# QUARTILE DEVIATION

Another measure of dispersion is provided by quartile deviation or semi-inter–quartile range.

The formula to find quartile deviation is given by

## Coefficient of Quartile Deviation

A relative measure of dispersion using quartiles is given by coefficient of quartile deviation.

Formula to find coefficient of quartile deviation :

## Properties 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

## 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

=  3rd observation

= 56

Third Quartile (Q1) :

= [³⁽ⁿ ⁺ ¹⁾⁄₄]th observation

= [³⁽¹¹ ⁺ ¹⁾⁄₄]th observation

= [³⁽¹²⁾⁄₄]rd observation

= 9th 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

=  2.75th observation

= 2nd observation + 0.75 × difference between the 3rd and the 2nd 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

= 8.25th observation

= 8th observation + 0.25 × difference between the 9th  and the 8th 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.

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