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(A) The given numbers are $a, 2a, 3a, \dots, 50a$. The number of terms $n = 50$.Since $n$ is even,the median $M$ is the average of the $25^{th}$ and $

Vedclass · Accepted Answer

$4$

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(A) The given numbers are $a, 2a, 3a, \dots, 50a$. The number of terms $n = 50$.Since $n$ is even,the median $M$ is the average of the $25^{th}$ and $26^{th}$ terms:$M = \frac{25a + 26a}{2} = 25.5a$.The mean deviation about the median is given by $M.D. = \frac{1}{n} \sum_{i=1}^{n} |x_i - M|$.$M.D. = \frac{1}{50} \sum_{i=1}^{50} |ia - 25.5a| = \frac{|a|}{50} \sum_{i=1}^{50} |i - 25.5|$.Expanding the sum: $\sum_{i=1}^{50} |i - 25.5| = |1 - 25.5| + |2 - 25.5| + \dots + |50 - 25.5|$.$= 24.5 + 23.5 + \dots + 0.5 + 0.5 + \dots + 23.5 + 24.5$.$= 2 	imes (0.5 + 1.5 + \dots + 24.5)$.This is an arithmetic progression with $25$ terms where the first term $a_1 = 0.5$ and last term $l = 24.5$.Sum $= 2 	imes \left[ \frac{25}{2} (0.5 + 24.5) ight] = 25 	imes 25 = 625$.Given $M.D. = 50$,so $\frac{|a|}{50} 	imes 625 = 50$.$|a| 	imes 12.5 = 50 \Rightarrow |a| = \frac{50}{12.5} = 4$.

$x_i$	$5, 15, 25, 35, 45$
$f_i$	$8, 48, 56, 32, 16$

Class interval	Frequency
$0-2$	$1$
$2-4$	$3$
$4-6$	$5$
$6-8$	$3$
$8-10$	$1$

Class interval	$0-4$	$4-8$	$8-12$	$12-16$	$16-20$
Frequency	$4$	$6$	$8$	$5$	$2$

If the mean deviation about the median of the numbers $a, 2a, 3a, \dots, 50a$ is $50$,then what is the value of $|a|$?

Similar Questions

If the mean deviation from the mean of the observations $\{1+K \alpha\}$,where $K=0, 1, 2, \ldots, 100$ and $\alpha > 0$ is $255$,then $\alpha =$

The mean deviation of the following data about its mean is
$x_i$ $5, 15, 25, 35, 45$
$f_i$ $8, 48, 56, 32, 16$

Calculate the mean deviation about the mean of the set of first $n$ natural numbers when $n$ is an even number.

The mean deviation about the mean for the following data is
Class interval Frequency
$0-2$ $1$
$2-4$ $3$
$4-6$ $5$
$6-8$ $3$
$8-10$ $1$

Calculate the mean deviation about the mean for the following frequency distribution:
Class interval $0-4$ $4-8$ $8-12$ $12-16$ $16-20$
Frequency $4$ $6$ $8$ $5$ $2$

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