250380111-Measures-of-Dispersion-ppt.ppt

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1. Measures of Dispersion
Measures of variation measure
the variation present among the
values of a data set, so measures
of variation are measures of
spread of values in the data.
1

2. 2
Absolute Measures of
Dispersion
Range
QuartileDeviation
MeanDeviation
VarianceandStandardDeviation

3. Range
Difference between the
largest and the smallest
observations
3Largest Smallest
RangeXX

4. Disadvantages of the Range
Ignores the way in which data are
distributed
Sensitive to outliers
4
7 8 9 10 11 12
Range = 12 -7 = 5
7 8 9 10 11 12
Range = 12 -7 = 5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120
Range = 5 -1 = 4
Range = 120 -1 = 119

5. Variance
Variance is the
average of the
squared deviations
taken from the mean
value.2
22
2
22
22
() 102
( ) 17
6
702 102
( ) 17
66
xx
i S cm
n
XX
ii S cm
nn

  


     





8:14 PM 5
X cm (X-Mean)^2 X
2
4 36 16
6 16 36
9 1 81
12 4 144
13 9 169
16 36 256
60 102 702

6. Standard Deviation
Standard deviation is the positive square
root of the mean-square deviations of the
observations from their arithmetic mean.varianceSD  
1
2




N
xx
s
i  
N
x
i


2


Population Sample

7. Standard Deviation for Group
Data
SD is :
Simplified formula2
2










N
fx
N
fx
s  
N
xxf
s
ii


2 


i
ii
f
xf
x
Where

8. Example-1: Find Standard
Deviation of Ungroup Data
Family
No.
12345678910
Size (x
i)3344556677

9. ix xx
i  
2
xx
i
 2
ix Family No.1 2 3 4 5 6 7 8 910Total
3 3 4 4 5 5 6 6 7 7 50
-2-2-1-10 0 1 1 2 2 0
4 4 1 1 0 0 1 1 4 4 20
9 916162525363649492705
10
50


n
x
x
i  
2
10
20
2
2




n
xx
s
i 41.12s
Here,

10. Comparing Standard
Deviations
10
Mean = 15.5
S = 3.33811 12 13 14 15 16 17 18 19 20 21
Data A
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
S = 4.567
Data C
•The smaller the standard deviation, the more tightly
clustered the scores around mean
•The larger the standard deviation, the more spread out
the scores from mean
11 12 13 14 15 16 17 18 19 20 21
Data B
Mean = 15.5
S = 0.926

11. Coefficient of Variation (CV)
Can be used to compare two or
more sets of data measured in
different units or same units but
different average size.
11100%
X
S
CV 










12. Use of Coefficient of Variation
Stock A:
◦Average price last year = $50
◦Standard deviation = $5
Stock B:
◦Average price last year = $100
◦Standard deviation = $5
12
but stock B is
less variable
relative to its
price10%100%
$50
$5
100%
X
S
CV
A









 5%100%
$100
$5
100%
X
S
CV
B










Both stocks
have the
same
standard
deviation

13. valuesof 68%about contains1SX valuesof 99.7%about contains3SX 13
The Empirical
RuleX
68%1SX valuesof 95%about contains2SX
95%X 2S X 3S
99.7%

14. 14
Adistributioninwhichthevalues
equidistantfromthecentrehaveequal
frequenciesisdefinedtobesymmetrical
andanydeparturefromsymmetryiscalled
skewness.
1.LengthofRightTail=LengthofLeftTail
2.Mean=Median=Mode
3.Sk=0
a)Sk=(Mean-Mode)/SD
b)Sk=(Q3-2Q2+Q1)/(Q3-Q1)
Measures of
Skewness

15. 15
Adistributionispositivelyskewed,ifthe
observationstendtoconcentratemoreatthe
lowerendofthepossiblevaluesofthevariable
thantheupperend.Apositivelyskewed
frequencycurvehasalongertailontheright
handside
1.LengthofRightTail>LengthofLeftTail
2.Mean>Median>Mode
3.SK>0
Measuresof
Skewness

16. 16
Adistributionisnegativelyskewed,ifthe
observationstendtoconcentratemoreat
theupperendofthepossiblevaluesofthe
variablethanthelowerend.Anegatively
skewedfrequencycurvehasalongertail
ontheleftside.
1.LengthofRightTail<LengthofLeftTail
2.Mean<Median<Mode
3.SK<0
Measures of
Skewness

17. 17
TheKurtosisisthedegreeofpeakednessorflatnessofa
unimodal(singlehumped)distribution,
•Whenthevaluesofavariablearehighlyconcentrated
aroundthemode,thepeakofthecurvebecomesrelatively
high;thecurveisLeptokurtic.
•Whenthevaluesofavariablehavelowconcentration
aroundthemode,thepeakofthecurvebecomesrelatively
flat;curveisPlatykurtic.
•Acurve,whichisneitherverypeakednorveryflat-toped,
itistakenasabasisforcomparison,iscalled
Mesokurtic/Normal.
Measures of Kurtosis

18. 18
Measures of Kurtosis