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.
Formula to find coefficient of quartile deviation :
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
Q.D of y = |b| ⋅ Q.D of x
Problem 1 :
Following are the wages of the 11 labourers :
$82, $56, $90, $50, $48, $99, $77, $75, $80, $97, $65
Find quartile deviation and its coefficient.
Solution :
Arrange the given data in ascending order :
$48, $50, $56, $65, $75, $77, $80, $82, $90, $97, $99
To get the first quartile, find the value of ⁽ⁿ ⁺ ¹⁾⁄₄.
= ⁽ⁿ ⁺ ¹⁾⁄₄
= ⁽¹¹ ⁺ ¹⁾⁄₄
= ¹²⁄₄
= 3
First Quartile (Q_{1}) :
= [⁽ⁿ ⁺ ¹⁾⁄₄]^{th} observation
= [⁽¹¹ ⁺ ¹⁾⁄₄]^{th} observation
= (¹²⁄₄)^{rd} observation
= 3^{rd} observation
= 56
Third Quartile (Q_{1}) :
= [³⁽ⁿ ⁺ ¹⁾⁄₄]^{th} observation
= [³⁽¹¹ ⁺ ¹⁾⁄₄]^{th} observation
= [³⁽¹²⁾⁄₄]^{rd} observation
= 9^{th} observation
= 80
Quartile Deviation (Q.D) :
= 12
Coefficient of Quartile Deviation :
= 17.91
Problem 2 :
Following are the marks of the 10 students :
56, 48, 65, 35, 42, 75, 82, 60, 55, 50
Find quartile deviation and its coefficient.
Solution :
After arranging the marks in an ascending order of magnitude, we get
35, 42, 48, 50, 55, 56, 60, 65, 75, 82
First quartile (Q_{1}) :
= [⁽ⁿ ⁺ ¹⁾⁄₄]^{th} observation
= [⁽¹⁰ ⁺ ¹⁾⁄₄]^{th} observation
= 2.75^{th} observation
= 2^{nd} observation + 0.75 × difference between the 3^{rd} and the 2^{nd} observations
= 42 + 0.75 (48 - 42)
= 42 + 0.75 x 6
= 42 + 4.5
Q_{1} = 46.5
Third quartile (Q_{3}) :
= [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} observations
= 65 + 0.25 ( 75 - 65)
= 65 + 0.25 x 10
= 65 + 2.5
Q_{3} = 67.5
Quartile Deviation (Q.D) :
= 10.5
Coefficient of Quartile Deviation :
= 18.42
Problem 3 :
If the quartile deviation of x is 6 and 3x + 6y = 20, what is the quartile deviation of y?
Solution :
Write the equation 3x + 6y = 20 in the form y = a + bx.
3x + 6y = 20
Subtract 6x from both sides.
6y = 20 - 3x
6y = 20 + (-3x)
Divide both sides by 6.
⁶ʸ⁄₆ = ²⁰⁄₆ + (-³⁄₆)x
y = ¹⁰⁄₃ + (-½)x
When x and y are related as y = a + bx, then
Q.D of y = |b| ⋅ Q.D of x
= |-½| ⋅ 6
= ½ ⋅ 6
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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