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Consider three observations $a, b$ and $c$ such that $b = a + c .$ If the standard deviation of $a +2$ $b +2, c +2$ is $d ,$ then which of the following is true ?
$b^{2}=3\left(a^{2}+c^{2}\right)+9 d^{2}$
$b^{2}=a^{2}+c^{2}+3 d^{2}$
$b^{2}=3\left(a^{2}+c^{2}+d^{2}\right)$
$b ^{2}=3\left( a ^{2}+ c ^{2}\right)-9 d ^{2}$
Solution
For $a, b, c$
mean $=\frac{a+b+c}{3}(=\bar{x})$
$b = a + c$
$\Rightarrow \quad \bar{x}=\frac{2 b}{3}$ $…..(1)$
S.D. $(a+2, b+2, c+2)=$ S.D. $(a, b, c)=d$
$\Rightarrow \quad d ^{2}=\frac{ a ^{2}+ b ^{2}+ c ^{2}}{3}-(\overline{ x })^{2}$
$\Rightarrow \quad d^{2}=\frac{a^{2}+b^{2}+c^{2}}{3}-\frac{4 b^{2}}{9}$
$\Rightarrow 9 d^{2}=3\left(a^{2}+b^{2}+c^{2}\right)-4 b^{2}$
$\Rightarrow \quad b^{2}=3\left(a^{2}+c^{2}\right)-9 d^{2}$
Similar Questions
Calculate mean, variance and standard deviation for the following distribution.
| Classes | $30-40$ | $40-50$ | $50-60$ | $60-70$ | $70-80$ | $80-90$ | $90-100$ |
| ${f_i}$ | $3$ | $7$ | $12$ | $15$ | $8$ | $3$ | $2$ |
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.]
