Question

Match List-I with List-II \[\begin{array}{|c|c|} \hline \textbf{Concepts} & \textbf{Formula} \\ \hline \text{(A) Weighted Mean} & \text{(I) \(\frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}\)} \\ \hline \text{(B) Grand Mean of Combined Data} & \text{(II) \(\frac{\sum_{i=1}^{n} x_i}{n}\)} \\ \hline \text{(C) Harmonic Mean} & \text{(III) \(\frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}\)} \\ \hline \text{(D) Geometric Mean} & \text{(IV) \(\left( \prod_{i=1}^{n} x_i \right)^{1/n}\)} \\ \hline \end{array}\] Choose the correct answer from the options given below:

• (A) - (IV), (B) - (III), (C) - (II), (D) - (I)
• (A) - (I), (B) - (II), (C) - (III), (D) - (IV)
• (A) - (I), (B) - (II), (C) - (IV), (D) - (III)
• (A) - (III), (B) - (IV), (C) - (II), (D) - (I)

Hint:

The weighted mean accounts for varying degrees of importance (weights) of different data points, whereas the harmonic mean is useful for rates and ratios.

Answer 1

The Correct Option is B

Step 1: Concept Definitions and Formulas.
1. Weighted Mean (A): Applied when data points have varying significance. Formula: \[\text{Weighted Mean} = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}\] where \(w_i\) are weights and \(x_i\) are values. Matches (I) in List-II.
2. Grand Mean of Combined Data (B): The weighted average of group means. Formula: \[\text{Grand Mean} = \frac{\sum_{i=1}^{n} x_i}{n}\] where \(x_i\) are values and \(n\) is the total count. Matches (II) in List-II.
3. Harmonic Mean (C): The reciprocal of the arithmetic mean of reciprocals. Formula: \[\text{Harmonic Mean} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}\] where \(x_i\) are values and \(n\) is the total count. Matches (III) in List-II.
4. Geometric Mean (D): The \(n\)-th root of the product of \(n\) values. Formula: \[\text{Geometric Mean} = \left( \prod_{i=1}^{n} x_i \right)^{1/n}\] where \(x_i\) are values and \(n\) is the total count. Matches (IV) in List-II.

Step 2: Concept-Formula Matching.
- (A) Weighted Mean maps to formula (I).
- (B) Grand Mean of Combined Data maps to formula (II).
- (C) Harmonic Mean maps to formula (III).
- (D) Geometric Mean maps to formula (IV).

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