Question

Particulars regarding income of two villages are given below.
Village
A
B
No. of people
600
500
Average income
175
186
Standard deviation of income
10
9
Compute combined mean and combined Standard deviation.​

Answer 1

Answer:

Standard deviation of income 10 9

Answer 2

Combined mean = 180 and Combined Standard deviation = 11.02.

Explanation:

Given:

$N_{1}$=600, $N_{2}$=500

$X_{1}$=175, $X_{2}$=186

$\sigma_1$=10, $\sigma_2$=9

Using combined mean $X_{12}$ = $\frac{N_1X_1+\;N_2X_2}{N_1+N_2}$

On substituting the values in the formula,

Here, combined mean of income $X_{12}$ = $\frac{600(175)\;+\;500(186)}{600+500}$ = $\frac{198000}{1100}$

                                                                 = 180

Using combined standard deviation $\sigma_1_2$= $\sqrt{\frac{N_1\sigma_1^2+N_2\sigma_2^2+N_1d_1^2+N_2d_2^2}{N_1+N_2}}$

where ${d_1=X_1-\;X_{12}}$ = 175-180 = -5

${d_2=X_2-\;X_{12}}$ = 186-180 = 6

On substituting the values in the formula,

Here, combined mean of income $\sigma_1_2$= $\sqrt{\frac{600(100)+500(81)+600(25)+500(36)}{600+500}}$

                                                                = $\sqrt{\frac{133500}{1100}}$ = $\sqrt{121.364}$  = 11.02

                   

Therefore, Combined mean = 180 and Combined Standard deviation = 11.02.