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Part 3:
Measure of Dispersion
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Why to analyze measure ofdispersion
To evaluate central tendency of a data set.
To compare distribution of two or more data set.
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Measures of Dispersion
(Range)
(Mean deviation)
(Variance)
(Standard deviation)
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Non-grouped datameasure of dispersion
(Range): The difference between maximum andminimum values in a data serie.
R = Maximum value
Minimum value
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Range
Series 1 Series 2
2 5
3 5
6 57 6
8 7
10 8X = 6 X = 6
R = 10 2 = 8 R = 8 5 = 3
____
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Grouped data setRange
Hourly wage
YTL
Frequency
5-10 10
10-15 21
15-20 9
20-25 5
R = 25 5 = 20 YTL
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(Mean Deviation or Mean Absolute Deviation)
Absolute deviations of all values in a population
from the populations aritmetic mean.
X- X
M.D. = N
___________
__
X X = absolute deviation__
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Mean deviation
(Non-grouped data)
Calculate absolute deviation of the following value serie:
15, 16, 18, 21, 25
16TOTAL
625-1925
221-1921
118-1918
316-1916
415-1915X - XX - XValues
____
16
M.D. = = 3.2
5
____
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Mean Deviation
(Grouped data set)
486.2169551Total
10921.855-33.227555550-60
106.211.845-33.240545940-50
28.81.835-33.2560351630-40
114.88.225-33.2350251420-30
127.418.215-33.210515710-20
f X XX - XX - X
fXGroup Mid-
point ( X)
Frequency
( f)
Group _ __
f X - X 486.2M.D. = = = 9.53
N 51
___________ _____
_
fX 1695X= = = 33.2
N 51
___________
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(Variance and Standard Deviation)
Variance , is the square aritmetic mean of all
deviations of values from the mean)
( X X )
=
N
__
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Standard Deviation
is the square root of variance of a data set.
( X X )
=
N
__
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Characteristics of Standard Deviationand Variance
( cX ) = c ( X )
( cX ) = c ( X )
( X a ) = ( X )
( X + a ) = ( X )
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Standard Deviation and Variance(Non-grouped data)
Calculate standard deviation and variance of the following
values: 22, 25, 28, 30 ve 35
( X X ) = 98( X X ) = 0 X = 140
49735
4230
0028
9-325
36-622
( X X )( X X )X__
__
98
= = 19.65
= 19.6 = 4.43
140
X = = 285
_
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Standard Deviation and Variance(Grouped data set)
14088001180032Total
65340010890033042007006600-800
1183001690013035005007400-600
539004900-70330030011200-400
58320072900-27080010080-200
f (X X)(X X)(X X)
fXGroup Mid-
point ( X)
Frequen.
( f)
groups _ __
1408800
= = 44025
32
_______11800
X= = 370
32
______
= 44025 = 209.82
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Variance
Its difficult to comment on asingle data set. Likely
average mean deviaiton or range, variance is
used to compare variation in two data sets.
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Relation between standard deviation and aritmeticmean in symmetric frequency distributions
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Relative Dispersion
Coeffient of variation : is expressed in %. Standard
deviation divided by aritmetic mean.
to benchmark two data sets which have different units (cm, Rp.)
to compare data which have same units same but means are
very different
COV (%) = * 100X
_
X = 500000, = 50000
X = 12000, = 2000_
_50000
COV = * 100 = 10%
500000
2000
COV = * 100 = 16.7%
12000
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Example (Relative Dispersion)
For a product A
T1 machine T2 machine
Length mean (X1) = 67 mm Length X2 = 64 mm
Standard deviation (1) = 2.5 mm 2 = 2.4 mm
X1 > X2 1 > 2 (as seen). It can be said that theres
much variation in first machine, however this comment is
wrong.
V1 = (2.5 / 67) * 100 = % 3.73 In fact, much variation in machine
V2 = (2.4 / 64) * 100 = % 3.75 T2
_
_
COV (%) = * 100X_
_
_
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Skewness
Expresses the skewness grade of frequency distributions.
Skewness is zero at symmetric frequency distributions.
Skewness usual vary between -3 and +3 .
3 (X median)Skewness =
_
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Skewness
Negative
Skewness
Positive
Skewness
Symmetric
Skewness = 0