Question

Calculate the mean deviation about median age for the age distribution of $100$ persons given below :

Age	Number
16-20	5
21-25	6
26-30	12
31-35	14
36-40	26
41-45	12
46-50	16
51-55	9

Answer 1

Clearly, we can see that given data is not continuous. So, to convert into continuous frequency distribution, we need to subtract $0.5$ from the lower limit and add $0.5$ to the upper limit of each class interval

Class	$f_i$	Cumulative frequency	$x_i$	$|x_i-M|$	$f_i|x_i-M|$
15.5-20.5	5	5	18	20	100
20.5-25.5	6	11	23	15	90
25.5-30.5	12	23	28	10	120
30.5-35.5	14	37	33	5	70
35.5-40.5	26	63	38	0	0
40.5-45.5	12	75	43	5	60
45.5-50.5	16	91	48	10	160
50.5-55.5	9	100	53	15	135
Total	100				735

Here, $N=100$

$\frac{N}{2}=\frac{100}{2}=50$

which occurs in the cumulative frequency $63$.

Hence, the median class is $35.5-40.5$.

Median $M=l+\frac{\frac{N}{2}-C_{f_{-1}}}{f_m}\times h$

$=35.5+\frac{50-37}{26}\times 5$

$=35.5+\frac{13}{26}\times 5$

$\Rightarrow M=35.5+2.5=38$

Mean deviation about the median is

$M.D.\left(M\right)=\frac{1}{N}\sum _{i=1}^8{f}_i\mid x_i-M\mid$

$=\frac{1}{100}\times 735=7.35$