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The eight numbers 3, 5, 7, 2k, 12, 16, 21, 24 are arranged in ascending order. If the mean deviation about the median is 6, find the value of the median.

1. 11.5
2. 10.5
3. 12
4. 11

Correct answer: 11

Solution

For ascending order with 3.5<=k<=6, the sorted list is 3, 5, 7, 2k, 12, 16, 21, 24. Median = (2k+12)/2 = k+6. Mean deviation = (1/8)*sum|xi - M| = 6, so sum = 48. The deviations are: |3-(k+6)|=k+3, |5-(k+6)|=k+1, |7-(k+6)|=k-1, |2k-(k+6)|=6-k (for k<=6), |12-(k+6)|=6-k, |16-(k+6)|=10-k, |21-(k+6)|=15-k, |24-(k+6)|=18-k. Sum = (k+3)+(k+1)+(k-1)+(6-k)+(6-k)+(10-k)+(15-k)+(18-k) = 58-2k = 48 => k = 5. Median = k+6 = 11.

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