. . 10 1 8.254
. . 8 0.25(9 value 8 value)i.e. 90 0.25(120 90)i.e. 90.5
th
th th th
Q n th value
e th value value
i e value
= + ×
= + × =
= + −= + −=
Arrange the wages in ascending order as : 50, 56, 65, 75, 79, 80, 82, 90, 120, 130
![Page 98: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/98.jpg)
Partition Values for Ungrouped Data:
For unclassified data the pth Decile is given by (n+1)pth value.
Take p=1/10 for first decile, p=2/10 for second decile, . . . p=9/10 for ninth decile.
![Page 99: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/99.jpg)
Example
Find D7, for the wages of labourers:
• Rs. 82, 56, 90, 50, 120, 75, 79, 80, 130, and 65.
![Page 100: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/100.jpg)
Solution
( )
( )
7
7
7110
7i. . 10 1 7.710
. . 7 0.7 (8 value 7 value)i.e. 82 0.7(90 82)i.e. D 87.6
th
th th th
D n th value
e th value value
i e value
= + ×
= + × =
= + −= + −
=
Arrange the wages in ascending order as : 50, 56, 65, 75, 79, 80, 82, 90, 120, 130
![Page 101: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/101.jpg)
Partition Values for Ungrouped Data:
For unclassified data the pth Percentile is given by (n+1)pth value.
Take p=1/100 for first percentile, p=2/100 for second percentile, . . . p=99/100 for ninety ninth percentile.
![Page 102: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/102.jpg)
Example
Find P73, for the wages of labourers: • Rs. 82, 56, 90, 50, 120, 75,
79, 80, 130, and 65.
![Page 103: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/103.jpg)
Solution
( )
( )
73
73
731100
73i. . 10 1 8.03100
. . 8 0.03(9 value 8 value)i.e. 90 0.03(120 90)i.e. 90.9
rd
th th th
P n th value
e th value value
i e value
P
= + ×
= + × =
= + −= + −
=
Arrange the wages in ascending order as : 50, 56, 65, 75, 79, 80, 82, 90, 120, 130
![Page 104: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/104.jpg)
Question Time M.C.Q's. on Median and other
Partition Values
![Page 105: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/105.jpg)
Question 1
What is the Median of 5, 8, 6, 9, 11 and 4.
a) 6
b) 7
c) 8
d) None of these
Answer: (b) 7
![Page 106: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/106.jpg)
Question 2
What is the value of the first Quartile for the observations 15, 18,10, 20, 23, 28, 12, 16?
(a) 17
(b) 15.75
(c) 12
(d) 12.75
Answer: (d) 12.75
![Page 107: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/107.jpg)
Question 3.
(a) 13
(b) 10.70
(c) 11
(d)11.50
Answer: (b) 10.70
The third decile for the numbers 15, 10, 20, 25, 18, 11, 9, 12 is
![Page 108: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/108.jpg)
Question 4
Find the median of 12, 4, 36, 45, 3, 58, 65?
(a) 45
(b) 36
(c) 58
(d) 42
Answer: (b) 36
![Page 109: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/109.jpg)
Question 5
Find the 73rd percentile of 4, 12, 36, 45, 3, 58, 65?
(a) 54.92
(b) 55.92
(c) 56.92
(d) 57.92
Answer: (b) 55.92
![Page 110: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/110.jpg)
Question 6 Find the missing frequency if the median mark is 23?
X : 0-10 10-20 20-30 30-40 40-50 f : 5 8 ? 6 3
(a) 9
(b) 11
(c) 10
(d) 7
Answer: (c) 10
![Page 111: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/111.jpg)
Question 7
The absolute sum of deviations is minimum when it is measured about
(a) Mean
(b) Median
(c) Mode
(d) None of these
Answer: (b) Median
![Page 112: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/112.jpg)
Question 8
The variables X and Y are given by Y= 3X+6. If the median of X is 10, what is the median of Y?
(a) 3
(b) 6
(c) 18
(d) 36
Answer: (d) 36
![Page 113: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/113.jpg)
Question 9
What is the value of the first quartile for observations 15, 18, 10, 20, 23, 28, 12, 16?
(a) 17
(b) 16
(c) 15.75
(d) 12
Answer: (c) 15.75
![Page 114: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/114.jpg)
Question 10
The third decile for the numbers 15, 10, 20, 25, 18, 11, 9, 12 is
(a) 13
(b) 10.70
(c) 11
(d) 11.50
Answer: (b) 10.70
![Page 115: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/115.jpg)
Question 11 Find the 27th percentile for the following frequency distribution. X: 0-9 10-19 20-29 30-39 39-49 f: 5 7 12 10 6
(a) 13
(b) 17.79
(c) 11
(d) 11.50
Answer: (b) 17.79
![Page 116: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/116.jpg)
Mode
![Page 117: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/117.jpg)
Mode (Mo):
This is an industrial average.
The most frequent value of the variable is called mode.
![Page 118: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/118.jpg)
Determination of Mode:
Case (i) When data is unclassified.
• e.g. Mo of 5, 3, 8, 9, 5, 6 is 5 which is the most frequent value.
![Page 119: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/119.jpg)
0 1
0 1 1
0
1
1
2
limit of modalclassf modal classf modal classf modal
(note that class intervals must be continuous)
f fMo l hf f f
wherel lower
frequency of thefrequency of pre
frequency of post classh class width
−
−
−
−= + × − −
=== −= −=
Case (ii) For grouped data i.e. if frequencies corresponding to class intervals are given, then Mo is given as
![Page 120: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/120.jpg)
Example:
Find the Mo for the following frequency distribution.
Class Intervals: 350-369, 370-389, 390-409, 410-429, 430-449, 450-469, 470-489 Frequency : 23, 38, 58, 82, 65, 31, 11
![Page 121: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/121.jpg)
Solution Class Intervals Continuous Class Intervals Frequency
350-369 349.5-369.5 23
370-389 369.5-389.5 38
390-409 389.5-409.5 58
410-429 409.5-429.5 82
430-449 429.5-449.5 65
450-469 449.5-469.5 31
470-489 469.5-489.5 11
![Page 122: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/122.jpg)
Solution
0 1
0 1 12
82 58409.5 202 82 58 65
. . 409.5 11.71 421.21
f fMo l h
f f f
Mo
i e Mo
−
−
−= + × − −
− = + × × − − = + =
![Page 123: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/123.jpg)
Relation between Mean, Median And Mode:
Mode =3 Median-2 Mean • (OR)
Mean-Mode=3(Mean-Median)
![Page 124: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/124.jpg)
MCQ’ on Mode
Question Time
![Page 125: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/125.jpg)
Question 1 What is the Modal value for the numbers 5, 8, 6, 4, 10,15, 18, 10, 5, 10?
(a) 5
(b) 10
(c) 14
(d) None of these
Answer: (b) 10
![Page 126: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/126.jpg)
Question 2 The Mode for the following frequency distribution:
(a) 390
(b) 394.86
(c) 395.86
(d) 396
Answer: (c) 395.86
Class Intervals: 350-369, 370-389, 390-409, 410-429, 430-449, 450-469 Frequency : 15 27 31 19 13 6
![Page 127: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/127.jpg)
Question 6 Following is an incomplete distribution having modal mark as 44.
(a) 45
(b) 46
(c) 47
(d) 48
Answer : (d) 48
Marks: 0-20, 20-40, 40-60, 60-80, 80-100 No. of students: 5 18 X 12 5 What would be the mean marks?
![Page 128: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/128.jpg)
Question 7 Measures of Central Tendency for given set of observations measures
(a) The scatterness of observations
(b) the central location of observations
(c) Both (a) and (b)
(d) None of these
Answer: (b) the central location of observations
![Page 129: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/129.jpg)
Question 8
(a) The data are not classified
(b) The data are put in the form of grouped frequency distribution
(c) All the observations are not of equal importance
(d) Both (i) and (iii)
Answer: (c) All the observations are not of equal importance
Weighted averages are considered when
![Page 130: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/130.jpg)
Geometric Mean
![Page 131: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/131.jpg)
Geometric Mean (G)
GM is defined if no observation is zero or negative.
![Page 132: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/132.jpg)
Geometric Mean (G)
GM of n positive observations is defined as the nth root of
their product.
![Page 133: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/133.jpg)
Geometric Mean (G)
GM is used to calculate average growth rate or average of ratios.
![Page 134: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/134.jpg)
Geometric Mean (G)
If Z=X.Y then GM(Z)=GM(X).GM(Y)
![Page 135: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/135.jpg)
Geometric Mean (G)
If Z=X/Y then GM(Z)=GM(X)/GM(Y)
![Page 136: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/136.jpg)
1
1 2 n(X .X ...X )1log log
nG
or G Xn
=
= ∑
GM of X1, X2,…Xn is given by
GM for Individual series
![Page 137: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/137.jpg)
GM for Discrete Series
( )1 2
1
1 2. ...
1log log
nff f NnG X X X
or G f XN
=
= ∑
GM for X1-f1, X2-f2,…Xn-fn type data
![Page 138: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/138.jpg)
Example 1
( ) ( )1 13 32 3 4 64 4G = × × = =
GM of 2, 3, and 4 is given as
![Page 139: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/139.jpg)
Example 2.
• X : 2 4 8 16 • f : 2 3 3 2
Find the GM of the following frequency distribution.
![Page 140: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/140.jpg)
Solution ( )
( )
( )
( )
1 2
1
1 2
12 3 3 2 (2 3 3 2)
12 6 9 8 10
125 2.510
. ...
2 .4 .8 .16
2 .2 .2 .2
2 2 4 2
5.66
nff f NnG X X X
or G
or G
or G
G
+ + +
=
=
=
= = =
=
![Page 141: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/141.jpg)
Harmonic Mean
![Page 142: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/142.jpg)
Harmonic Mean (H)
HM is defined if no observation is zero.
![Page 143: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/143.jpg)
Harmonic Mean (H)
HM of n observations is defined as the reciprocal of Arithmetic Mean of the
reciprocals of the observations.
![Page 144: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/144.jpg)
Harmonic Mean (H)
This is used to calculate the average speeds.
![Page 145: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/145.jpg)
Harmonic Mean (H) Combined HM of two groups having n1 and n2
observations and H1 and H2 as their respective Harmonic Means is given by
( )1 2
1 2
1 2
n nH
n nH H
+=
+
![Page 146: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/146.jpg)
Case (i)
1nH
X
=
∑
HM of X1, X2, … Xn is
![Page 147: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/147.jpg)
Example 1
3 3 36 3.27111 1 1 1 11122 4 6
nH
X
= = = = = + +
∑ ∑
HM of 2, 4, and 6 is
![Page 148: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/148.jpg)
Find the HM of the following data X : 2 4 8 16 f : 2 3 3 2
Example 2
( )2 3 3 24.44
2 3 3 22 4 16
:
8
NHf
Solut
X
ion+ + +
= = = + + +
∑ ∑
![Page 149: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/149.jpg)
Case(ii) If the data is of X1-f1, X2-f2,…Xn-fn type
data, then HM is given by
NH where N ffX
= = ∑∑
![Page 150: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/150.jpg)
Relation Between AM, GM and HM: 2( ) ( ) ( )
.
G A H
MoreoverAM GM HMNote that equality holds if all the observations are equal
= ×
≥ ≥
![Page 151: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/151.jpg)
Weighted Average
![Page 152: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/152.jpg)
Weighted Average:
If all the observations are not equally important i.e. weights are assigned to them, then Weighted AM, Weighted GM and Weighted
HM are calculated.
![Page 153: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/153.jpg)
Weighted AM is
WX
AMW
= ∑∑
![Page 154: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/154.jpg)
Weighted GM is
log
logW X
GM AnteW
=
∑∑
![Page 155: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/155.jpg)
Weighted HM is
WHM
WX
= ∑∑
![Page 156: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/156.jpg)
Example 1
Find the weighted AM and weighted HM for the following data
X : 1 2 3 4 5 6 7 8 9 10 W : 12 22 32 42 52 62 72 82 92 102
![Page 157: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/157.jpg)
Solution 2 2 2
2 2 2
2
3 3 3
2 2 2
1 1 2 2 ...1 2 ...
(n 1)1 2 ... 2
(n 1)(2n 1)1 2 ...6
3n(n 1)2(2n 1)
WX n nWeighted AMW n
nn
nn
or AM
× + × + + ×= =
+ + +
+ + + + = =+ ++ + +
+=
+
∑∑
![Page 158: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/158.jpg)
Solution 2 2 2
2 2 2
2 2 2
1 2 ...1 2 ...1 2
(n 1)(2n 1)1 2 ... 6
(n 1)1 2 ...2
(2n 1)3
W nWeighted HMW nX n
nn
nn
or HM
+ + += =
+ + +
+ + + + + = =
++ + +
+=
∑∑
![Page 159: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/159.jpg)
Question Time MCQ’ on GM & HM
![Page 160: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/160.jpg)
Question 1
(a) AM
(b) GM
(c) HM
(d) Both (b) & (c)
Answer: (b) GM
Which measure of central tendency is used to find average rates.
![Page 161: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/161.jpg)
Question 2
(a) AM ≥ GM ≥ HM
(b) HM ≥ GM ≥ AM
(c) AM > GM > HM
(d) GM > AM > HM
Answer: (a) AM ≥ GM ≥ HM
Which of the following results hold for a set of distinct positive observations?
![Page 162: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/162.jpg)
Question 3
(a) 2.00
(b) 3.33
(c) 2.90
(d) -3.33
Answer: (c) 2.90
The Harmonic Mean for the numbers 2,3,5 is
![Page 163: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/163.jpg)
Question 4
(a) 24
(b) 12
(c) 36
(d) 18
Answer: (b) 12
What is the Geometric Mean of the numbers 4,12, 36?
![Page 164: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/164.jpg)
Question 5
(a) 6 and 7
(b) 9 and 4
(c) 10 and 3
(d) 8 and 5
Answer (b) 9 and 4
If the AM and GM for the two numbers are 6.50 and 6 respectively then the two numbers are
![Page 165: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/165.jpg)
Question 6
(a) 16
(b) 4.10
(c) 4.05
(d) 4.00
Answer: (d) 4
If the AM and HM for two numbers are 5 and 3.2 respectively then the GM will be
![Page 166: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/166.jpg)
Question 7
(a) 150
(b) log 10 x log15
(c) log 150
(d) none of these
Answer: (a) 150
If GM of X is 10 and GM of Y is 15, then the value of GM of XY is
![Page 167: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/167.jpg)
Question 8
(a) 40
(b) 50
(c) 20
(d) 30
Answer: (a) 40
If AM of a variable X is 10 and GM of X is 20, then the HM of X is
![Page 168: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/168.jpg)
Question 9
(a) n
(b) 2n
(c) 2/(n+1)
(d) n(n+1)/2
Answer: (c) 2/(n+1)
The HM of 1, 1/2, 1/3 … 1/n is
![Page 169: Measures of Central Tendency - ICAI Knowledge …. Measures of Central Tendency. The tendency of a given set of observations to cluster around a single central or middle value and](https://reader031.vdocuments.net/reader031/viewer/2022011801/5af663237f8b9a954690b451/html5/thumbnails/169.jpg)
Thank You