Quartile deviation (QD) :
Another measure of dispersion is provided by quartile- deviation or semi - inter –quartile range
The formula to find quartile-deviation is given by
A relative measure of dispersion using quartiles is given by coefficient of quartile-deviation which is
1) Quartile-deviation provides the best measure of dispersion for open-end classification.
2) It is less affected due to sampling fluctuations.
3) Like other measures of dispersion, quartile-deviation remains unaffected due to a change of origin but is affected in the same ratio due to change in scale.
i.e. if y = a + bx, a and b being constants,
then QD of y = |b| × QD of x
Problem 1 :
Following are the marks of the 10 students :
56, 48, 65, 35, 42, 75, 82, 60, 55, 50
Find QD and also its coefficient.
After arranging the marks in an ascending order of magnitude, we get
35, 42, 48, 50, 55, 56, 60, 65, 75, 82
First quartile (Q₁) = (n+1)/4 th observation
= (10+1)/4 th observation
= 2.75 th observation
= 2nd observation + 0.75 × difference between the 3 rd and the 2 nd observation
= 42 + 0.75 (48 - 42)
= 42 + 0.75 x 6
= 42 + 4.5
Q₁ = 46.5
Third quartile (Q₃) = 3(n+1)/4 th observation
= 3(10+1)/4 th observation
= 8.25 th observation
= 8 th observation + 0.25 × difference between the 9 th and the 8 th observation
= 65 + 0.25 ( 75 - 65)
= 65 + 0.25 x 10
= 65 + 2.5
Q₃ = 67.5
The formula to find QD is given by
QD = (Q₃ - Q₁) / 2
QD = (67.50 - 46.50) / 2
QD = 21 / 2
QD = 10.5
The formula to find coefficient of QD is given by
= [ (Q₃ - Q₁) / (Q₃ - Q₁) ] x 100
= [ (67.50 - 46.50) / (67.50 + 46.50) ] x 100
= [ 21 / 114] x 100
Coefficient of QD = 18.42
Problem 2 :
If the QD of x is 6 and 3x + 6y = 20, what is the QD of y ?
Let us write the given given equation 3x + 6y = 20 as y = ax + b,
So, we get
y = (20/6) + (-3/6)x
When "x" and "y" are related as y = a + bx, then
QD of "y" = |b| x QD of "x"
QD of "y" = |-3/6| x 6
QD of "y" = 3
Quartile-deviation is also rigidly defined, easy to compute and not much affected by sampling fluctuations.
The presence of extreme observations has no impact on quartile-deviation since quartile-deviation is based on the central fifty-percent of the observations.
However, quartile-deviation is not based on all the observations and it has no desirable mathematical properties.
Nevertheless, quartile deviation is the best measure of dispersion for open-end classifications.
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