Measures of dispersion - Unitedworld School of Business

2. Measures of Dispersion
Are measures of scatter ( spread) about an
average
i.e. extent to which individual items vary
Measures of Dispersion
Absolute Measures – measure value in same units
– age
Relative Measures - % or coefficient of absolute
measures

3. Measures of Dispersion
1. Range
2. Inter-quartile range
3. Quartile deviation
4. Mean deviation
5. Standard deviation

4. A. 1. Range = Xmax – X min = L-S
Xmax – X min L-S
2. Coefficient of Range = ---------------- = ------
Xmax + X min L+S

5. Measures of Dispersion
Q1. Calculate range & co-efficient of range from
following information
480,562,570,322,435,497,675,732,375,482,791,820,275

6. B. Quartiles
1.Inter quartile range = Q3 – Q1
2. Quartile déviation
or semi inter quartile range = ( Q3 – Q1)/2
a. In a normal distribution
Q1 < Q2 < Q3
Q2 = M
b. In a symmetrical distribution
Q2 + Quartile Déviation = Q3
Q2 - Quartile Déviation = Q1

7. Q1 = first quartile or lower quartile
Q2 = second / middle Quartile or median
Q3 = third quartile or upper quartile
Q3 – Q1
Coefficient of Quartile deviation = -----------
Q3 + Q1
Coefficient of Quartile Deviation
Deviation by Quartiles =---------------------- x 100
Median

8. Calculation of Quartile deviation under
continuous series
1. If inclusive class intervals , convert to
exclusive class intervals
2. Size of class intervals should be equal
throughout distribution
3. L2 of first class interval should be equal to
L1 of next class interval
4. If mid values are given , it is necessary to
determine class intervals
5. If it is open end type of frequency
distribution , coefficient of variation is
suitable measure

9. Calculation of Quartile deviation
N+1
Q1=size of (---------) th item of the series
4
3(N+1)
Q3=size of ---------) th item of the series
4

10. Q2. Calculate quartile deviation & its co-efficient
for the data given below
168
147 150 169 170 154 156 171 162 159 174 173 166 164 172

11. Q3. Compute quartile deviation & its coefficient
for following data
X 10 12 14 16 18 20 22 24 28 30 34 36 38
F 3 6 10 15 20 24 30 22 18 14 10 6 6
Soln. calculate cumulative frequency
calculate Q1=N+1/4 &Q3=3(N+1)/4 th observation

12. Procedure:
Compute cumulative frequency
N
Find out Q1 & Q3 classes by m (Q1)=---------
4
3N
& m(Q3)= ---------
4

13. After locating l1, l2 , f & c substitute values in
l2-l1 N
Q1= l1+ --------- ( m-c) where m =---------
f 4
N/4 - C
Q1 =l1+ --------- (l2-l1)
f
l1= lower limit of quartile class
l2 = upper limit of quartile class
f =frequency of quartile class
c =cumulative frequency before quartile class
M = quartile position

14. After locating l1, l2 , f & c substitute values in
l2-l1 3N
Q3= l1+ --------- ( m-c) where m =---------
f 4
3N/4 - C
Q3 =l1+ --------- (l2-l1)
f

15. Q4. Compute quartile deviation & its s
coefficient for marks of 215 student
Marks 0-
10
10-
20
20-
30
30-
40
40-
50
50-
60
60-
70
70-
80
80-
90
90-
100
Students 10 15 28 32 40 35 26 14 10 5
Soln. condition if class interval inclusive convert into
exclusive, class size equal
calculate cumulative frequency
calculate m (Q1)=N/4 &m(Q3)=3N/4 th observation
inter quartile range = (Q3-Q1)
quartile deviation = (Q3-Q1)/2

16. X 10 12 14 16 18 20 22 24 28 30 34 36 38
F 3 6 10 15 20 24 30 22 18 14 10 6 6
cf 3 9 19 34 54 78 10
8
130 148 162 172 178 184
Soln. calculate cumulative frequency
calculate Q1=N+1/4 &Q3=3(N+1)/4 th observation
inter quartile range = (Q3-Q1)
quartile deviation = (Q3-Q1)/2
Q3-Q1
coefficient of quartile deiation = -----------------
Q3+Q1

17. l2-l1 N
Q1 = l1+--------------* (m-c) m= ------------
m 4
l1- lower limit of Q1 class , l2= upper limit of Q1 class
f = frequency of Q1 class , c= cumulative frequency
before Q1 class

18. l2-l1 3N
Q3 = l1+--------------* (m-c) m= -------------
f 4
l1- lower limit of Q3 class , l2= upper limit of Q3 class
f = frequency of Q3 class , c= cumulative frequency
before Q3 class

19. Inter quartile range = (Q3-Q1)
Quartile deviation = (Q3-Q1)/2
Q3-q1
Coefficient of quartile deiation = ---------------
--
q3+q1

20. Mean Deviation = sum of absolute deviations
from an average divided by total number of
items
Coefficient of Mean Deviation = mean
Deviation / Mean

21. Σ f(x-a)mod Σ f dmod
Mean deviation = ------------- = ---------------
Σ fx N

22. Q5A. calculate mean deviation & coefficient of
mean for the following two series
A
105 112 110 125 138 149 161 175 185 190
B 22 24 26 28 30 32 34 40 44 50

23. Standard deviation of a series is the square
root of the average of the squared deviations
from the mean ( Average – Arithmatic mean)

24. Standard deviation σ – positive square root of
arithmetic mean of squares of deviations
Σ dx2 Σ fdx2
σ = √ (-------) = (--------)
N N
For frequencies of a value
σ
Coefficient of Standard deviation = ------------------
average
σ
Coefficient of variation = --------------- x 100
average

25. Q5Calculate standard deviation & coefficient of
variation
X 65 67 68 68 69 71 72 72

26. Q6.Calculate standard deviation & coefficient of
variation
X 95 10
0
10
5
11
5
12
5
13
0
13
5
14
0
15
0
16
0
170
f 5 8 12 15 35 40 30 20 10 10 10

27. Q6.Calculate standard deviation & coefficient of
variation
X 95 100 105 115 125 130 135 140 150 160 170
f 5 8 12 15 35 40 30 20 10 10 10
dx=(x-
130)
-35 -30 -25 -15 -05 0 5 10 20 30 40

28. Standard deviation σ – positive square root of
arithmetic mean of squares of deviations
Σ dx Σ dx2 Σ dx
σ = √ (-------)2 = √ ------- - (------------) 2
N N N
Σ fdx Σ fdx2 Σ fdx
σ = √ (-------)2 = √ ------- - (------------) 2
Σ f Σ f Σ f

29. A 158 160 163 165 167 170 172 175 177 181
B 163 158 167 170 160 180 170 175 172 175
By using standard deviation find out which series
is more variable

30. A 15
8
16
0
16
3
16
5
16
7
17
0
17
2
17
5
17
7
18
1
1688 168.8
B 16
3
15
8
16
7
17
0
16
0
18
0
17
0
17
5
17
2
17
5
1690 169
dxA -12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2

31. A 15
8
16
0
16
3
16
5
16
7
17
0
17
2
17
5
17
7
18
1
1688 168.8
B 16
3
15
8
16
7
17
0
16
0
18
0
17
0
17
5
17
2
17
5
1690 169
dxA -12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2
dx2
A
14
4
10
0
49 25 9 0 4 25 49 12
1
526

32. A 158 160 163 165 167 170 172 175 177 181 1688 168.8
B 163 158 167 170 160 180 170 175 172 175 1690 169
dxA -12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2
dx2A 144 100 49 25 9 0 4 25 49 121 526
dxB -7 -12 -3 0 -10 10 0 5 2 5 10 (dxB)2

33. A 158 16
0
16
3
16
5
16
7
17
0
17
2
17
5
17
7
181 1688 168.8
B 163 15
8
16
7
17
0
16
0
18
0
17
0
17
5
17
2
175 1690 169
dx
A
-12 -10 -7 -5 -3 0 2 5 7 11 -12 (dxA)2
dx2
A
144 10
0
49 25 9 0 4 25 49 121 526
dx
B
-7 -12 -3 0 -10 10 0 5 2 5 10 (dxB)2
dx2
B
49 14
4
9 0 10
0
10
0
0 25 4 25 456

34. Σ dx Σ dx2 Σ dx 526 12
σ = √ (-------)2 = √ ------- - (------------) 2 =√ [ ----------- - ( ------)
2
N N 10
10
=√ [ 52.6- 1.2*1.2] =√ 52.6-1.44 =√ 51.46 =7.2
Coefficient of variation = σ / x bar = 7.2*100/ 166.8 = 4.26%

35. Σ dx Σ dx2 Σ dx
Σ dx Σ dx2 Σ dx 456 10
σ = √ (-------)2 = √ ------- - (------------) 2 =√ [ ----------- - ( ------
) 2
N N 10
10
=√ [ 45.6- 1] =√ 44.6 = 6.7
Coefficient of variation = σ / x bar = 6.7*100 / 169 = 3.96%

36. Σ fd’x Σ fd’x2 Σ fd’x
σ = √ (-------)2 x i = [√ ------- - (------------) 2 ]xi
Σ f Σ f Σ f
i= class interval

37. Σ fd’x Σ fd’x2 Σ fd’x
σ = √ (-------)2 x i = [√ ------- - (------------) 2 ]xi
Σ f Σ f Σ f
i= class interval

38. class 80-
84
75-
79
70-
74
65-
69
60-
64
55-
59
50-
54
45-
49
40-
44
35-
39
30-
34
25-
29
Mid v 82 77 72 67 62 57 52 47 42 37 32 27
frequenc
y
1 1 1 4 4 7 6 6 6 3 0 1
dx=x-52 30 25 20 15 10 5 0 -5 -10 -15 -20 -25
d’x=(x-
52/5)
6 5 4 3 2 1 0 -1 -2 -3 -4 -5
fd’x 6 5 4 12 8 7 0 -6 -12 -9 0 -5
d’x2 36 25 16 9 4 1 0 1 4 9 16 25
fd’x2 36 25 16 36 16 7 0 6 24 27 0 25
σ = i* σA = 5* σ
Σ dx Σ fd’x2 Σ fd’x 218 10
σ = √ (-------)2 = √ ------- - (------------) 2
=√ [ --------- - ( ------) 2
N N N 40 40
=√ [ 5.45- 0.25 ] =√ 5.20 = 2.32
σ = i* σA = 5* σ = 5*2.32 = 11.60

39. Standard deviation σ – positive square root of
arithmetic mean of squares of deviations
Properties
Standard deviation σ is independent of change of
origin but not of scale
If dx = x-A σx=σd
x-A
If d’x =------ σx= i.σd
i

40. Standard deviation σ – positive square root of
arithmetic mean of squares of deviations
N1σ1
2 + N2σ2
2 + N3σ3
2 + …….Nnσn
2
σ12...n = √ --------------------------------------------
N1 + N2+ N3 +……….. Nn

41. compute coefficient of variation & comment which
factory profits are more consistent
Particular
s
Factory A Factory B
Average
profits
19.7 21
Standard
deviation
6.5 8.64

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