Given that the arithmetic mean and standard deviation of a sample of 15 observations are 24 and 0 respectively. Then which one of the following is the arithmetic mean of the smallest five observation in the data?

Options

A. 0
B. 8
C. 16
D. 24

Explanation

To solve the problem, we need to analyze the information given about the sample of 15 observations.

### Step-by-Step Solution:

1. **Understanding the Given Data**:
- We have a sample of 15 observations.
- The arithmetic mean (average) of these observations is 24.
- The standard deviation is 0.

2. **Implication of Standard Deviation**:
- A standard deviation of 0 indicates that all observations in the sample are identical. This means every observation in the sample is equal to the mean, which is 24.

3. **Identifying the Observations**:
- Since all observations are the same and equal to the mean, we can conclude that:
- Observation 1 = 24
- Observation 2 = 24
- Observation 3 = 24
- Observation 4 = 24
- Observation 5 = 24
- Observation 6 = 24
- Observation 7 = 24
- Observation 8 = 24
- Observation 9 = 24
- Observation 10 = 24
- Observation 11 = 24
- Observation 12 = 24
- Observation 13 = 24
- Observation 14 = 24
- Observation 15 = 24

4. **Calculating the Arithmetic Mean of the Smallest Five Observations**:
- The smallest five observations are simply the first five observations since they are all identical and equal to 24.
- Therefore, the smallest five observations are: 24, 24, 24, 24, 24.

5. **Finding the Mean of These Observations**:
- To find the arithmetic mean of these five observations:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} = \frac{24 + 24 + 24 + 24 + 24}{5} = \frac{120}{5} = 24 \]

### Conclusion:
The arithmetic mean of the smallest five observations in the data is **24**.

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