Let x ₁, x ₂, ldots ldots x _{10} be ten observations such that Σ

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13.Statistics

hard

Let $x _1, x _2, \ldots \ldots x _{10}$ be ten observations such that $\sum_{i=1}^{10}\left(x_i-2\right)=30, \sum_{i=1}^{10}\left(x_i-\beta\right)^2=98, \beta>2$ and their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^2$ are respectively the mean and the variance of $2\left( x _1-1\right)+4 \beta, 2\left( x _2-1\right)+$ $4 \beta, \ldots . ., 2\left(x_{10}-1\right)+4 \beta$, then $\frac{\beta \mu}{\sigma^2}$ is equal to :

A$100$

B$110$

C$120$

D$90$

(JEE MAIN-2025)

Solution

$\frac{4}{5}=\frac{\Sigma x_{i}^2}{10}-\left(\frac{\Sigma x_{i}}{10}\right)^2$
$\frac{4}{5}=\frac{\Sigma x_{i}^2}{10}-25$
$\Rightarrow \Sigma x_{i}^2=258$
$\text { Now } \sum_{i=1}^{10}\left(x_{i}-\beta\right)^2=98$
$\sum_{i=1}^{10}\left(x_{i}^2-2 \beta \cdot x_{i}+\beta^2\right)=98$
$258-2 \beta(50)+10 \beta^2=98$
$(\beta-8)(\beta-2)=0$
$\beta=\text { or } \beta=2 \quad(\text { as } \beta>2)$
$\therefore \beta=8$
Now,
$=2\left(x_1-1\right)+4 \beta, 2\left(x_2-1\right)+4 \beta, \ldots .2\left(x_{10}-1\right)+4 \beta$
$=2 x_1+30,2 x_2+30, \ldots . .2 x_{10}+30$
$\mu=2(5)+30=40$
$\sigma^2=2^2\left(\frac{4}{5}\right)=\frac{16}{5}$
$\because \frac{B \mu}{2}=\frac{8 \times 40}{}=100$

Standard 11

Mathematics

For $(2n+1)$ observations ${x_1},\, – {x_1}$, ${x_2},\, – {x_2},\,…..{x_n},\, – {x_n}$ and $0$ where $x$’s are all distinct. Let $S.D.$ and $M.D.$ denote the standard deviation and median respectively. Then which of the following is always true

easy

Calculate the mean, variance and standard deviation for the following distribution:

Class $30-40$ $40-50$ $50-60$ $60-70$ $70-80$ $80-90$ $90-100$

$f_i$ $3$ $7$ $12$ $15$ $8$ $3$ $2$

hard

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

medium

The mean and variance of $7$ observations are $8$ and $16,$ respectively. If five observations are $2, 4, 10,12,14,$ then the absolute difference of the remaining two observations is

hard

(JEE MAIN-2020)

The mean and standard deviation of a group of $100$ observations were found to be $20$ and $3,$ respectively. Later on it was found that three observations were incorrect, which were recorded as $21,21$ and $18 .$ Find the mean and standard deviation if the incorrect observations are omitted.

medium

Class	$30-40$	$40-50$	$50-60$	$60-70$	$70-80$	$80-90$	$90-100$
$f_i$	$3$	$7$	$12$	$15$	$8$	$3$	$2$

Solution

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Calculate the mean, variance and standard deviation for the following distribution:

Class $30-40$ $40-50$ $50-60$ $60-70$ $70-80$ $80-90$ $90-100$

$f_i$ $3$ $7$ $12$ $15$ $8$ $3$ $2$