If the standard deviation of the first $n$ natural numbers is $2$,then the value of $n$ is

For $10$ observations $x_1, x_2, ..., x_{10}$,if $\sum_{i=1}^{10} (x_i + 2)^2 = 180$ and $\sum_{i=1}^{10} (x_i - 1)^2 = 90$,then their standard deviation is:

Let $X = \{x \in N : 1 \leq x \leq 17\}$ and $Y = \{ax + b : x \in X \text{ and } a, b \in R, a > 0\}$. If the mean and variance of the elements of $Y$ are $17$ and $216$ respectively,then $a + b$ is equal to:

$\text{Assertion (A):}$ Variance of $4x_1, 4x_2, \ldots, 4x_n$ is $16$ times the variance of $x_1, x_2, \ldots, x_n$. $\text{Reason (R):}$ If $y = ax + b$,then variance of $y$ is $a(\text{variance of } x) + b$. The correct option among the following is

In an experiment with $15$ observations for $x$,the following results were available: $\sum x^2 = 2830$ and $\sum x = 170$. One observation $20$ was found to be wrong and was replaced by the correct value $30$. Then the corrected variance is:

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

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