Calculate the mean deviation about the median of the following frequency distribution:
$x_{i}$ $5$ $7$ $9$ $11$ $13$ $15$ $17$
$f_{i}$ $2$ $4$ $6$ $8$ $10$ $12$ $8$

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Solution

We will first calculate the median for the data.
$x_{i}$ $f_{i}$ Cumulative Frequency $(C F)$ $d_{i} = x_{i} - A$
$= x_{i} - 13$ $f_{i} d_{i}$
$5$ $2$ $2$ $8$ $16$
$7$ $4$ $6$ $6$ $24$
$9$ $6$ $12$ $4$ $24$
$11$ $8$ $20$ $2$ $16$
$13$ $10$ $30$ $0$ $0$
$15$ $12$ $42$ $2$ $24$
$17$ $8$ $50$ $4$ $32$
$N = \sum f_{i} = 50$ $\sum f_{i} d_{i} = 136$

Here,
$\frac{N}{2} = \frac{50}{2} = 25$
The cumulative frequency just greater than $25$ is $30$ and the corresponding value of $x$ is $13.$
$∴$ Median, $M = 13$
Mean deviation about median, $M D = \frac{\sum f i d i}{N}$
$= \frac{136}{50} = 2.72$

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$x_{i}$	$5$	$7$	$9$	$11$	$13$	$15$	$17$
$f_{i}$	$2$	$4$	$6$	$8$	$10$	$12$	$8$