Question

Calculate the mean deviation about the median for the following data:

$\begin{array}{ccccccc}\text{Height (in cm)}& 95-105& 105-115& 115-125& 125-135& 135-145& 145-155\\ \text{Number of boys}& 9& 13& 25& 30& 13& 10\end{array}$

Answer 1

First we find the median.

$\begin{array}{cccc}\text{Class}& \text{Frequency}{f}_i& \text{Cumulative frequency (cf)}& \text{Midpoint}{x}_i\\ 95-105& 9& 9& 100\\ 105-115& 13& 22& 110\\ 115-125& 25& 47& 120\\ 125-135& 30& 77& 130\\ 135-145& 13& 90& 140\\ 145-155& 10& 100& 150\\ & N=\sum f_i=100& & \end{array}$

Thus N = 100 and therefore, $\frac{N}{2}=50$

$\Rightarrow$ median class is 125 - 135

$\Rightarrow$ L = 125, f = 30, h = 10 and c = 47.

$\therefore \text{median}=L+\frac{(\frac{N}{2}-c)}{f}\times h$

$=\{125+\frac{(50-47)}{30}\times 10\}=(125+1)=126$

$\therefore M=126$

Now, we prepare the table given below.

$\begin{array}{cccc}{f}_i& x_i& |x_i-M|& f_i\times |x_i-M|\\ 9& 100& 26& 234\\ 13& 110& 16& 208\\ 25& 120& 6& 150\\ 30& 130& 4& 120\\ 13& 140& 14& 182\\ 10& 150& 24& 140\\ N=100& & & 1134\end{array}$

$\therefore MD\left(M\right)=\frac{\sum f_i\times |x_i-M|}{N}=\frac{1134}{100}=11.34$

Hence, the mean deviation about the median is 11.34