Question

Calculate the mean deviation about median age for the age distribution of 100persons given below:12.Age 16-20 21-25 26-30 31-35 36-40 41-45 46-50 51-55(in years)Number512142612 16 9

Answer 1

The given data is:

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

Since the given data is not continuous, therefore it has to be made continuous by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of ach class interval. The midpoint of the intervals is calculated by adding first and last terms and dividing it by 2. The cumulative frequencies is given by writing first term of discrete frequency in the first row of cumulative frequency then adding the corresponding terms of cumulative frequency and frequency. The formula to calculate the absolute deviation of the respective observations of the data is,

| x i −M |.

Now, make a table using the above formula and continuous frequency.

Age	Number, f i	c.f.	Midpoint, x i	| x i −M |	f i | x i −M |
15.5-20.5	5	5	18	20	100
20.5-25.5	6	5+6=11	23	15	90
25.5-30.5	12	11+12=23	28	10	120
30.5-35.5	14	23+14=37	33	5	70
35.5-40.5	26	37+26=63	38	0	0
40.5-45.5	12	63+12=75	43	5	60
45.5-50.5	16	75+16=91	48	10	160
50.5-55.5	9	91+9=100	53	15	135
	∑ i=1 n f i =100				∑ i=1 n f i | x i −M | =735

Identify the observation whose cumulative frequency is equal to or just greater than N 2 , where N is the sum of frequency.

Since the sum of the frequencies is 100 and it is an even number, therefore the observations which are calculated by N 2 are 50 th . Since the value of N 2 is 50 and the frequency that is equal to or just greater than 50 is 63, its corresponding observation is 35.5-45.5.

Thus, the median class of the given data is 35.5-45.5.

The formula to calculate the median is,

M=l+ N 2 −C f ×h(1)

Here, l is the lower limit of the median class, C is the number previous to the number of actual cumulative frequency, f is the frequency corresponding to the median class and h is the interval gap.

Substitute 100 for N, 35.5 for l, 37 for C, 26 for f and 5 for h in equation (1).

M=35.5+ 100 2 −37 26 ×5 =35.5+ 50−37 26 ×5 =35.5+ 13×5 26 =38

Thus, the median of the given data is 38.

The formula to calculate the mean deviation about the median is,

M.D= ∑ i=1 n f i | x i −M | ∑ i=1 n f i M.D= 1 N ∑ i=1 n f i | x i −M | (2)

Substitute 100 for N and 735 for ∑ i=1 n f i | x i −M | in equation (2).

M.D.= 735 100 =7.35

Thus, the mean deviation of the given data is 7.35.