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Question
Calculate the mean deviation about the mean of the set of first n natural numbers when n is an even number.
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Solution
First n natural numbers are 1, 2, 3, 4, 5, 6, …, n (even)
∴ Mean `barx = (1 + 2 + 3 + 4 + ... + n)/n`
= `(n(n + 1))/(2n)`
= `(n + 1)/2`
∴ M.D. = `1/n[|1 - (n + 1)/2| + |2 - (n + 1)/2| + |3 - (n + 1)/2| + ... + |(n - 2)/2 - (n + 1)/2| + |n/2 - (n + 1)/2| + |(n + 2)/2 - (n + 1)/2| ... + |n - (n + 1)/2|]`
= `1/n[|(1 - n)/2| + |(3 - n)/2| + |(5 - n)/2| + ... + |(-3)/2| + |- 1/2| + |1/2| + ... + |(n - 1)/2|]`
= `1/n[1/2 + 3/2 + ... + (n - 1)/2] (n/2)` terms
= `1/n (n/2)^2`
= `1/n * n^2/4`
= `n/4` ....[∵ Sum of first odd n natural numbers = n2]
Hence, the required M.D. = `n/4`.
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