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: 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: \[ \text{Weighted Mean} = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i} \] where \(w_i\) are the weights, and \(x_i\) 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: \[ \text{Grand Mean} = \frac{\sum_{i=1}^{n} x_i}{n} \] where \(x_i\) are the values and \(n\) 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: \[ \text{Harmonic Mean} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}} \] where \(x_i\) are the values and \(n\) is the total number of observations. This corresponds to **(III)** in List-II.
4. **Geometric Mean (D):** The geometric mean is the \(n\)-th root of the product of \(n\) values. The formula is: \[ \text{Geometric Mean} = \left( \prod_{i=1}^{n} x_i \right)^{1/n} \] where \(x_i\) are the values and \(n\) 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)**.