Question

Diane has the following data:

54 62 m 58 64
If the median is 62, which number could m be?

Answer 1

To find a value for \( m \) such that the median of the data set \( 54, 62, m, 58, 64 \) is 62, we need to consider how the median is calculated. The median is the middle value when the numbers are arranged in order.

1. First, let's arrange the numbers without considering \( m \):
- The current numbers are \( 54, 58, 62, 64 \).

2. Depending on the value of \( m \), we will need to position it correctly in the ordered list to see how it affects the median.

The ordered list has 5 numbers, so the median will be the 3rd number when they are sorted.

### Case Analysis:
1. **If \( m < 54 \):**
- The order would be \( m, 54, 58, 62, 64 \)
- Median = 58 (not 62)

2. **If \( 54 \leq m < 58 \):**
- The order would be \( 54, m, 58, 62, 64 \)
- Median = 58 (not 62)

3. **If \( 58 \leq m < 62 \):**
- The order would be \( 54, 58, m, 62, 64 \)
- Median = \( m \) (needs to be 62, so \( m \) must be 62)

4. **If \( m = 62 \):**
- The order would be \( 54, 58, 62, 62, 64 \)
- Median = 62 (this works)

5. **If \( 62 < m < 64 \):**
- The order would be \( 54, 58, 62, m, 64 \)
- Median = 62 (this works)

6. **If \( m = 64 \):**
- The order would be \( 54, 58, 62, 62, 64 \)
- Median = 62 (this works)

7. **If \( m > 64 \):**
- The order would be \( 54, 58, 62, 64, m \)
- Median = 62 (this works)

### Conclusion:
The potential values for \( m \) that would allow the median to equal 62 are:
- \( m = 62 \)
- Any \( m \) such that \( 58 \leq m \leq 64 \)

Thus the possible values for \( m \) could be 62, or any number in the range 58 to 64 (inclusive).