Topic: Measures of Central Tendency
Understanding the Question:
Calculate the Geometric Mean (GM) for the set $\{2, 4, 8\}$.
Key Formulas and Approach:
The Geometric Mean of $n$ numbers is the $n$-th root of their product:
\[ GM = \sqrt[n]{x_1 \cdot x_2 \cdot \dots \cdot x_n} \]
Detailed Solution:
Step 1: Identify the values. $n = 3$. The values are $x_1 = 2$, $x_2 = 4$, and $x_3 = 8$.
Step 2: Multiply the values.
\[ \text{Product} = 2 \times 4 \times 8 = 64 \]
Step 3: Take the n-th root. Since there are 3 numbers, we take the cube root:
\[ GM = \sqrt[3]{64} \]
Step 4: Simplify. Since $4^3 = 64$, the cube root is 4.
Conclusion: The Geometric Mean is 4.