3 Dispersion

8/11/2019 3 Dispersion

1/30

1


2/30

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

2


3/30

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

3


4/30

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

4

The upper distributionhas more dispersion

because the scoresare more spread out

That is, they are lesssimilar to each other


5/30

The following are the main methods ofmeasuring Dispersion:-

Range

Interquartile Range and Quartile Deviation

Mean Deviation

Standard Deviation

Coefficient of Variation

Lorenz Curve

5


6/30

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

6


7/30

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

7


8/30

Interquartile range(IR) is defined as thedifference of the Upper and Lowerquartiles

Example:-

Upper quartile =Q1

Lower quartile =Q3Interquartile Range= Q3 Q1

8


9/30

Quartile Deviation also, called semi-interquaetile range is half of thedifference between the upper and lower

quartiles

Example:-

Quartile Deviation = Q3 -Q1/ 2

9


10/30


11/30

The relative measures of quartile deviationalso called the Coefficient of QuartileDeviation

Example:-

Coefficient of (Q.D)= Q3Q1/ Q3+ Q1

11


12/30

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

12


13/30


14/30

14

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


15/30

solution :

MD = d

N (deviation taken frommean)

=30.714/21= 1.462

15


16/30

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

16


17/30

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.

17


18/30

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

18


19/30

Standard deviation = variance

Variance = standard deviation2

19

S.D


20/30

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

20

2is the population variance, X is a score, is

the population mean, and N is the number of


21/30

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

21


22/30

2

6

126

294306

6

6306

N

N

42

XX

2

2

2

2

2

6

12

N

X 2

2

22


23/30

Varianceis defined as the averageof the square deviations:

23

N

X 2

2


24/30

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

24


25/30

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!


26/30

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

26


27/30

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

27


28/30

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

28

s2is the sample variance, X is a score, X is the

sample mean, and N is the number of scores


29/30


30/30

30

Documents

3 Dispersion