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

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. Thescatterofthevaluesabouttheircentreiscalled
dispersionandanymeasureindicatingtheamountof
scatteraboutthecentreiscalledaMeasureof
Dispersion.
Theindividualobservationsofavariabletendtoscatter
abouttheircentre.Thehighestdegreeof
concentrationisthatalltheobservationsareofsame
size.Thescatterinthiscasewouldbezeroandmean
willbeexactlysameastheindividualvaluesofthe
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
Theabsolutemeasureofdispersion
measuresthevariationpresentamongthe
observationsintheunitofthevariableor
squareoftheunitofthevariable.
4

5. Relative Measure of Dispersion
Therelativemeasureofdispersion
measuresthevariationpresentamongthe
observationsrelativetotheiraverage.It
isexpressedintheformofaratio,
coefficientorpercentage.Itis
independentoftheunitofmeasurement.
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. IfX1,X2,…,Xnarenobservationsofa
variableX,withX1andXnasthesmallest
andlargestobservationsrespectively.
Thenitsrangeisdefinedas:
Range=Xn-X1
8

9. Example:Thefollowingdatasetshowsthe
weeklyTVviewingtimes,inhours.
Calculaterangeandrangecoefficientof
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
Range ofefficient -Co
5.35
2
Range Mid
5.30
2
566
22
Range
Range Semi
61hours566XXRange
1
1
1n












XX
hours
XX
n
n
9

10. If X1, X2, …, Xn are n observations of a
variable X, with Q1and Q3as their first
and third quartiles respectively, then
their Quartile Deviation (QD) is as:2
QQ
MIQR
2
QQ
2
IQR
QD SIQR
QQ IQR
13
13
13





10

11. Example:
CalculateQDandcoefficientofQDof
abovedatasetshowstheweeklyTV
viewingtimes,inhours.0.248
QQ
QQ
MIQR
SIQR
Q.D ofEfficient -Co
h 29.25
2
QQ
MIQR
h 7.25
2
QQ
2
IQR
QD SIQR
h 14.5 22.0-36.5 IQRh 36.5 Qh 0.22
13
13
13
13
31










Q
11

12. If X1, X2, …, Xnare 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
4110.75 10.7526 -4.25 4.25
27 -3.25 3.25 32 1.75 1.75
32 1.75 1.75 38 7.75 7.75
4312.75 12.7516-14.25 14.25
6635.75 35.7530 -0.25 0.25
35 4.75 4.75 38 7.75 7.75
31 0.75 0.75 30 -0.25 0.25
1515.25 15.2520-10.25 10.25
5 -25.25 25.2521 -9.25 9.25
Continue 605 0 175.00
13

14. Example:
Calculate MD and coefficient of MD.h 30.25
20
605
n
X
X 
 289.0
25.30
75.8
UsedAverage
M.D.
MD oft Coefficien
h 8.75
20
175mX
MD





n
14

15. 15
TheVarianceisdefinedasthemeanofthe
squareddeviationsfrommean.The
populationvarianceisdenotedbyσ2where
assamplevarianceisdenotedbyS2and
definedas
Forungroupeddata sampleFor
n
)x - (x
= S
population For
N
) - (x
=
2
2
2
2




16. 16
Forgroupeddata sampleFor
n
)x-(x f
= S
population For
N
)-(x f
=
2
2
2
2




17. 17
Standard deviation:
Thepositivesquarerootofthevarianceiscalled
StandardDeviation.Itisdenotedbyσ(Sforsample)

18. 




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
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

22
n
x
n
x
S 18
Itisverymuchmorestraight-forward
tousetheshortcutformulagiven
below:

19. X X
2
4 16
6 36
2 4
0 0
3 9
5 25
8 64
Total28 154 19 
fatalities45.26
1622
7
28
7
154
S
2



















Therefore

20. Theformulaethatwehavejustdiscussedare
validincaseofrawdata.
Incaseofgroupeddatai.e.afrequency
distribution,eachsquareddeviationroundthemean
mustbemultipliedbytheappropriatefrequency
figurei.e.
n
xxf
S
2


Andtheshortcutformulaincaseofa
frequencydistributionis:

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














22
n
fx
n
fx
S

21. whichisagainpreferredfromthecomputational
standpoint.
Forexample,thestandarddeviationlifeofa
batchofelectriclightbulbswouldbecalculatedas
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 67.5506.25
10 – 20 38 15.0570.08550.0
20 – 40 33 30.0990.029700.0
40 and over16 50.0800.040000.0
100 2437.578781.25
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 Sample100%
μ
σ
CV 










25. 25
Example:FindVariance,S.DandCo-efficientofVariation.
X 23681130
(X-6)
2
169042554% 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 price10%100%
$50
$5
100%
x
s
CV
A









 5%100%
$100
$5
100%
x
s
CV
B











27. 27
Example:-FindVariance,S.DandCo-efficientofVariation.
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% = 100 x
39
6.85
= C.V
6.85 = 47 =
n
)x -(x f
= S
47
50
2350
n
)x -(x f
S
39X
2
2
2




