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Question:

If the mean deviation about the median of the numbers a, 2a, ..., 50a is 50, then |a| is equal?

2

3

4

5

Solution:

Median is the mean of 25th and 26th observation. So, median = (25a + 26a)/2 = 25.5a Mean deviation = 1/50 Σ50i=1 |ai - 25.5a| = |a| /50 Σ50i=1 |i - 25.5| = 2|a|/50[(25.5 - 1) + (25.5 - 2) + … + (25.5 - 25) + (26 - 25.5) + … + (50 - 25.5)] = |a|/25[24.5 + 23.5 + … + 0.5 + 0.5 + … + 24.5] = |a|/25 × 2[24.5 + 23.5 + … + 0.5] = |a|/25[25/2(24.5 + 0.5)] = 12.5 |a| Given, mean deviation = 50 ⇒ 12.5 |a| = 50 ⇒ |a| = 4