The mean deviation about the mean of the set | vedclass

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(D) The mean $\bar{x}$ of the first $n$ natural numbers is given by $\bar{x} = \frac{1+2+\ldots+n}{n} = \frac{n(n+1)}{2n} = \frac{n+1}{2}$.Given that

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$\frac{n^2-1}{4n}$

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(D) The mean $\bar{x}$ of the first $n$ natural numbers is given by $\bar{x} = \frac{1+2+\ldots+n}{n} = \frac{n(n+1)}{2n} = \frac{n+1}{2}$.Given that $n$ is an even number,the mean deviation about the mean is calculated as:$M.D.(\bar{x}) = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}| = \frac{1}{n} \sum_{i=1}^{n} |i - \frac{n+1}{2}|$.Since $n$ is even,the sum splits into two equal parts of absolute differences:$M.D.(\bar{x}) = \frac{1}{n} \left[ \sum_{i=1}^{n/2} (\frac{n+1}{2} - i) + \sum_{i=n/2+1}^{n} (i - \frac{n+1}{2}) \right]$.Evaluating this sum,we get $M.D.(\bar{x}) = \frac{n^2-1}{4n}$.

$x_i$	$5, 15, 25, 35, 45$
$f_i$	$8, 48, 56, 32, 16$

Income per day	Number of persons
$0-100$	$4$
$100-200$	$8$
$200-300$	$9$
$300-400$	$10$
$400-500$	$7$
$500-600$	$5$
$600-700$	$4$
$700-800$	$3$

The mean deviation about the mean of the set of first $n$ natural numbers,when $n$ is an even number,is equal to:

Similar Questions

If the mean deviation about the median of the numbers $a, 2a, 3a, \dots, 50a$ is $50$,then what is the value of $|a|$?

What is the mean deviation of the numbers $1, 2, 3, 4, 5$?

The mean deviation from the median is

The mean deviation of the following data about its mean is
$x_i$ $5, 15, 25, 35, 45$
$f_i$ $8, 48, 56, 32, 16$

Find the mean deviation about the mean for the following data:
Income per day Number of persons
$0-100$ $4$
$100-200$ $8$
$200-300$ $9$
$300-400$ $10$
$400-500$ $7$
$500-600$ $5$
$600-700$ $4$
$700-800$ $3$

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