Mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted a an

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13.Statistics

hard

The mean and variance of $7$ observations are $8$ and $16$ respectively. If one observation $14$ is omitted a and $b$ are respectively mean and variance of remaining $6$ observation, then $a+3 b-5$ is equal to $..........$.

$36$

$35$

$34$

$37$

(JEE MAIN-2023)

Solution

$\frac{x_1+x_2+\ldots .+x_7}{7}=8$

$\frac{x_1+x_2+x_3 \ldots .+x_6+14}{7}=8$

$\Rightarrow x_1+x_2+\ldots .+x_6=42$

$\therefore \frac{x_1+x_2 \ldots .+x_6}{6}=\frac{42}{6}=7=a$

$\frac{\sum x_i^2}{7}-8^2=16$

$\Rightarrow x^2=560$

$\Rightarrow x_1^2+x_2^2+\ldots+x_6^2=364$

$b=\frac{x_1^2+x_2^2+\ldots . .+x_6^2}{6}-7^2$

$=\frac{364}{6}-49$

$b=\frac{70}{6}$

$a+3 b-5=7+3 \times \frac{70}{6}-5$

$=37$

Standard 11

Mathematics

The diameters of circles (in mm) drawn in a design are given below:

Diameters $33-36$ $37-40$ $41-44$ $45-48$ $49-52$

No. of circles $15$ $17$ $21$ $22$ $25$

Calculate the standard deviation and mean diameter of the circles.

[ Hint : First make the data continuous by making the classes as $32.5-36.5,36.5-40.5,$ $40.5-44.5,44.5-48.5,48.5-52.5 $ and then proceed.]

medium

If the variance of the frequency distribution is $160$ , then the value of $\mathrm{c} \in \mathrm{N}$ is

$X$ $c$ $2c$ $3c$ $4c$ $5c$ $6c$

$f$ $2$ $1$ $1$ $1$ $1$ $1$

medium

(JEE MAIN-2024)

If the mean and variance of eight numbers $3,7,9,12,13,20, x$ and $y$ be $10$ and $25$ respectively, then $\mathrm{x} \cdot \mathrm{y}$ is equal to

hard

(JEE MAIN-2020)

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)

Find the mean and variance for the following frequency distribution.

Classes $0-30$ $30-60$ $60-90$ $90-120$ $120-150$ $50-180$ $180-210$

$f_i$ $2$ $3$ $5$ $10$ $3$ $5$ $2$

hard

Diameters	$33-36$	$37-40$	$41-44$	$45-48$	$49-52$
No. of circles	$15$	$17$	$21$	$22$	$25$

Classes	$0-30$	$30-60$	$60-90$	$90-120$	$120-150$	$50-180$	$180-210$
$f_i$	$2$	$3$	$5$	$10$	$3$	$5$	$2$

The mean and variance of $7$ observations are $8$ and $16$ respectively. If one observation $14$ is omitted a and $b$ are respectively mean and variance of remaining $6$ observation, then $a+3 b-5$ is equal to $..........$.

Solution

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