Question

Find the mean deviation about the median for the data.

$x_i$	5	7	9	10	12	15
$f_i$	8	6	2	2	2	6

Answer 1

The given observations are already in ascending order.

Adding a column corresponding to cumulative frequencies of the given data, we obtain the following table

$x_i$	$f_i$	$c.f$
5	8	8
7	6	14
9	2	16
10	2	18
12	2	20
15	6	26

Here, N = 26, which is even.

Median is the mean of 13th and 14th observations. Both of these observations lie in the cumulative frequency 14, for which the corresponding observation is 7.

Median,M= $\frac{\frac13^{th}observation+14^{14}observation}{2}=\frac{7+7}{2}=7$

The absolute values of the deviations from median, i.e $|x_i-M|,$ are

$|x_i,M|$| 2| 0| 2| 3| 5| 8
---|---|---|---|---|---|---
$f_i$| 8| 6| 2| 2| 2| 6
$f_i|x_iM|$| 16| 0| 4| 6| 10| 48

$\sum_{I=1}^{6}f_i=26$ and $\sum_{I=1}^{6}f_i|x_i-M|=84$

$=MD(M)=\frac{1}{N}\sum_{i=1}^{6}f_i|x_i-M|=\frac{1}{26}X84=3.23$