Combined Variance of Two Sets

Let $$n_1=10,\quad \bar{x}_1=5,\quad \sigma_1^2=4$$ and $$n_2=20,\quad \bar{x}_2=2,\quad \sigma_2^2=9$$

First find the mean of the combined group.

$$\bar{x}=\frac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2}$$

$$=\frac{10(5)+20(2)}{30}$$

$$=\frac{90}{30}$$

$$=3$$

Now use the combined variance formula:

$$\sigma^2=\frac{n_1\left(\sigma_1^2+(\bar{x}_1-\bar{x})^2\right)+n_2\left(\sigma_2^2+(\bar{x}_2-\bar{x})^2\right)}{n_1+n_2}$$

Substituting the values,

$$\sigma^2=\frac{10\left(4+(5-3)^2\right)+20\left(9+(2-3)^2\right)}{30}$$

$$=\frac{10(4+4)+20(9+1)}{30}$$

$$=\frac{10(8)+20(10)}{30}$$

$$=\frac{80+200}{30}$$

$$=\frac{280}{30}$$

$$=\frac{28}{3}$$

Final Answer :

$$\frac{28}{3}$$