Question

Calculate the mean deviation about the median of the following observations:
(i) 3011, 2780, 3020, 2354, 3541, 4150, 5000
(ii) 38, 70, 48, 34, 42, 55, 63, 46, 54, 44
(iii) 34, 66, 30, 38, 44, 50, 40, 60, 42, 51
(iv) 22, 24, 30, 27, 29, 31, 25, 28, 41, 42
(v) 38, 70, 48, 34, 63, 42, 55, 44, 53, 47

Answer 1

Formula used for mean deviation:
$$\begin{gathered} MD=\frac{1}{n}\sum _{i=1}^n\left|d_i\right| \\ Here, \\ {d}_i=x_i-M \end{gathered}$$
M = Median

i) Arranging the data in ascending order:
2354, 2780, 3011, 3020, 3541, 4150, 5000

Here, median $\left(M\right)=3020$ and n = 7.

$x_i$	$\left|d_i\right|$ = $\left|x_i-3020\right|$
3011	9
2780	240
3020	0
2354	666
3541	521
4150	1130
5000	1980
Total	4546

$MD=\frac{1}{n}\sum _{i=1}^n\left|d_i\right|$

$\Rightarrow MD=\frac{1}{7}\times 4546=649.42$

ii) Arranging the data in ascending order:
34, 38, 42, 44, 46, 48, 54, 55, 63, 70

Here, n is equal to 10.
Median is the arithmetic mean of the fifth and the sixth observation.

$Median,M=\frac{46+48}{2}=47$

xi	$\left|d_i\right|=\left|x_i-M\right|$
38	9
70	23
48	1
34	13
42	5
55	8
63	16
46	1
54	7
44	3
Total	86

$MD=\frac{1}{10}\times 86=8.6$

iii) Arranging the data in ascending order:
30, 34, 38, 40, 42, 44, 50, 51, 60, 66

Here,
$n=10$
Also, median is the AM of the fifth and the sixth observation.

$Median,M=\frac{42+44}{2}=43$

xi	$\left|d_i\right|=\left|x_i-M\right|$
34	9
66	23
30	13
38	5
44	1
50	7
40	3
60	17
42	1
51	8
Total	87

$MD=\frac{1}{10}\times 87=8.7$

iv) Arranging the data in ascending order.
22, 24, 25, 27, 28, 29, 30, 31, 41, 42

$\mathrm{Here},n=10.$
Also, median is the AM of the fifth and the sixth observation.

$Median,M=\frac{28+29}{2}=28.5$

xi	$\left|d_i\right|=\left|x_i-M\right|$
22	6.5
24	4.5
30	1.5
27	1.5
29	0.5
31	2.5
25	3.5
28	0.5
41	12.5
41	13.5
Total	47

$MD=\frac{1}{10}\times 47=4.7$

v) Arranging the data in ascending order:
34, 38, 42, 44, 47, 48, 53, 55, 63, 70

Here, $n=10$.
Also, median is the AM of the fifth and the sixth observation.

$Median,M=\frac{47+48}{2}=47.5$

xi	$\left|d_i\right|=\left|x_i-M\right|$
38	9.5
70	22.5
48	0.5
34	13.5
63	15.5
42	5.5
55	7.5
44	3.5
53	5.5
47	0.5
Total	84

$MD=\frac{1}{10}\times 84=8.4$