The mean deviation about the mean of the set | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

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

Vedclass · Answer

(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 $M.D.(\bar{x}) = \frac{n^2-1}{4n}$.

Marks obtained	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$	$60-70$	$70-80$
No of students	$2$	$3$	$8$	$14$	$8$	$3$	$2$

Age	$15$	$16$	$17$	$18$	$19$	$20$
No. of students	$5$	$8$	$5$	$12$	$X$	$Y$

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 $x, 2x, 3x, 4x, 5x, 6x, 7x, 8x, 9x, 10x$ is $30$,then $|x|$ equals:

The mean deviation from the mean for the data $ 3, 10, 10, 4, 7, 10, 5 $ is:

If $\bar{x}$ is the mean of $n$ observations $x_1, x_2, \ldots, x_n$,then the mean of the absolute deviations of these observations from $\bar{x}$ is

Find the mean deviation about the mean for the following data.
Marks obtained $10-20$ $20-30$ $30-40$ $40-50$ $50-60$ $60-70$ $70-80$
No of students $2$ $3$ $8$ $14$ $8$ $3$ $2$

The frequency distribution of the age of students in a class of $40$ students is given below.

Age $15$ $16$ $17$ $18$ $19$ $20$

No. of students $5$ $8$ $5$ $12$ $X$ $Y$

If the mean deviation about the median is $1.25$,then $4X + 5Y$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module