MEASURES OF DISPERSION

Presented by:

Sonia gupta

Persentation outline

Introduction of measure of dispersion Defination Method of dispersion:• Range

• Quartile deviation

• Mean deviation

• Standard deviation

• Lorenz curve

Measure of skewness

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What u want to know?

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DEFINITION

According to Bowley “Dispersion is the measure of the variation of the items”

According to Conar “Dispersion is a measure of the extent to which the individual items vary”

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Measures of Dispersion

Which of the distributions of scores has the larger 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

The upper distribution has more dispersion because the scores are more spread out

That is, they are less similar to each other

METHODS OF MEASURING DISPERSION

Range

Quartile Deviation

Mean Deviation

Standard Deviation

Lorenz curve

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RANGE

It is defined as the difference between the smallest and the largest observations in a given set of data.

Formula is R = L – SThe relative measure corresponding to range, called

the coefficient of range.

Formula is Coefficient of R = L - S L + S

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Ex. Find out the range of the given distribution: 1, 3, 5, 9, 11

Soluion: Range = L – S

L= 11 And S= 1

Range = 11 – 1 =10

Coefficient of range = L – S

L + S

= 11 – 1 =10 = 1.2

11+1 128

QUARTILE DEVIATION

It is the second measure of dispersion, no doubt improved version over the range. It is based on the quartiles so while calculating this may require upper quartile (Q3) and lower quartile (Q1) and then is divided by 2. Hence it is half of the deference between two quartiles it is also a semi inter quartile range.

The formula of Quartile Deviation is

(Q D) = Q3 - Q1

2 9

Roll no. 1 2 3 4 5 6 7

marks 12 15 20 28 30 40 50

Solution:-

Q1= Size of N+1th item = Size of 7+1 = 2nd item

4 4Size of 2nd item is 15. Thus Q1=15

Q3= Size of 3 N+1 th item = Size of 3x8 th item =6th item

4 4Size of 6th item is 40. Thus Q3= 40

Q.D. =Q3-Q1/2 =40-15/2 =12.5 10

Ex:-

MEAN DEVIATION

Mean Deviation is also known as average deviation. In this case deviation taken from any average especially Mean, Median or Mode. While taking deviation we have to ignore negative items and consider all of them as positive. The formula is given below:

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MEAN DEVIATION

The formula of MD is given below

MD = d

N (deviation taken from mean)

MD = m

N (deviation taken from median)

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The Mean Deviation (cont.)EX:

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Data(X1)

Deviation(X1 – X)

Absolute deviationX - X

13 -1 1

17 +3 3

14 0 0

11 -3 3

15 +1 1

TOTAL = 70 D = 0 D = 8

MEAN = 70/5 , = 14 M.D.= 8/5, = 1.6

MEDIAN= N+1/2th item =6/2th item , = 14

M.D.= 8/5, = 1.6

STANDARD DEVIATION

The concept of standard deviation was first introduced by Karl Pearson in 1893. The standard deviation is the most useful and the most popular measure of dispersion. Just as the arithmetic mean is the most of all the averages, the standard deviation is the best of all measures of dispersion.

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STANDARD DEVIATION

The standard deviation is represented by the Greek letter (sigma). It is always calculated from the arithmetic mean, median and mode is not considered. While looking at the earlier measures of dispersion all of them suffer from one or the other demerit i.e.

Range –it suffer from a serious drawback considers only 2 values and neglects all the other values of the series.

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STANDARD DEVIATION

Quartile deviation considers only 50% of the item and ignores the other 50% of items in the series.

Mean deviation no doubt an improved measure but ignores negative signs without any basis.

Karl Pearson after observing all these things has given us a more scientific formula for calculating or measuring dispersion. While calculating SD we take deviations of individual observations from their AM and then each squares. The sum of the squares is divided by the number of observations. The square root of this sum is knows as standard deviation.

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MERITS OF STANDARD DEVIATION

Very popular scientific measure of dispersion

From SD we can calculate Skewness, Correlation etc

It considers all the items of the seriesThe squaring of deviations make them positive and the difficulty about algebraic signs which was expressed in case of mean deviation is not found here.

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STANDARD DEVIATION

The formula of SD is = √∑d2

N EX: Calculate Standard Deviation of the following series X – 40, 44, 54, 60, 62, 64, 70, 80, 90, 96

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Solution :

NO OF YOUNG ADULTS VISIT TO THE LIBRARY

IN 10 DAYS (X) d=X - A.M d2

40 -26 676

44 -22 484

54 -12 144

60 -6 36

62 -4 16

64 -2 4

70 4 16

80 14 196

90 24 596

96 30 900

N=10X=660

d2 = 3048

Standard deviation

AM = X

N

= 660 = 66 AM

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  SD = √∑d2

N

SD =√3048 = 17.46

10 20

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Measure of Skew

Skew is a measure of symmetry in the distribution of scores

Positive Skew

Negative Skew

Normal (skew = 0)

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Measure of Skew

The following formula can be used to determine skew:

N

N

XX

XX

s 2

3

3

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Measure of Skew

If s3 < 0, then the distribution has a negative skew

If s3 > 0 then the distribution has a positive skew

If s3 = 0 then the distribution is symmetrical

The more different s3 is from 0, the greater the skew in the distribution

Thanku

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