The commonly used measures of absolute dispersion are: 1. Range 2. Quartile Deviation 3. Mean (Average) Deviation 4. Variance and Standard Deviation

2. 2

3. The scatter of the values about their centre is called
dispersion and any measure indicating the amount of
scatter about the centre is called a Measure of
Dispersion.
The individual observations of a variable tend to scatter
about their centre. The highest degree of
concentration is that all the observations are of same
size. The scatter in this case would be zero and mean
will be exactly same as the individual values of the
variable.
3

4. There are two main types of measures of
dispersion:
1. Absolute Measure of Dispersion
2. Relative Measure of Dispersion
Absolute Measure of Dispersion
The absolute measure of dispersion
measures the variation present among the
observations in the unit of the variable or
square of the unit of the variable.
4

5. Relative Measure of Dispersion
The relative measure of dispersion
measures the variation present among the
observations relative to their average. It
is expressed in the form of a ratio,
coefficient or percentage. It is
independent of the unit of measurement.
5

6. The commonly used measures of absolute
dispersion are:
1. Range
2. Quartile Deviation
3. Mean (Average) Deviation
4. Variance and Standard Deviation
6

7. Their corresponding measures of relative
dispersion are:
1. Coefficient of Range
2. Coefficient of Quartile Deviation
3. Coefficient of Mean (Average)
Deviation
4. Coefficient of Variation (CV)
7

8. If X1, X2, …, Xn are n observations of a
variable X, with X1 and Xn as the smallest
and largest observations respectively.
Then its range is defined as:
Range = Xn - X1
8

9. Example: The following data set shows the
weekly TV viewing times, in hours.
Calculate range and range coefficient of
variation.
25, 41, 27, 32, 43, 66, 35, 31, 15, 5,
34, 26, 32, 38, 16, 30, 38, 30, 20, 21.
   
   
   
859.0
566
5-66
Rangeofefficient-Co
5.35
2
RangeMid
5.30
2
566
22
Range
RangeSemi
61hours566XXRange
1
1
1n












XX
hours
XX
n
n
9

10. If X1, X2, …, Xn are n observations of a
variable X, with Q1 and Q3 as their first
and third quartiles respectively, then
their Quartile Deviation (QD) is as:
2
QQ
MIQR
2
QQ
2
IQR
QDSIQR
QQIQR
13
13
13





10

11. Example:
Calculate QD and coefficient of QD of
above data set shows the weekly TV
viewing times, in hours.
0.248
QQ
QQ
MIQR
SIQR
Q.DofEfficient-Co
h29.25
2
QQ
MIQR
h7.25
2
QQ
2
IQR
QDSIQR
h14.522.0-36.5IQRh36.5Qh0.22
13
13
13
13
31










Q
11

12. If X1, X2, …, Xn are n observations of a
variable X, with m as their average
(mean, median or mode), then their
mean deviation, denoted by MD, is
defined as:
n
 

mX
MD
12

13. X X-Mean |X-Mean| X X-Mean |X-Mean|
25 -5.25 5.25 34 3.75 3.75
41 10.75 10.75 26 -4.25 4.25
27 -3.25 3.25 32 1.75 1.75
32 1.75 1.75 38 7.75 7.75
43 12.75 12.75 16 -14.25 14.25
66 35.75 35.75 30 -0.25 0.25
35 4.75 4.75 38 7.75 7.75
31 0.75 0.75 30 -0.25 0.25
15 15.25 15.25 20 -10.25 10.25
5 -25.25 25.25 21 -9.25 9.25
Continue 605 0 175.00 13

14. Example:
Calculate MD and coefficient of MD.
h30.25
20
605
n
X
X 

289.0
25.30
75.8
UsedAverage
M.D.
MDoftCoefficien
h8.75
20
175mX
MD





n
14

15. 15
The Variance is defined as the mean of the
squared deviations from mean. The
population variance is denoted by σ2 where
as sample variance is denoted by S2 and
defined as
For ungrouped data
sampleFor
n
)x-(x
=S
populationFor
N
)-(x
=
2
2
2
2

 


16. 16
For grouped data
sampleFor
n
)x-(xf
=S
populationFor
N
)-(xf
=
2
2
2
2

 


17. 17
Standard deviation:
 The positive square root of the variance is called
Standard Deviation. It is denoted by σ (S for sample)

18. 















22
n
x
n
x
S
18
It is very much more straight-forward
to use the short cut formula given
below:

19. X X
2
4 16
6 36
2 4
0 0
3 9
5 25
8 64
Total 28 154
19
 
fatalities45.26
1622
7
28
7
154
S
2
















Therefore

20. The formulae that we have just discussed are
valid in case of raw data.
In case of grouped data i.e. a frequency
distribution, each squared deviation round the mean
must be multiplied by the appropriate frequency
figure i.e.
 
n
xxf
S
2
 

And the short cut formula in case of a
frequency distribution is:
















22
n
fx
n
fx
S

21. which is again preferred from the computational
standpoint.
For example, the standard deviation life of a
batch of electric light bulbs would be calculated as
follows:
Life (in
Hundreds
of Hours)
No. of
Bulbs
f
Mid-
point
x
fx fx
2
0 – 5 4 2.5 10.0 25.0
5 – 10 9 7.5 67.5 506.25
10 – 20 38 15.0 570.0 8550.0
20 – 40 33 30.0 990.0 29700.0
40 and over 16 50.0 800.0 40000.0
100 2437.5 78781.25
EXAMPLE

22. Therefore,
standard deviation:















2
100
5.2437
100
25.78781
S
= 13.9 hundred hours
= 1390 hours
















22
n
fx
n
fx
S

23. 100100
..

X
s
Mean
DS
CV
23
Co-efficient of Variation.
The standard deviation is an absolute measure
of dispersion its relative measure of dispersion
is called co-efficient of variation (CV) and is
defined by :

24.  Measures relative variation
 Always in percentage (%)
 Shows variation relative to mean
 Is used to compare two or more sets of data
measured in different units
100%
x
s
CV 








Population Sample
100%
μ
σ
CV 







25. 25
Example: Find Variance, S.D and Co-efficient of Variation.
X 2 3 6 8 11 30
(X-6)2 16 9 0 4 25 54
%54.76=100
6
3.286
=100
x
S
=C.V
3.286=10.=
n
)x-(x
=S
10.8=
5
54
=
n
)x-(x
=S
2
2
2



8

26.  Stock A:
 Average price last year = $50
 Standard deviation = $5
 Stock B:
 Average price last year = $100
 Standard deviation = $5
Both stocks have
the same standard
deviation, but
stock B is less
variate relative to
its price
10%100%
$50
$5
100%
x
s
CVA 








5%100%
$100
$5
100%
x
s
CVB 









27. 27
Example:- Find Variance, S.D and Co-efficient of Variation.
Class f X ( X-X ) ( X-X )2 f ( X-X )2
20---24 1 22 -17 289 289
25---29 4 27 -12 144 576
30---34 8 32 -7 49 392
35---39 11 37 -2 4 44
40---44 15 42 3 9 135
45---49 9 47 8 64 576
50---54 2 52 13 169 338
TOTAL 50 2350

28. 28
Example:-
17.56%=100x
39
6.85
=C.V
6.85=47=
n
)x-(xf
=S
47
50
2350
n
)x-(xf
S
39X
2
2
2




