Question

The variable x takes two values x1 and x2 with frequencies f1 and f2 respectively. If sigma denotes the standard deviation of x then (sigma)^2 is equal to

Answer 1

Answer:

$f_1f_2\frac{(x_1-x_2)^2}{(f_1+f_2)^2}$

Step-by-step explanation:

We denote ,

n = sample size

$n=(f_1+f_2)$

$\sigma =Standard\:Deviation\\ \\ \sigma^2 = Variance$

We know that,

$\sigma^2 = \frac{\sum x_i^2}{n}-(\frac{\sum x_i}{n})^2 \\ \\ => \frac{x_1^2f_1+x_2^2f_2}{f_1+f_2}-(\frac{x_1f_1+x_2f_2}{f_1+f_2})^2\\ \\ =>\frac{f_1f_2x_2^2+f_1f_2x_1^2}{(f_1+f_2)^2}-\frac{2f_1f_2x_1x_2}{(f_1+f_2)^2}\\ \\=>\frac{f_1f_2(x_1-x_2)^2}{(f_1+f_2)^2}$