If each observation in a data set for number of employees in different divisions is doubled then the coefficient of quartile deviation:

The correct answer is that it remains same.

Key Points 

 The formulae we need for solving the problem are as follows:

$$\begin{gathered} \mathbf{\text{Quartile formula for Q1,grouped data}}=Q_1=I_1+\frac{(N/4)-c}{f}\ast (I_2-I_1) \\ \mathbf{\text{Quartile formula for Q3,grouped data}}=Q_3=I_1+\frac{3\ast (N/4)-c}{f}\ast (I_2-I_1) \\ \mathbf{\text{Coefficient of quartile deviation}}=\frac{Q_3-Q_1}{Q_3+Q_1} \end{gathered}$$

Here, since all the observations are doubled the value for Quartiles becomes:

$$\begin{gathered} \text{Quartile formula for Q1,grouped data}=Q_1=I_1+\frac{(2N/4)-2c}{2f}\ast (I_2-I_1) \\ ⟹\text{Quartile formula for Q1,grouped data}=Q_1=I_1+\frac{(N/4)-c}{f}\ast (I_2-I_1) \\ \text{Quartile formula for Q3,grouped data}=Q_3=I_1+\frac{3\ast (2N/4)-2c}{2f}\ast (I_2-I_1) \\ ⟹\text{Quartile formula for Q3,grouped data}=Q_3=I_1+\frac{3\ast (N/4)-c}{f}\ast (I_2-I_1) \\ \text{Coefficient of quartile deviation}=\frac{Q_3-Q_1}{Q_3+Q_1} \end{gathered}$$

This is the same value as the coefficient of quartile deviation when the values of the observation of the data set were unchanged.

Thus the correct answer is- If each observation in a data set for the number of employees in different divisions is doubled then the coefficient of quartile deviation remains same.

Additional Information

•  The formula in this case used is for grouped data as the question specifically says observation of different employees for different departments.
• The same result can be derived for ungrouped data as well.