Question

Find the mean deviation from the median for the following data :

(i) $\begin{array}{cccccc}{x}_i& 15& 21& 27& 30& 35\\ f_i& 3& 5& 6& 7& 8\end{array}$

(ii) $\begin{array}{cccccccc}{x}_i& 74& 89& 42& 54& 91& 94& 35\\ f_i& 20& 12& 2& 4& 5& 3& 4\end{array}$

(iii) $\begin{array}{cccccc}\text{Marks obtained}& 10& 11& 12& 14& 15\\ \text{No. of students}& 2& 3& 8& 3& 4\end{array}$

Answer 1

(i) $\begin{array}{ccccc}{x}_i& f_i& \text{Cumulative frequency}& |d_i|-|x_i-30|& f_i|x_i-30|\\ 15& 3& 3& 15& 45\\ 21& 5& 8& 9& 45\\ 27& 6& 14& 3& 18\\ 30& 7& 21& 0& 0\\ 35& 8& 29& 5& 40\\ & N=\sum f_i=29& & & {\sum }_{i=1}^n{f}_i|x_i-30|=148\end{array}$

Here, $\frac{N}{2}=\frac{29}{2}=14.5$

The cumulative frequency greater than 14.5 is 21 and the corresponding value of x is 30 Median, M = 30

$M.D.=\frac{1}{N}{\sum }_{i=1}^n{f}_i|x_i-M|=\frac{1}{29}\times 148=5.10$

$\begin{array}{ccccc}{x}_i& f_i& \text{Cumulative frequency}& |x_i-74|& f_i|x_i-74|\\ 35& 4& 4& 39& 156\\ 42& 2& 6& 32& 64\\ 54& 4& 10& 20& 80\\ 74& 20& 30& 0& 0\\ 89& 12& 42& 15& 180\\ 91& 5& 47& 17& 85\\ 94& 3& 50& 20& 60\\ & N=\sum f_i=50& & & {\sum }_{i=1}^n{f}_i|x_i-74|=625\end{array}$

Here, $\frac{N}{2}=\frac{50}{2}=25$

The cumulative frequency greter than 25 is 30 and the corresponding value of x is 74 Median, M= 89

$M.D.=\frac{1}{N}{\sum }_{i=1}^n{f}_i|x_i-M|=\frac{1}{50}\times 775=12.5$

$\begin{array}{ccccc}{x}_i& f_i& \text{Cumulative frequency}& |d_i|=|x_i-12|& f_i|d_i|\\ 10& 2& 2& 2& 4\\ 11& 3& 5& 1& 3\\ 12& 8& 13& 0& 0\\ 14& 3& 16& 2& 6\\ 15& 4& 20& 3& 12\\ & 20& & & \text{Total = 25}\end{array}$

$\frac{N}{2}=10$

Median, = 12

$M.D.=\frac{25}{20}=1.25$