1. The problem involves understanding the variance formula given by $$\operatorname{Var}(\bar{X}) = \sigma^2 \cdot \frac{n^2}{m n^2} = \sigma^2 \cdot \frac{1}{m}$$ This shows how the variance of the sample mean depends on the variance of individual observations $\sigma^2$ and the sample size $m$. 2. Next, given the expression $$\frac{(m n^2 - 1)}{(m^2 n^2)}, \quad \sigma_3 = \frac{1}{2}, \quad n=1$$ we substitute $n=1$ to simplify: $$\frac{(m \cdot 1^2 - 1)}{(m^2 \cdot 1^2)} = \frac{m - 1}{m^2}$$ which expresses a ratio involving $m$ only. 3. The parameter $\sigma_3 = \frac{1}{2}$ is given, but there is no direct relation to rewrite. It may be a given constant or standard deviation related to a part of the problem. 4. Regarding the geometric description, a blue square of side approximately 2 cm contains a smaller black square of side about 1 cm located near the bottom-left corner of the larger square. This spatial arrangement may represent a scaling or nested region scenario. Final answers and key expressions: - Variance of the mean: $$\operatorname{Var}(\bar{X}) = \sigma^2 \cdot \frac{1}{m}$$ - Simplified ratio expression: $$\frac{m - 1}{m^2}$$