Variance and standard deviation

The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If $\sigma$ is the standard deviation of the data after omitting the two wrong observations from the data, then $38\sigma^2$ is equal to ______.

The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is ______.

Let the mean and the variance of 20 observations $x_1,x_2,\dots,x_{20}$ be 15 and 9, respectively. For α ∈ R, if the mean of $(x_1+\alpha)^2,(x_2+\alpha)^2,\dots,(x_{20}+\alpha)^2$ is 178, then the square of the maximum value of α is equal to _______.

The sum of 100 observations and the sum of their squares are 400 and 2475, respectively. Later on, three observations, 3, 4 and 5, were found to be incorrect. If the incorrect observations are omitted, then the variance of the remaining observations is :

If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true ?

If the mean deviation of the numbers 1, 1 + d, ..., 1 + 100d from their mean is 255, then a value of d is :

The mean of 5 observations is 5 and their variance is 124. If three of the observations are 1, 2 and 6 ; then the mean deviation from the mean of the data is :

The variance of first 50 even natural numbers is :

Let $\bar x$, M and $\sigma^2$ be respectively the mean, mode and variance of n observations $x_1,x_2,\dots,x_n$ and $d_i=-x_i-a$, $i=1,2,\dots,n$, where a is any number.

Statement I : Variance of $d_1,d_2,\dots,d_n$ is $\sigma^2$.

Statement II : Mean and mode of $d_1,d_2,\dots,d_n$ are $-\bar x-a$ and $-M-a$, respectively