3 dispersion
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In the words of Bowley Dispersion is themeasure of the variation of the items
According to Conar Dispersionis a measure ofthe extent to which the individual items vary
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Measures of dispersionare descriptivestatistics that describe how similar a set ofscores are to each other
The more similar the scores are to each other,the lower the measure of dispersion will be
The less similar the scores are to each other,
the higher the measure of dispersion will be In general, the more spread out a distribution
is, the larger the measure of dispersion willbe
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Which of thedistributions ofscores has thelarger dispersion?
0
25
50
75
100
125
1 2 3 4 5 6 7 8 9 10
0
25
50
75
100
125
1 2 3 4 5 6 7 8 9 10
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The upper distributionhas more dispersion
because the scoresare more spread out
That is, they are lesssimilar to each other
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The following are the main methods ofmeasuring Dispersion:-
Range
Interquartile Range and Quartile Deviation
Mean Deviation
Standard Deviation
Coefficient of Variation
Lorenz Curve
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TheRangeis defined as the differencebetween the largest score in the set ofdata and the smallest score in the set of
data, XL- XS
What is the range of the following data:4 8 1 6 6 2 9 3 6 9 ?
The largest score (XL) is 9; the smallestscore (XS) is 1; the range is XL- XS= 9 - 1= 8
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The range is used when you have ordinal data or you are presenting your results to people
with little or no knowledge of statistics The range is rarely used in scientific work as it
is fairly insensitive It depends on only two scores in the set of
data, XLand XS Two very different sets of data can have the
same range:1 1 1 1 9 vs 1 3 5 7 9
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Interquartile range(IR) is defined as thedifference of the Upper and Lowerquartiles
Example:-
Upper quartile =Q1
Lower quartile =Q3Interquartile Range= Q3 Q1
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Quartile Deviation also, called semi-interquaetile range is half of thedifference between the upper and lower
quartiles
Example:-
Quartile Deviation = Q3 -Q1/ 2
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The relative measures of quartile deviationalso called the Coefficient of QuartileDeviation
Example:-
Coefficient of (Q.D)= Q3Q1/ Q3+ Q1
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Mean Deviation is also known as averagedeviation. In this case deviation taken from anyaverage especially Mean, Median or Mode.
While taking deviation we have to ignorenegative items and consider all of them aspositive. The formula is given below
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xi fi xi.fi x-x x -x.fi
10-15 12.5 3 37.5 9.286 27.85
15-20 17.5 5 87.5 4.286 21.43
20-25 22.5 7 157.5 .714 4.99
25-30 27.5 4 110 5.714 22.85
30-35 32.5 2 65 10.714 21.42
21 457.5 30.714 98.57
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solution :
MD = d
N (deviation taken frommean)
=30.714/21= 1.462
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When the deviate scores are squared in variance,their unit of measure is squared as well
E.g. If peoples weights are measured in pounds,then the variance of the weights would beexpressed in pounds2(or squared pounds)
Since squared units of measure are often awkwardto deal with, the square root of variance is often
used instead The standard deviation is the square root of
variance
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Very popular scientific measure of dispersion
From SD we can calculate Skewness,Correlation etc
It considers all the items of the series The squaring of deviations make them positive
and the difficulty about algebraic signs whichwas expressed in case of mean deviation is notfound here.
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Calculation is difficult not as easier as Rangeand QD
It always depends on AM
Extreme items gain great importanceThe formula of SD is = d2
N
Problem: Calculate Standard Deviation of the following
series X 40, 44, 54, 60, 62, 64, 70, 80, 90, 96
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Standard deviation = variance
Variance = standard deviation2
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S.D
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When calculating variance, it is often easierto use a computational formula which is
algebraically equivalent to the definitionalformula:
NN
N XX
X 2
2
2
2
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2is the population variance, X is a score, is
the population mean, and N is the number of
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X X2 X- (X-)2
9 81 2 4
8 64 1 16 36 -1 1
5 25 -2 4
8 64 1 1
6 36 -1 1
= 42 = 306 = 0 = 12
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2
6
126
294306
6
6306
N
N
42
XX
2
2
2
2
2
6
12
N
X 2
2
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Varianceis defined as the averageof the square deviations:
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N
X 2
2
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First, it says to subtract the mean from each ofthe scores
This difference is called a deviateor a deviation
score
The deviate tells us how far a given score isfrom the typical, or average, score
Thus, the deviate is a measure of dispersionfor a given score
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Why cant we simply take the averageof the deviates? That is, why isnt
variance defined as:
25
N
X2
This is not theformula for
variance!
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One of the definitions of the meanwas that italways made the sum of the scores minus themean equal to 0
Thus, the average of the deviates must be 0since the sum of the deviates must equal 0
To avoid this problem, statisticians square thedeviate score prior to averaging them
Squaring the deviate score makes all thesquared scores positive
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Variance is the mean of the squareddeviation scores
The larger the variance is, the more thescores deviate, on average, away fromthe mean
The smaller the variance is, the less thescores deviate, on average, from themean
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Because the sample mean is not a perfectestimate of the population mean, the formulafor the variance of a sample is slightly different
from the formula for the variance of apopulation:
1N
XX
s
2
2
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s2is the sample variance, X is a score, X is the
sample mean, and N is the number of scores
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