Measures of dispersion unitedworld school of business
1.
2. Designation: PGDM Student
Batch: 2012-2014.
Specialization: Major – Marketing
Minor – Finance
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3. 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
4. Measures of Dispersion
1. Range
2. Inter-quartile range
3. Quartile deviation
4. Mean deviation
5. Standard deviation
5. A. 1. Range = Xmax – X min = L-S
Xmax – X min L-S
2. Coefficient of Range = ---------------- = ------
Xmax + X min L+S
6. 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
7. 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
8. 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
9. 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
10. 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
11. 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
12. 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
14. 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
15. 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
16. Q4. Compute quartile deviation & its
coefficient for marks of 215 students
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
18. 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
19. 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
20. Inter quartile range = (Q3-Q1)
Quartile deviation = (Q3-Q1)/2
Q3-q1
Coefficient of quartile deiation = ---------------
--
q3+q1
21. Mean Deviation = sum of absolute deviations
from an average divided by total number of
items
Coefficient of Mean Deviation = mean
Deviation / Mean
22. Σ f(x-a)mod Σ f dmod
Mean deviation = ------------- = ---------------
Σ fx N
23. 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
24. Standard deviation of a series is the square
root of the average of the squared deviations
from the mean ( Average – Arithmatic mean)
25. 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
29. 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
30. 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
40. 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
41. 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
42. 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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