The reciprocal of the mean of the reciprocals of n observations is

Options

A. A.M.
B. G.M.
C. H.M.
D. Median

Answer

To solve the problem, we need to find the reciprocal of the mean of the reciprocals of \( n \) observations. Let's denote the observations as \( x_1, x_2, \ldots, x_n \).

### Step-by-Step Solution:

1. **Identify the Observations**:
Let the observations be \( x_1, x_2, \ldots, x_n \).

2. **Calculate the Reciprocals**:
The reciprocals of these observations are \( \frac{1}{x_1}, \frac{1}{x_2}, \ldots, \frac{1}{x_n} \).

3. **Find the Mean of the Reciprocals**:
The mean of the reciprocals is calculated as:
\[ \text{Mean of the reciprocals} = \frac{\frac{1}{x_1} + \frac{1}{x_2} + \ldots + \frac{1}{x_n}}{n} \]

4. **Reciprocal of the Mean of the Reciprocals**:
We need to find the reciprocal of the mean calculated in the previous step:
\[ \text{Reciprocal of the mean of the reciprocals} = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \ldots + \frac{1}{x_n}} \]

5. **Recognize the Harmonic Mean**:
The expression we derived in the previous step represents the harmonic mean of the observations \( x_1, x_2, \ldots, x_n \). The harmonic mean \( H \) is defined as:
\[ H = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \ldots + \frac{1}{x_n}} \]

6. **Conclusion**:
Therefore, the reciprocal of the mean of the reciprocals of \( n \) observations is the harmonic mean of those observations.

### Final Answer:
The reciprocal of the mean of the reciprocals of \( n \) observations is the **harmonic mean**.

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