If Σ limits_{i=1}^{n}left(x_{i}-aright)=n and Σ limits_{i=1}^{n}left(x_{i}-aright)^{2}=n a,(n, a&g

English

Gujarati

Hindi

13.Statistics

medium

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

A

$n \sqrt{ a -1}$

B

$\sqrt{a-1}$

C

$a-1$

D

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

(JEE MAIN-2020)

Solution

$S.D =\sqrt{\frac{\sum_{i=1}^{n}\left( x _{ i }- a \right)}{ n }-\left(\frac{\sum_{i=1}^{ n }\left( x _{ i }- a \right)}{ n }\right)^{2}}$

$=\sqrt{\frac{ na }{ n }-\left(\frac{ n }{ n }\right)^{2}}$

$\left\{\right.$ Given $\left.\sum_{i=1}^{n}\left(x_{i}-a\right)=n \sum_{i=1}^{n}\left(x_{i}-a\right)^{2}=n a\right\}$

$=\sqrt{a-1}$

Standard 11

Mathematics

Share

Similar Questions

The mean and standard deviation of $20$ observations are found to be $10$ and $2$, respectively. On respectively, it was found that an observation by mistake was taken $8$ instead of $12$ . The correct standard deviation is

medium

(JEE MAIN-2024)

The mean of $5$ observations is $4.4$ and their variance is $8.24$. If three observations are $1, 2$ and $6$, the other two observations are

medium

Let $9 < x_1 < x_2 < \ldots < x_7$ be in an $A.P.$ with common difference $d$. If the standard deviation of $x_1, x_2 \ldots$, $x _7$ is $4$ and the mean is $\overline{ x }$, then $\overline{ x }+ x _6$ is equal to:

hard

(JEE MAIN-2023)

Determine the mean and standard deviation for the following distribution:

$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|} \hline \text { Marks } & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ \hline \text { Frequency } & 1 & 6 & 6 & 8 & 8 & 2 & 2 & 3 & 0 & 2 & 1 & 0 & 0 & 0 & 1 \\ \hline \end{array}$

medium

If $\sum\limits_{i = 1}^{18} {({x_i} – 8) = 9} $ and $\sum\limits_{i = 1}^{18} {({x_i} – 8)^2 = 45} $ then the standard deviation of $x_1, x_2, …… x_{18}$ is :-

normal