The frequency distribution:
$\begin{array}{|l|l|l|l|l|l|l|} \hline X & A & 2 A & 3 A & 4 A & 5 A & 6 A \\ \hline f & 2 & 1 & 1 & 1 & 1 & 1 \\ \hline \end{array}$
where $A$ is a positive integer, has a variance of $160 .$ Determine the value of $A$.
The frequency distribution:
$\begin{array}{|l|l|l|l|l|l|l|} \hline X & A & 2 A & 3 A & 4 A & 5 A & 6 A \\ \hline f & 2 & 1 & 1 & 1 & 1 & 1 \\ \hline \end{array}$
where $A$ is a positive integer, has a variance of $160 .$ Determine the value of $A$.
$\begin{array}{|c|c|c|c|} \hline x & f_{i} & f_{1} x_{i} & f x_{i}^{2} \\ \hline A & 2 & 2 A & 2 A^{2} \\ \hline 2 A & 1 & 2 A & 4 A^{2} \\ \hline 3 A & 1 & 3 A & 9 A^{2} \\ \hline 4 A & 1 & 4 A & 16 A^{2} \\ \hline 5 A & 1 & 5 A & 25 A^{2} \\ \hline 6 A & 1 & 6 A & 36 A^{2} \\ \hline \text { Total } & n=7 & \Sigma f_{i}=22 A & \Sigma f_{i}^{2}=92 A^{2} \\ \hline \end{array}$
$\therefore \quad \sigma^{2}=\frac{\Sigma f_{t} x_{1}^{2}}{n}-\left(\frac{\Sigma f_{1} x_{1}}{n}\right)^{2}$
$\Rightarrow \quad 160=\frac{92 A^{2}}{7}-\left(\frac{22 A}{7}\right)^{2} \Rightarrow 160=\frac{92 A^{2}}{7}-\frac{484 A^{2}}{49}$
$\Rightarrow \quad 160=(644-484) \frac{A^{2}}{49} \Rightarrow 160=\frac{160 A^{2}}{49}$
$\Rightarrow \quad A^{2}=49 \quad \therefore \quad A=7$
Similar Questions
The mean and standard deviation of $20$ observations were calculated as $10$ and $2.5$ respectively. It was found that by mistake one data value was taken as $25$ instead of $35 .$ If $\alpha$ and $\sqrt{\beta}$ are the mean and standard deviation respectively for correct data, then $(\alpha, \beta)$ is :
The mean and standard deviation of $20$ observations were calculated as $10$ and $2.5$ respectively. It was found that by mistake one data value was taken as $25$ instead of $35 .$ If $\alpha$ and $\sqrt{\beta}$ are the mean and standard deviation respectively for correct data, then $(\alpha, \beta)$ is :
- [JEE MAIN 2021]
For the frequency distribution :
Variate $( x )$
$x _{1}$
$x _{1}$
$x _{3} \ldots \ldots x _{15}$
Frequency $(f)$
$f _{1}$
$f _{1}$
$f _{3} \ldots f _{15}$
where $0< x _{1}< x _{2}< x _{3}<\ldots .< x _{15}=10$ and
$\sum \limits_{i=1}^{15} f_{i}>0,$ the standard deviation cannot be
For the frequency distribution :
| Variate $( x )$ | $x _{1}$ | $x _{1}$ | $x _{3} \ldots \ldots x _{15}$ |
| Frequency $(f)$ | $f _{1}$ | $f _{1}$ | $f _{3} \ldots f _{15}$ |
where $0< x _{1}< x _{2}< x _{3}<\ldots .< x _{15}=10$ and
$\sum \limits_{i=1}^{15} f_{i}>0,$ the standard deviation cannot be
- [JEE MAIN 2020]
The variance of first $50$ even natural numbers is
The variance of first $50$ even natural numbers is
- [JEE MAIN 2014]
Let $\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{N}$ and $\mathrm{a}<\mathrm{b}<\mathrm{c}$. Let the mean, the mean deviation about the mean and the variance of the $5$ observations $9$,$25$, $a$, $b$, $c$ be $18$,$4$ and $\frac{136}{5}$, respectively. Then $2 \mathrm{a}+\mathrm{b}-\mathrm{c}$ is equal to ..............
Let $\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{N}$ and $\mathrm{a}<\mathrm{b}<\mathrm{c}$. Let the mean, the mean deviation about the mean and the variance of the $5$ observations $9$,$25$, $a$, $b$, $c$ be $18$,$4$ and $\frac{136}{5}$, respectively. Then $2 \mathrm{a}+\mathrm{b}-\mathrm{c}$ is equal to ..............
- [JEE MAIN 2024]
Calculate the mean, variance and standard deviation for the following distribution:
Class
$30-40$
$40-50$
$50-60$
$60-70$
$70-80$
$80-90$
$90-100$
$f_i$
$3$
$7$
$12$
$15$
$8$
$3$
$2$
Calculate the mean, variance and standard deviation for the following distribution:
| Class | $30-40$ | $40-50$ | $50-60$ | $60-70$ | $70-80$ | $80-90$ | $90-100$ |
| $f_i$ | $3$ | $7$ | $12$ | $15$ | $8$ | $3$ | $2$ |