Average

Step 1: Formula for geometric mean.
The geometric mean of values is given by:

Step 2: Apply the formula.
For the values 2, 4, and 8, the geometric mean is:

Step 3: Conclusion.
The geometric mean of 2, 4, and 8 is 4, so the correct answer is (C).

Step 1: Understanding the concepts and formulas.
1. **Weighted Mean (A):** The weighted mean is used when each value in a data set has a different level of importance or weight. The formula is: where are the weights, and are the values. This corresponds to **(I)** in List-II.
2. **Grand Mean of Combined Data (B):** The grand mean is the weighted average of the means of several groups. The formula is: where are the values and is the total number of observations. This corresponds to **(II)** in List-II.
3. **Harmonic Mean (C):** The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. The formula is: where are the values and is the total number of observations. This corresponds to **(III)** in List-II.
4. **Geometric Mean (D):** The geometric mean is the -th root of the product of values. The formula is: where are the values and is the total number of observations. This corresponds to **(IV)** in List-II.

Step 2: Matching the concepts with formulas.
- (A) **Weighted Mean** corresponds to formula **(I)**. - (B) **Grand Mean of Combined Data** corresponds to formula **(II)**. - (C) **Harmonic Mean** corresponds to formula **(III)**. - (D) **Geometric Mean** corresponds to formula **(IV)**.

Step 3: Conclusion.
The correct answer is **2. (A) - (I), (B) - (II), (C) - (III), (D) - (IV)**.