If $\sum \limits_{i=1}^{n}\left(x_{i}-a\right)=n$ and $\sum \limits_{i=1}^{n}\left(x_{i}-a\right)^{2}=n a,(n, a>1)$ then the standard deviation of $n$ observations $x _{1}, x _{2}, \ldots, x _{ n }$ is

  • [JEE MAIN 2020]
  • A

    $n \sqrt{ a -1}$

  • B

    $\sqrt{a-1}$

  • C

    $a-1$

  • D

    $\sqrt{n(a-1)}$

Similar Questions

The variance of $20$ observations is $5 .$ If each observation is multiplied by $2,$ find the new variance of the resulting observations.

If the variance of the first $n$ natural numbers is $10$ and the variance of the first m even natural numbers is $16$, then $m + n$ is equal to

  • [JEE MAIN 2020]

The $S.D$ of $15$ items is $6$ and if each item is decreased or increased by $1$, then standard deviation will be

Find the variance of the following data: $6,8,10,12,14,16,18,20,22,24$

The mean and standard deviation of marks obtained by $50$ students of a class in three subjects, Mathematics, Physics and Chemistry are given below:

Subject  Mathematics Physics Chemistty
Mean $42$ $32$ $40.9$
Standard deviation $12$ $15$ $20$

Which of the three subjects shows the highest variability in marks and which shows the lowest?