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Question: If a variable takes the discrete values $$\alpha - 4, \alpha - \frac{7}{2}, \alpha - 3, \alpha - \fr...

If a variable takes the discrete values α−4,α−72,α−3,α−52,α−2,α−12,α+12,α+5(α>0),\alpha - 4, \alpha - \frac{7}{2}, \alpha - 3, \alpha - \frac{5}{2}, \alpha - 2, \alpha - \frac{1}{2}, \alpha + \frac{1}{2}, \alpha + 5 \quad (\alpha > 0), then its median is

A

α−94\alpha - \frac{9}{4}

B

α−12\alpha - \frac{1}{2}

C

α−2\alpha - 2

D

α+14\alpha + \frac{1}{4}

Answer

α−94\alpha - \frac{9}{4}

Explanation

Solution

To find the median of the given discrete values, we first need to arrange them in ascending order. The given values are: α−4,α−72,α−3,α−52,α−2,α−12,α+12,α+5\alpha - 4, \alpha - \frac{7}{2}, \alpha - 3, \alpha - \frac{5}{2}, \alpha - 2, \alpha - \frac{1}{2}, \alpha + \frac{1}{2}, \alpha + 5 There are 8 values. We can determine the order by comparing the constant terms: −4,−72,−3,−52,−2,−12,12,5-4, -\frac{7}{2}, -3, -\frac{5}{2}, -2, -\frac{1}{2}, \frac{1}{2}, 5 Converting these to decimals for easier comparison: −4.0,−3.5,−3.0,−2.5,−2.0,−0.5,0.5,5.0-4.0, -3.5, -3.0, -2.5, -2.0, -0.5, 0.5, 5.0 Arranging these constants in ascending order gives: −4.0,−3.5,−3.0,−2.5,−2.0,−0.5,0.5,5.0-4.0, -3.5, -3.0, -2.5, -2.0, -0.5, 0.5, 5.0 So, the values arranged in ascending order are: α−4,α−72,α−3,α−52,α−2,α−12,α+12,α+5\alpha - 4, \alpha - \frac{7}{2}, \alpha - 3, \alpha - \frac{5}{2}, \alpha - 2, \alpha - \frac{1}{2}, \alpha + \frac{1}{2}, \alpha + 5 The number of values is n=8n=8, which is an even number. For an even number of data points, the median is the average of the n2\frac{n}{2}-th and (n2+1)(\frac{n}{2} + 1)-th values in the sorted list. Here, n=8n=8, so n2=82=4\frac{n}{2} = \frac{8}{2} = 4. The median is the average of the 4th and 5th values.

From the sorted list: The 4th value is α−52\alpha - \frac{5}{2}. The 5th value is α−2\alpha - 2.

The median is the average of these two values: Median=(α−52)+(α−2)2\text{Median} = \frac{(\alpha - \frac{5}{2}) + (\alpha - 2)}{2} Median=α−2.5+α−22\text{Median} = \frac{\alpha - 2.5 + \alpha - 2}{2} Median=2α−4.52\text{Median} = \frac{2\alpha - 4.5}{2} Median=2α−922\text{Median} = \frac{2\alpha - \frac{9}{2}}{2} Median=2α2−922\text{Median} = \frac{2\alpha}{2} - \frac{\frac{9}{2}}{2} Median=α−94\text{Median} = \alpha - \frac{9}{4}

Comparing this result with the given options, the calculated median α−94\alpha - \frac{9}{4} matches option (a).