If the mean deviation about the median of the numbers a, 2a,,50a is 50, then |a| equals

Question

If the mean deviation about the median of the numbers $a, 2a,\ldots,50a$ is 50, then $|a|$ equals

Accepted Answer

NOTE :

If total no of terms are even then median

$$ = {1 \over 2}$$ [ $${n \over 2}$$th term $$ + \left( {{n \over 2} + 1} \right)$$ th term]

Here total terms $= 50,$ which is even

$$\therefore$$ $$\,\,\,$$ Median $$ = {1 \over 2}$$ [ $${{50} \over 2}$$ th term $$ + \left( {{{50} \over 2} + 1} \right)$$ th term]

$$ = {1 \over 2}$$ [ $25$ th term $+$ $26$ th term ]

$$ = {1 \over 2}$$ [ $25a$ $+$ $26a$ ]

$=25.5a$

Mean deviation (M.D.) about the median

$$ = {{\sum\limits_{i = 1}^{50} {\left| {{x_i} - Median} \right|} } \over N} = 50$$ (given)

$$\therefore$$ $${1 \over {50}}\left[ {2 \times \left| a \right| \times \left( {0.5 + 1.5 + 2.5 + ....24.5} \right)} \right] = 50$$

$$ \Rightarrow 2\left| a \right| \times {{25} \over 2} \times 25 = 2500$$

$$ \Rightarrow \left| a \right| = {{2500} \over {25 \times 25}} = 4$$

Solutions · 1