Answer
Here, the variance is:
$(1-3.5)^2*(\frac{1}{6})+(2-3.5)^2*(\frac{1}{6})+(3-3.5)^2*\frac{1}{6}+(4-3.5)^2*(\frac{1}{6})+(5-3.5)^2*(\frac{1}{6})+(6-3.5)^2*(\frac{1}{6})=2.92$
The standard deviation is the square root of the variance:
The standard deviation here equals to $\sqrt{2.92}=1.71$
Work Step by Step
In order to calculate the standard deviation, we have to get the expected value of the probability distribution.
Here, we can calculate it as:
$p_{1}\times X_1+p_2 \times X_2 +...$
Where $p$ is the probability, $X$ is the value for every possible value of the distribution.
By rolling a fair die, we have a probability distribution from 1 to 6 with even probabilities, all equal to $\frac{1}{6}\approx.167$
The expected value is: $(\frac{1}{6})*1+(\frac{1}{6})*2+(\frac{1}{6})*3+(\frac{1}{6})*4+(\frac{1}{6})*5+(\frac{1}{6})*6=3.5$
The variance of X can be calculated as:
$(X_1-E(X))^2*p_1+(X_2-E(X))^2*p_2+...$
Here, the variance is:
$(1-3.5)^2*(\frac{1}{6})+(2-3.5)^2*(\frac{1}{6})+(3-3.5)^2*\frac{1}{6}+(4-3.5)^2*(\frac{1}{6})+(5-3.5)^2*(\frac{1}{6})+(6-3.5)^2*(\frac{1}{6})=2.92$
The standard deviation is the square root of the variance:
The standard deviation here equals to $\sqrt{2.92}=1.71$
