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 ?
$b^{2}=3\left(a^{2}+c^{2}\right)+9 d^{2}$
$b^{2}=a^{2}+c^{2}+3 d^{2}$
$b^{2}=3\left(a^{2}+c^{2}+d^{2}\right)$
$b ^{2}=3\left( a ^{2}+ c ^{2}\right)-9 d ^{2}$
There are 60 students in a class. The following is the frequency distribution of the marks obtained by the students in a test:
$\begin{array}{|l|l|l|l|l|l|l|} \hline \text { Marks } & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Frequency } & x-2 & x & x^{2} & (x+1)^{2} & 2 x & x+1 \\ \hline \end{array}$
where $x$ is a positive integer. Determine the mean and standard deviation of the marks.
The mean and standard deviation of $15$ observations are found to be $8$ and $3$ respectively. On rechecking it was found that, in the observations, $20$ was misread as $5$ . Then, the correct variance is equal to......
The mean and variance of $5$ observations are $5$ and $8$ respectively. If $3$ observations are $1,3,5$, then the sum of cubes of the remaining two observations is
The mean and variance of eight observations are $9$ and $9.25,$ respectively. If six of the observations are $6,7,10,12,12$ and $13,$ find the remaining two observations.
Mean and standard deviation of 100 observations were found to be 40 and 10 , respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, find the correct standard deviation.