standard deviation can not be $6$. For the frequency distribution :
Variate x : $x_1 , x_2  , x_3 ,  x_ 4 ........,  x_{15}$
Frequency f : $f_1 ,  f_2  , f_3  , f_ 4 ........, f_{15}$   where $0 < x_1 < x_2  < x_3 <  x_ 4 ........<  x_{15} = 10$ and $\sum f_i >0$.
Then the standard deviation  can not be $6$.
Can anyone please help me by giving some hints? I have no idea.
 A: There is a standard result that $$\sigma_x^2 \leq \frac{R_x^2}{4}$$ where $\sigma_x^2$ is the variance of $X$ and $R_x$ is the range of $X$
Note that here $R_x = 10 - 0 = 10$
Using this result, $$\sigma_x^2 \leq \frac{10^2}{4} = 25 $$
Or, $$\sigma_x \leq 5$$
Proof:
Let $\min_{1\leq i \leq n} x_i = a$ and $\max_{1\leq i \leq n} x_i = b$
Then $R_x = b - a$
$\sigma_x^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2$
Now $\sum_{i=1}^n (x_i - c)^2$ is least when $c = \bar{x}$
Hence $\sum_{i=1}^n (x_i - \bar{x})^2 \leq \sum_{i=1}^n (x_i - \frac{a+b}{2})^2 = \sum_1 (x_i - \frac{a+b}{2})^2 + \sum_2 (x_i - \frac{a+b}{2})^2$
where $\sum_1$ includes the values of $x$ less than or equal to $\frac{a+b}{2}$ and $\sum_2$ includes values of $x$ greater than $\frac{a+b}{2}$
Or, $\sum_{i=1}^n (x_i - \bar{x})^2 \leq \sum_1 (a - \frac{a+b}{2})^2 + \sum_2 (b - \frac{a+b}{2})^2 = \sum_1 \frac{R_x^2}{4} + \sum_2 \frac{R_x^2}{4} = n \frac{R_x^2}{4}$
Or, $\sigma_x^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2 \leq \frac{R_x^2}{4}$
