Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Business Mathematics and Statistics

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QUARTILE DEVIATION 
Quartiles are such values which divide the total number of observations into 4 equal parts. Obviously, there are 3 quartiles—
(i) First quartile (or Lower quartile): Q1
(ii) Second quartile, (or Middle quartile) : Q2
(iii) Third quartile (or Upper quartile): Q3 
The number of observations smaller than Q, is the same as the number lying between Q1 and Q2, or between Q2 and Q3, or larger than Q3. For data of continuous type, one-quarter of the observations is smaller than Q1, two-quarters are smaller than Q2 , and three-quarters are smaller than Q3. This means that Q1, Q2, Qare values of the variable corresponding to ‘less-than’ cumulative frequencies N/4, 2N/4, 3N/ 4 respectively. Since, 2N/4 = N/2, it is evident that the second quartile Q2 is the same as median.

Q1 < Q2 < Q3 ; Q2 = Median. 
Quartiles are used for measuring central tendency, dispersion and skewness. For instance, the second quartile Q2 is itself taken as a measure of central tendency, where it is known as Median. Quartile deviation is defined as half the difference between the upper and the lower quartiles.

Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
The difference Q3- Q1 being the distance between the quartiles can also be called inter quartile range; half of this Semi- inter quartile Range. Thus the name ‘Semi - inter quartile Range’ itself gives the definition of Quartile Deviation.

Quartile Deviation (Q.D.) is dependent on the two quartiles and does not take into account the variability of the largest 25% or the smallest 25% of observations. It is therefore unaffected by extreme values. Since in most cases the central 50% of observation tend to be fairly typical, Q.D. affords a convenient measure of dispersion. It can be calculated from frequency distributions with open-end classes. Q.D. is thus superior to Range in many ways. Its unpopularity lies in the fact that Q.D. does not depend on the magnitudes of all observations. The calculation of Q.D. only depends on that of the two quartiles, Q1 and Q3 which can be found from a cumulative frequency distribution using simple interpolation.

Example and Solution.
Example 1. Calculate the quartile deviation from the following:
Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
Solution. In order to compute Quartile Deviation, we have to find QI and Q3 i.e. values of the variable corresponding to Cumulative frequencies N/4 and 3N/4. Here, total frequency N = 82. Therefore, N/4 = 20.5 and 3N/4 = 61.5 

Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev


Example 2. Calculate the appropriate measure of dispersion from the following data :
Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
Solution. Since there are open-end classes in the frequency distribution. Quartile deviation would be the most appropriate measure of dispersion for the data. So, we have to determine the quartiles QI and Q3 which can be done from cumulative frequency distribution using simple interpolation.

Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Applying Simple interpolation,

Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRevQuartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Example 3. Consider the following scores of 10 CFA Level 1 candidates
78   56   67   51   43   89   57   67   78   50
Solution. Range = 89 – 43 = 46

Example 4. 6 Investment analysts attain the following returns on six different investments:
{12%   4%   23%   8%   9%   16%}
Calculate the mean absolute deviation and interpret it.
Solution. First, we have to calculate the arithmetic mean:
Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
Next, we can now compute the MAD:
Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
= 30/6
= 5%
Interpretation: It means that on average, an individual return deviates 5% from the mean return of 12%

Example 5. Working with data from example 2 above, the variance will be calculated as follows:
Solution.
Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
Thus, the average variation from the mean (0.12) is 0.003767.
The standard deviation is 0.0037671/2 = 0.06137 or 6.14%
Analysts use the standard deviation to interpret returns as opposed to the variance since it is much easier to comprehend.

Example 6. Assume that the returns realized in example 2 above were sampled from a population comprising 100 returns. Compute the sample mean and the corresponding sample variance.
Solution. The sample mean will still be 12%
Hence,
Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev
Therefore,
Quartile Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

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