Outcome of each of 30 items was observed; 10 items gave an outcome frac{1}{2}

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13.Statistics

hard

The outcome of each of $30$ items was observed; $10$ items gave an outcome $\frac{1}{2} - d$ each, $10$ items gave outcome $\frac {1}{2}$ each and the remaining $10$ items gave outcome $\frac{1}{2} + d$ each. If the variance of this outcome data is $\frac {4}{3}$ then $\left| d \right|$ equals

$\frac {2}{3}$

$2$

$\frac {\sqrt 5}{2}$

$\sqrt 2$

(JEE MAIN-2019)

Solution

Variance remains some if same number is subracted from each observation. (subtract $10$ from each observation)

$\therefore \frac{{1{{\left( { – d} \right)}^2} + 10{{\left( 0 \right)}^2} + 10{{\left( d \right)}^2}}}{{30}} – {\left( {\frac{{10\left( { – d} \right) + 10\left( 0 \right) + 10\left( d \right)}}{{30}}} \right)^2} = \frac{4}{3}$

$\frac{{20{d^2}}}{{30}} = \frac{4}{3}$

$ \Rightarrow {d^2} = 2$

$\left( d \right) = \sqrt 2 $

Standard 11

Mathematics

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

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

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hard

In a series of $2n$ observations, half of them equal to $a$ and remaining half equal to $-a$. If the standard deviation of the observations is $2$, then $|a|$ equals

hard

(AIEEE-2004)

Two sets each of 20 observations, have the same standard derivation 5. The first set has a mean 17 and the second a mean 22. Determine the standard deviation of the set obtained by combining the given two sets.

medium

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medium

(JEE MAIN-2024)

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

The outcome of each of $30$ items was observed; $10$ items gave an outcome $\frac{1}{2} - d$ each, $10$ items gave outcome $\frac {1}{2}$ each and the remaining $10$ items gave outcome $\frac{1}{2} + d$ each. If the variance of this outcome data is $\frac {4}{3}$ then $\left| d \right|$ equals

Solution

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

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

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