Quartile Deviation

Dispersion means scatterings, variability, deviation or fluctuation of data.

The various measures of dispersion are:

1. An absolute measure of dispersion: It includes range, quartile deviation, mean deviation, standard deviation.
2. A relative measure of dispersion: It includes coefficient of range, coefficient of quartile deviation, coefficient of mean deviation, coefficient of standard deviation and coefficient of variation.

Half of the difference between upper quartile (Q₃) and lower quartile (Q₁) is called quartile deviation. It is also called as semi-interquartile range. Its formula is:
Quartile deviation (QD) = $\frac{Q_3 - Q_1}{2}$

The relative measure based on lower and upper quartile is coefficient of quartile deviation. It is given by:
Coefficient of Quartile deviation = $\frac{Q_3 - Q_1}{Q_3 + Q_1}$

Quartile deviation is a half of the difference between upper quartile (Q₃) and lower quartile (Q₁) whereas coefficient of quartile deviation is the relative measure based on lower and upper quartile.
Formula to calculate quartile deviation is $\frac{Q_3 - Q_1}{2}$ and coefficient of quartile deviation is $\frac{Q_3 - Q_1}{Q_3 + Q_1}$

Here,
First quartile(Q₁) = 40
Third quartile(Q₃) = 60
Quartile deviation(QD) = ?

Now, we know,
QD = $\frac{Q_3 - Q_1}{2}$ = $\frac{60 - 40}{2}$ ∴ QD = 10

Here,
Lower quartile(Q₁) = 25
Quartile deviation(QD) = 10
Upper quartile(Q₃) = ?

Now, we know, QD = $\frac{Q_3 - Q_1}{2}$ or, 10 = $\frac{Q_3 - 25}{2}$
or, 20 + 25 = Q₃
∴ Q₃ = 45

Here,
Coefficient of quartile deviation = $\frac 12$
Upper quartile(Q₃) = 60
First quartile(Q₁) = ?

Now, we know, Coefficient of quartile deviation = $\frac{Q_3 \, - \, Q_1}{Q_3 \, + \, Q_1}$ or, $\frac 12$ = $\frac{60 \, - \, Q_1}{60 \, + \, Q_1}$
or, 60 + Q₁ = 120 - 2Q₁
or, 3Q₁ = 60
∴ Q₁ = 20

Here,
Coefficient of quartile deviation = $\frac 37$
First quartile(Q₁) = 20
Upper quartile(Q₃) = ?

Now, we know, Coefficient of quartile deviation = $\frac{Q_3 \, - \, Q_1}{Q_3 \, + \, Q_1}$ or, $\frac 37$ = $\frac{Q_3 \, - \, 20}{Q_3 \, + \, 20}$
or, 3Q₃ + 60 = 7Q₃ - 140
or, 140 + 60 = 4Q₃
∴ Q₃ = 50