Answer
Here, the variance is:
$(1-2.5)^2*(.25)+(2-2.5)^2*(.25)+(3-2.5)^2*(.25)+(4-2.5)^2*(.25)=1.25$
The standard deviation is the square root of the variance:
The standard deviation here equals to $\sqrt{1.25}=1.12$
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 choosing from the set at random, we have a probability distribution from 1 to 4 with even probabilities, all equal to $\frac{1}{4}=.25$
The expected value is: $(.25)*1+(.25)*2+(.25)*3+(.25)*4=2.5$
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:
$(1-2.5)^2*(.25)+(2-2.5)^2*(.25)+(3-2.5)^2*(.25)+(4-2.5)^2*(.25)=1.25$
The standard deviation is the square root of the variance:
The standard deviation here equals to $\sqrt{1.25}=1.12$