<p>For 50 observations \(a, 2a, 3a, \ldots, 50a\), the mean deviation about the median is minimized and \(\dfrac{1}{n}\sum|x_i - A|\) is minimized when \(A\) is the median. If \(625a = 2500\), find \(a\).</p>
Step-by-Step Solution
Key Concept: The mean deviation about any point A is minimized when A equals the median of the dataset. Since we have 50 observations in arithmetic progression (a, 2a, 3a, ..., 50a), the median is the average of the 25th and 26th terms, which equals 25.5a.
<p><strong>Step 1:</strong> Identify the data set. We have 50 observations: a, 2a, 3a, ..., 50a in arithmetic progression.</p><p><strong>Step 2:</strong> Find the median. For 50 observations (even number), the median is the average of the 25th and 26th terms.</p><p>25th term = 25a, 26th term = 26a</p><p>Median = (25a + 26a)/2 = 51a/2 = 25.5a</p><p><strong>Step 3:</strong> Apply the minimization property. The mean deviation is minimized when A equals the median, so A = 25.5a.</p><p><strong>Step 4:</strong> Use the given condition. We're given that 625a = 2500.</p><p><strong>Step 5:</strong> Solve for a.</p><p>625a = 2500</p><p>a = 2500/625 = 4</p><p>∴ Answer: a = 4 (Option B)</p>
Correct Answer: B