Question

(c) The standard deviation of X must always be positive. (d) Adding a constant to a random variable does not effect the standard deviation

Answer 1

c)   Standard Deviation of X is always positive or zero

     The formula for computing standard deviation is $\sigma=\sqrt{\frac{\sum\left ( x_{i}-\overline{x} \right )^{2}}{N}}$

      As can be seen, the numerator inside the squareroot is a sum of squares, hence cannot be negative (but can be zero). Denominator N is never zero as the population size is always greater than 1

Thus, Standard Deviation of X is always positive or zero

d)   Suppose we add a constant 'c' to xi.

     Let y = xi + c. To find standard deviation of y, we first find $\bar{y}$

    $\newline\bar{y} = \frac{\sum{y}}{N} \newline=\frac{\sum \left ( x_{i}+c \right )}{N}\newline=\frac{x_{i}}{N}+\frac{Nc}{N}\newline=\bar{x}+c\newline\newline\bar{y}=\bar{x}+c$

Standard Deviation of Y is given by

$\newline\sigma_{y}=\sqrt{\frac{\sum\left ( y_{i}-\overline{y} \right )^{2}}{N}}\newline =\sqrt{\frac{\sum\left ( x_{i}+c-\overline{x}-c \right )^{2}}{N}}\newline=\sqrt{\frac{\sum\left ( x_{i}-\overline{x} \right )^{2}}{N}}\newline$

which is the same as the standard deviation of X

Hence,

Adding a constant to a variable does not effect the standard deviation.