QUARTILE DEVIATION

Solved Problems

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 (Q₁) :

= [⁽ⁿ ⁺ ¹⁾⁄₄]^th observation

= [⁽¹¹ ⁺ ¹⁾⁄₄]^th observation

= (¹²⁄₄)^rd observation

=  3^rd observation

= 56

Third Quartile (Q₁) :

= [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 (Q₁) :

= [⁽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

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 observations

= 65 + 0.25 ( 75 - 65)

= 65 + 0.25 x 10

= 65 + 2.5

Q₃ = 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

Review

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.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.