In a series of 2n observations half of them equals a and remaining half equals -a . If standard devi

English

Gujarati

Hindi

13.Statistics

normal

In a series of $2n$ observations half of them equals $a$ and remaining half equals $-a$. If the standard deviation of observations is $2$ then $\left| a \right|$ equals

A

$2$

B

$\sqrt 2$

C

$\frac{1}{n}$

D

$\frac{{\sqrt 2 }}{n}$

Solution

mean $\bar{x}=0$

$\mathrm{x}_{1}=\mathrm{x}_{2}=\ldots=\mathrm{x}_{\mathrm{n}}=\mathrm{a}$

$x_{\mathrm{n}+1}=\mathrm{x}_{\mathrm{n}+2}=\ldots=\mathrm{x}_{2 \mathrm{n}}=-\mathrm{a}$

$2=\mathrm{S.D}=\sqrt{\frac{2 \mathrm{x}^{2}}{2 \mathrm{n}}-\overline{\mathrm{x}}^{2}}=\sqrt{\frac{2 \mathrm{na}^{2}}{2 \mathrm{n}}}=|\mathrm{a}|$

Standard 11

Mathematics

Share

Similar Questions

Let in a series of $2 n$ observations, half of them are equal to $a$ and remaining half are equal to $-a.$ Also by adding a constant $b$ in each of these observations, the mean and standard deviation of new set become $5$ and $20 ,$ respectively. Then the value of $a^{2}+b^{2}$ is equal to ……. .

hard

(JEE MAIN-2021)

For two data sets, each of size $5$, the variances are given to be $4$ and $5$ and the corresponding means are given to be $2$ and $4$, respectively. The variance of the combined data set is

hard

(AIEEE-2010)

The frequency distribution:

$\begin{array}{|l|l|l|l|l|l|l|} \hline X & 2 & 3 & 4 & 5 & 6 & 7 \\ f & 4 & 9 & 16 & 14 & 11 & 6 \\ \hline \end{array}$

Find the standard deviation.

medium

What is the standard deviation of the following series

class $0-10$ $10-20$ $20-30$ $30-40$

Freq $1$ $3$ $4$ $2$

medium

The mean and standard deviation of some data for the time taken to complete . a test are calculated with the following results:

Number of observations $=25,$ mean $=18.2$ seconds, standard deviation $=3.25 s$

Further, another set of 15 observations $x_{1}, x_{2}, \ldots, x_{15},$ also in seconds, is now available and we have $\sum_{i=1}^{15} x_{i}=279$ and $\sum_{i=1}^{15} x_{i}^{2}=5524 .$ Calculate the standard deviation based on all 40 observations.

hard