Measures of Variation.pdf

Variations

1. Measures of Variation / Dispersion/ Spread
•Althougharithmeticmeanisaconcisemethodof
presentationofastatisticaldatayetitis
inadequateforseveralreasons,forexample,it
givesnoindicationofitsreliability.
•Ameasureofdispersionexpressquantitativelythe
degreeofvariationordispersionofvalueofa
variableaboutanyaverage.

2. Measures of variation / dispersion/ spread
•Measuresofvariationmeasurethevariation
presentamongthevaluesinadatasetwitha
singlenumbersomeasuresofvariationare
summarymeasuresofspreadofvaluesinthe
data.
•Ameasureofcentraltendencyalongwitha
measureofdispersiongivesanadequate
descriptionofstatisticaldata.

3. Types of dispersion
•Absolute dispersion
The measure of dispersion which expressed in
terms of original units of data are termed as
absolute measures
•Relative measures of dispersion
This term also known as coefficients of
dispersion, are obtained as ratios or
percentages.

4. Methods of measures of dispersion

5. The Range & Coefficient of Range

6. Example
The marks obtained by 9 students are given below:-
45, 32, 37, 46, 39, 36, 41, 48, 36
Find the range and the Coefficient of Range.
Maximum Obs is 48 and Minimum 32, therefore
Range = 16 marks
Co-efficient of Range = 0.2

7. Semi Inter Quartile Range /
Quartile Deviation

8. The Mean Deviation OR Average
Deviation

9. Example
X
3 2 0 1.5
4 1 1 0.5
7 2 4 2.5
7 2 4 2.5
8 3 5 3.5
3 2 0 1.5
5 0 2 0.5
3 2 0 1.5
Mean=5
Mode=3
Median=4.5

10. Solution:

11. 11
Idea:-Selectsinglevalueasreferencevalue(idealreferencevalueis
meanofthedata)takedeviationsofvaluesfrommeanandtakesum
ofthesedeviations
Example:-Followingdatarepresenttheyieldperplotofthreewheat
veritiesA,BandC.Comparetheyieldperformanceofthreeverities
XAXB XC
4030 10
5050 50
6070 90
Center Base Measures

12. 12
Solution:Squaredthedeviationsandthensumthesquared
deviationstogetridofcancellingsignproblem2
()XX−
XAXB XC
4030 10
5050 50
6070 90
DevDevDev
-10-20-40
0 0 0
1020 40
Dev
2
Dev
2
Dev
2
1004001600
0 0 0
1004001600( )
( )
( )
2
2
2
2
2
2
2
2
2
67.1066
3
3200
67.266
3
800
67.6
3
200
Kg
n
XX
S
Kg
n
XX
S
Kg
n
XX
S
CC
C
BB
B
AA
A
==
−
=
==
−
=
==
−
=



Variance:AverageoftheSquareddeviationsfrommean

13. Variance
•Thevarianceisameasureofvariabilitythat
utilizesallthedata.
•Itisbasedonthesquareddifferencebetween
thevalueofeachobservation(x
i)andthe
meanofthedata
•Thevarianceisdenotedbys
2
.

14. Problem With Variance
14
Variance measures the variation in the data as the
square of the units of measurements of the data so it
is difficult to interpret it precisely
Solution:-Take positive square root of the variance
known as standard deviation denoted by S.
It has the same units as the measurements
themselves

15. Standard Deviation
•Thestandarddeviationofadatasetisthepositive
squarerootofthevariance.
•Itismeasuredinthesameunitsasthedata,
makingitmoreeasilycomparable,thanthe
variance,tothemean.
•ThestandarddeviationisdenotedS.KgS
KgS
KgS
C
B
A
66.3267.1066
33.1667.266
58.267.6
==
==
==

16. Coefficient of Variation (CV)
•Shows relative variability, that is, variability
relative to the magnitude of the data i.e variation
relative to mean
•Always in percentage (%)
•Unitfree measure of variation
•Can be used to compare two or more sets of data
–measured in different units
–same units but different average sizeCV= ×100
S
X

17. Coefficient of Variation
Thefollowingdatarepresentlength(ininches)and
weight(inKg)forasampleof10fishofsame
speciesafterusingaparticulartypeoffishfeedFish 1 2 3 4 5 6 7 8 9 10
Weight 1.8 1.9 2.1 2.4 2.5 2.6 2.7 2.8 3.1 3.2
Length 11 12 12 13 15 15 16 17 18 18

Whichcharacteristicweightorlengthisrelatively
morevariable

18. Standard deviation
(S)
Weight 0.472 kg
Length 2.584 inches
Mean CV
2.51 kg 18.82
14.70 inches 17.58

19. Standard Variable
•A variable that has mean “0” and Variance “1” is called
standard variable
•Values of standard variable is called standard scores
•Values of standard variable i.estandard scores are unit-less
•Constructionvariableofdeviation Standard
variableofMeanVarable
Z
−
=
19

20. X Z
3 25 -1.36241.8561
6 4 -0.54500.2970
11 9 0.817410.6682
12 16 1.08991.1879
32 54 0 4.00967.3
5.13
4
54
8
4
32
2
=
==
===

x
x
S
S
n
X
X 2
)(XX− 67.3
8−
=
−
=
X
Sx
XX
Z
201
4
4.009
S
0
n
Z
Z
2
z =
==
 2
)(ZZ−
Variable Z has mean “0” and variance “1” so Z is a standard variable3624.1
67.3
83
3atScoreStandard
−=
−
=
−
=
=
Sx
XX
Z
X