The average (Arithmetic Mean) and the Median of a set of numbers is the same which of the following must be true?

Options

A. All the numbers are odd in the set
B. All the numbers are even in the set
C. All the numbers are consecutive integer in the set
D. The data set have even numbers of observations

Answer

To solve the problem, we need to analyze the relationship between the average (arithmetic mean) and the median of a set of numbers. The question states that the average and the median of a set of numbers are the same, and we need to determine which of the following statements must be true.

### Step-by-Step Solution:

1. **Understanding Average and Median**:
- The average (arithmetic mean) of a set is calculated as the sum of all the numbers divided by the count of the numbers.
- The median is the middle value when the numbers are arranged in ascending order. If there is an even number of observations, the median is the average of the two middle numbers.

2. **Setting Up a Sample Set**:
- Let's consider a simple set of consecutive integers: {1, 2, 3, 4, 5}.
- Calculate the average:
\[ \text{Average} = \frac{1 + 2 + 3 + 4 + 5}{5} = \frac{15}{5} = 3 \]
- Calculate the median:
- Since there are 5 numbers (odd), the median is the middle number, which is 3.

3. **Exploring Other Sets**:
- Now, let's consider another set: {1, 2, 3, 4, 5, 6}.
- Calculate the average:
\[ \text{Average} = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = 3.5 \]
- Calculate the median:
- Since there are 6 numbers (even), the median is the average of the two middle numbers (3 and 4):
\[ \text{Median} = \frac{3 + 4}{2} = 3.5 \]

4. **Conclusion**:
- From the examples, we observe that when the average and median are equal, it often occurs in sets of consecutive integers.
- Therefore, if the average and median are the same, it implies that the data set consists of consecutive integers.

5. **Final Answer**:
- The statement that must be true is that the numbers in the set are consecutive integers.