If a variable takes the discrete values `alpha-4`,
`alpha -(7)/(2), alpha-(5)/(2), alpha-2,alpha+(1)/(2), alpha-(1)/(2), alpha+5(alpha gt 0)`, then the median is

Options

A. `alpha-(5)/(4)`
B. `alpha-(1)/(2)`
C. `alpha-2`
D. `alpha+(5)/(4)`

Explanation

To find the median of the given discrete values, we will follow these steps:

### Step 1: List the Values
The variable takes the following discrete values:
- \( \alpha - 4 \)
- \( \alpha - \frac{7}{2} \)
- \( \alpha - \frac{5}{2} \)
- \( \alpha - 2 \)
- \( \alpha + \frac{1}{2} \)
- \( \alpha - \frac{1}{2} \)
- \( \alpha + 5 \)

### Step 2: Arrange the Values in Ascending Order
We need to arrange these values in ascending order. To do this, we can compare them based on the value of \( \alpha \).

1. \( \alpha - 4 \)
2. \( \alpha - \frac{7}{2} \) (which is \( \alpha - 3.5 \))
3. \( \alpha - \frac{5}{2} \) (which is \( \alpha - 2.5 \))
4. \( \alpha - 2 \)
5. \( \alpha - \frac{1}{2} \) (which is \( \alpha - 0.5 \))
6. \( \alpha + \frac{1}{2} \) (which is \( \alpha + 0.5 \))
7. \( \alpha + 5 \)

Now, we can order them:
- \( \alpha - 4 \)
- \( \alpha - \frac{7}{2} \)
- \( \alpha - \frac{5}{2} \)
- \( \alpha - 2 \)
- \( \alpha - \frac{1}{2} \)
- \( \alpha + \frac{1}{2} \)
- \( \alpha + 5 \)

### Step 3: Identify the Number of Values
There are 7 values in total.

### Step 4: Find the Median
Since there are 7 values (an odd number), the median is the middle value. The middle value can be found using the formula:
\[ \text{Median} = \text{Value at position } \left(\frac{n + 1}{2}\right) \]
where \( n \) is the number of values.

Here, \( n = 7 \):
\[ \text{Median} = \text{Value at position } \left(\frac{7 + 1}{2}\right) = \text{Value at position } 4 \]

### Step 5: Determine the 4th Value
From our ordered list:
1. \( \alpha - 4 \)
2. \( \alpha - \frac{7}{2} \)
3. \( \alpha - \frac{5}{2} \)
4. \( \alpha - 2 \) (This is the 4th value)
5. \( \alpha - \frac{1}{2} \)
6. \( \alpha + \frac{1}{2} \)
7. \( \alpha + 5 \)

### Step 6: Conclusion
Thus, the median of the given discrete values is:
\[ \text{Median} = \alpha - 2 \]