Answer
Here, the variance is:
$(-5-(-.1))^2*(.2)+(-1-(-.1))^2*(.3)+(0-(-.1))^2*(.2)+(2-(-.1))^2*(.1)+(5-(-.1))^2*(.2)+(10-(-.1))^2*0=10.69$
The standard deviation is the square root of the variance:
The standard deviation here equals to $\sqrt{10.69}=3.27$
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.
The expected value is: $(.2)*-5+(.3)*-1+(.2)*0+(.1)*2+(.2)*5+0*10=-.1$
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:
$(-5-(-.1))^2*(.2)+(-1-(-.1))^2*(.3)+(0-(-.1))^2*(.2)+(2-(-.1))^2*(.1)+(5-(-.1))^2*(.2)+(10-(-.1))^2*0=10.69$
The standard deviation is the square root of the variance:
The standard deviation here equals to $\sqrt{10.69}=3.27$
