**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.

**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

= 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 (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 observation

= 65 + 0.25 ( 75 - 65)

= 65 + 0.25 x 10

= 65 + 2.5

**Q****₃ **** = 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 ?

**Solution :**

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.

After having gone through the stuff given above, we hope that the students would have understood "Quartile-deviation".

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