Quartile Deviation
The range as a measure of dispersion has certain limitations. This is basically based on two extreme items and it fails to take account of the scatter within the range. From this there is reason to believe that if the dispersion of the extreme items is discarded, then the limited range thus established might be more instructive. For this purpose there has been developed a measurement known as the inter quartile range. The range which includes the middle 50 percent of the distribution that is one fourth of the observations at the lower end, another quarter of the observations at the upper end of the distribution are excluded, in computing the inter quartile range. In another works, inter-quartile range shows the difference between the third quartile and the first quartile.
Symbolically,
Inter quartile range = Q3 - Q1
Very frequently the inter quartile range is reduced to the form of the semi-tier quartile range or quartile deviation by dividing it by 2
Symbolically
Quartile deviation or Q. D = (Q3 - Q1) / 2
Quartile deviation gives the average amount by which the two quartiles differ from the median. In a symmetrical distribution the two quartiles Q1 and Q3 are equidistant from the median med - 1 = Q3 - med. Therefore the difference can be taken as a measure of dispersion. The median ± covers exactly 50 per cent of the observations.
In reality, however one seldom finds a series in business and economic data that is perfectly symmetrical. Nearly all the distributions of social series are asymmetrical. In an asymmetrical distribution Q1and Q3 are not equidistant from the median. As a result an asymmetrical distribution includes only approximately 50 per cent of the observations.
When quartile deviation is very small it explains high uniformity or small variation of the central 50% items and a high quartile deviation means that the variation among the central items is very large.
The Quartile deviation is an absolute measurement of dispersion. The relative measure corresponding to this measure, known as the coefficient of quartile deviation is calculated as follows.
Coefficient of Q.D = (Q3 - Q1) / 2 / (Q3 + Q1)/2 = Q3 - Q1 / Q3 + Q1
The Coefficient of quartile deviation can be used to compare the degree of variation in various distributions.
In the computation of quartile deviation the process of computing quartile deviation is very simple. We just have to compute the values of the upper and lower quartiles; the following illustrations would clarify the calculations.
Illustration:
Find out the value of quartile deviation and its coefficient from the following data
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Roll no.
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1
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2
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3
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4
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5
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6
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7
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Marks
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20
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28
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40
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12
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30
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15
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50
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Solution: calculation of quartile deviation
Marks arranged in ascending order 12 15 20 28 30 40 50
Q1 = size of N+1 / th item = size of 7+1/4 = 2nd item
Size of 2nd item is 15. Thus Q1 = 15
Q3 = size of 3 (N+1 / 4) th item = size of (3x 8 / 4 ) th item = 6th item
Size of 6th item is 40. Thus Q3 = 40
Q.D = Q3- Q1 / 2 = 40-15 / 2 = 12.5
Coefficient of Q.D = Q3 - Q1 / Q3 + Q1 == 40-15 / 40+ 15 = 25 / 55 = 0.455.