Step 1: Under normality, both the sample mean and sample median are unbiased estimators of the population mean $\mu$.
Step 2: By the Cramer-Rao inequality, the minimum possible variance any unbiased estimator of $\mu$ can achieve is $\sigma^2/n$, and the sample mean actually attains this bound, making it the most efficient (minimum variance unbiased) estimator for a normal population.
Step 3: The sample median is a different, less efficient estimator: its asymptotic efficiency relative to the mean is only $2/\pi \approx 0.637$, meaning it uses information less effectively than the mean.
Step 4: Since the mean already achieves the lowest possible variance and the median is strictly less efficient, the median's variance must be strictly larger: $V(\text{Median}) > V(\text{Mean})$.
\[\boxed{V(\text{Median}) > V(\text{Mean})}\]