Question

For a normal distribution variance of mean V(Mean) and variance of median V(Median), which of the following is true?

• A. V(Median) < V(Mean)
• B. V(Median) = V(Mean)
• C. V(Median) > V(Mean)
• D. V(Median) &times; 1.57 = V(Mean)

Hint:

For normal populations, \(V(\text{Median}) = (\pi/2)\,V(\text{Mean})\), and \(\pi/2 > 1\).

Answer 1

The Correct Option is C

Step 1: For a random sample of size \(n\) drawn from a normal population with variance \(\sigma^2\), the sampling variance of the sample mean is \[V(\text{Mean}) = \frac{\sigma^2}{n}\]

Step 2: For large \(n\), the sampling variance of the sample median from a normal population is \[V(\text{Median}) = \frac{\pi}{2}\cdot\frac{\sigma^2}{n} \approx 1.5708 \times V(\text{Mean})\]

Step 3: Since \(\pi/2 \approx 1.5708 > 1\), it follows that \[V(\text{Median}) > V(\text{Mean})\]

Step 4: This means the sample median is a less efficient estimator of the population mean than the sample mean; the asymptotic relative efficiency of median to mean is \(2/\pi \approx 0.637\).

Final answer: \(\boxed{V(\text{Median}) > V(\text{Mean})}\)