<p>If a variable takes the discrete values \(\alpha - 4\), \(\alpha - \dfrac{7}{2}\), \(\alpha - \dfrac{5}{2}\), \(\alpha - 3\), \(\alpha - 2\), \(\alpha + \dfrac{1}{2}\), \(\alpha - \dfrac{1}{2}\), \(\alpha + 5\) \((\alpha > 0)\), then the median is</p>
<p>(1) \(\alpha - \dfrac{5}{4}\)</p>
<p>(2) \(\alpha - \dfrac{1}{2}\)</p>
<p>(3) \(\alpha - 2\)</p>
<p>(4) \(\alpha + \dfrac{5}{4}\)</p>
Step-by-Step Solution
Key Concept: To find the median of discrete data, first arrange all values in ascending order, then for an even number of observations (n=8), the median is the average of the (n/2)th and (n/2+1)th terms. Since all values are expressed in terms of α, the relative ordering is independent of α's value.
<p><strong>Step 1:</strong> List all 8 values and arrange in ascending order (subtracting constants from α preserves order):</p><p>α - 4, α - 7/2, α - 5/2, α - 3, α - 2, α - 1/2, α + 1/2, α + 5</p><p>Rewrite as: α - 4, α - 3.5, α - 2.5, α - 3, α - 2, α - 0.5, α + 0.5, α + 5</p><p><strong>Step 2:</strong> Order the constant terms: -4, -3.5, -3, -2.5, -2, -0.5, +0.5, +5</p><p>Ascending order: α - 4, α - 3.5, α - 3, α - 2.5, α - 2, α - 0.5, α + 0.5, α + 5</p><p><strong>Step 3:</strong> For n = 8 observations, median = (4th value + 5th value)/2</p><p>Median = [(α - 2.5) + (α - 2)]/2 = (2α - 4.5)/2 = α - 2.25 = α - 9/4</p><p>∴ Answer: A</p>
Correct Answer: A