Question

Find the M.D. & variance of the following distribution. x 2 3 4 5 6 7 8 9 10 f 1 1 2 4 4 3 7 5 3 

Answer 1

Mean:

$\bar{x}=\frac{\Sigma f_i x_i}{\Sigma f_i}$

$=\frac{(2\times1)+(3\times1)+(4\times2)+(5\times4)+(6\times4)+(7\times3)+(8\times7)+(9\times5)+(10\times3)}{1+1+2+4+4+3+7+5+3}$

$=\frac{2+3+8+20+24+21+56+45+30}{30}$

$=\frac{209}{30}=6.97$

Mean is approximately 7.

Variance:

$\sigma^{2}=\frac{\Sigma f_i (x-\bar{x})^{2}}{\Sigma f_i}$

$=\frac{1\times(2-7)^2+1\times(3-7)^2+2\times(4-7)^2+4\times(5-7)^2+4\times(6-7)^2+3\times(7-7)^2+7\times(8-7)^2+5\times(9-7)^2+3\times(10-7)^2}{30}$

$=\frac{25+16+18+16+4+0+7+20+27}{30}=4.43$

Mean Deviation:

$M.D = \frac{\Sigma f\lvert x-\bar{x}\lvert}{N}$

$=\frac{1\times(2-7)+1\times(3-7)+2\times(4-7)+4\times(5-7)+4\times(6-7)+3\times(7-7)+7\times(8-7)+5\times(9-7)+3\times(10-7)}{30}$

$=\frac{1}{30}=0.033$