Mean – Problems

Mean – Problems

1. Mean of 9 observations is 100 and mean of 6 observations is 80, then the mean of 15 observations is

Solutions: 9 observations is 100

6 observations is 80

n₁ = 9 and n₂ = 6

x̄₁ = 100 and x̄₂ = 80

We know that

\(\bar{x}=\frac{{{n}_{1}}{{{\bar{x}}}_{1}}+{{n}_{2}}{{{\bar{x}}}_{2}}}{m+n}\),

\(=\frac{9\times 100+6\times 80}{9+6}\),

\(=\frac{1380}{15}=92\).

2. Mean and made of the following data are respectively

Class	fᵢ
0 – 10	22
10 – 20	38
20 – 30	46
30 – 40	35
40 – 50	20

Solution:

Class	xᵢ	fᵢ	xᵢfᵢ
0 – 10	5	22	110
10 – 20	15	38	570
20 – 30	25	46	1150
30 – 40	35	35	1225
40 – 50	42	20	900
		161	3955

Mean \(=\frac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}=\frac{3955}{161}\) = 24.56

Median = 24.26 (approximate)

Mean = 3 (median) – 2 (mean)

Mode = 3 (24.46) – 2 (24.46)

= 24.26.

3. Find the mean deviation about mean for the data in following table

Income per Year	Number of Persons
0 – 100	4
100 – 200	8
200 – 300	9
300 – 400	10
400 – 500	7
500 – 600	5
600 – 700	4
700 – 800	3

Solution:

Income per Year	Number of Persons	Mid Value xᵢ	\({{d}_{i}}={{A}_{i}}+\frac{\sum{{{f}_{i}}{{d}_{i}}}}{\sum{{{f}_{i}}}}\times h\), A = 350 & h = 100	fᵢdᵢ	|xᵢ – x̄|	fᵢ |xᵢ – x̄|
0 – 100	4	50	-3	-12	308	1232
100 – 200	8	150	-2	-16	208	1664
200 – 300	9	250	-1	-9	108	972
300 – 400	10	350	0	0	8	80
400 – 500	7	450	1	7	92	644
500 – 600	5	550	2	10	192	960
600 – 700	4	650	3	12	292	1168
700 – 800	3	750	4	12	392	1176
		∑ fᵢ = 50				7896

Mean x̄ \(={{A}_{i}}+\frac{\sum{{{f}_{i}}{{d}_{i}}}}{\sum{{{f}_{i}}}}\times h\),

\(=350+\frac{4}{50}\times 100=350+8\),

x̄ = 358,

Mean deviation about the mean \(=\frac{\sum{{{f}_{i}}|{{x}_{i}}-\bar{x}|}}{\sum{{{f}_{i}}}}=\frac{7896}{50}=157.92\).