The mean of `25` observation is `36`. Out of these observations, if the mean of first `13` observations is `32` and that of the last `13` observations is `40`, the `13th` observation is

Options

A. 23
B. 36
C. 38
D. 40

Explanation

To find the 13th observation from the given data, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Total Sum of 25 Observations**:
The mean of 25 observations is given as 36.
\[ \text{Total Sum} = \text{Number of Observations} \times \text{Mean} = 25 \times 36 = 900 \]

2. **Calculate the Sum of the First 13 Observations**:
The mean of the first 13 observations is given as 32.
\[ \text{Sum of First 13 Observations} = 13 \times 32 = 416 \]

3. **Calculate the Sum of the Last 13 Observations**:
The mean of the last 13 observations is given as 40.
\[ \text{Sum of Last 13 Observations} = 13 \times 40 = 520 \]

4. **Set Up the Equation to Find the 13th Observation**:
The 13th observation is included in both the sum of the first 13 and the last 13 observations. Therefore, we can express the 13th observation (let's denote it as \( x \)) as:
\[ x = (\text{Sum of First 13 Observations} + \text{Sum of Last 13 Observations}) - \text{Total Sum} \]

5. **Substitute the Values**:
\[ x = (416 + 520) - 900 \]
\[ x = 936 - 900 = 36 \]

### Conclusion:
The 13th observation is \( 36 \).

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