The Standard Deviation Of a Discrete Random Variable
The standard deviation of a discrete random variable of a probability distribution is the square root of the variance of the random variable.
Lets once again revisit the newspaper boy problem.
| x | p(x) | x px) |
| 0 | 0.10 | 0.0 |
| 1 | 0.12 | 0.12 |
| 2 | 0.35 | 0.7 |
| 3 | 0.20 | 0.60 |
| 4 | 0.15 | 0.60 |
| 5 | 0.08 | 0.40 |
We have already calculated the expected value of the random variable. E(X)=2.42
The formula for calculating the variance of a random variable of a probability distribution is
We need to add three columns to our table, a deviation column, a deviation squared column, and a p(x) times squared deviation column.
| x | p(x) | x p(x) | (x-E(x))) |  |  |
| 0 | 0.10 | 0.00 | -2.42 | 5.86 | 0.59 |
| 1 | 0.12 | 0.12 | -1.42 | 2.02 | 0.24 |
| 2 | 0.35 | 0.70 | -0.42 | 0.18 | 0.06 |
| 3 | 0.20 | 0.60 | 0.58 | 0.34 | 0.07 |
| 4 | 0.15 | 0.60 | 1.58 | 2.50 | 0.38 |
| 5 | 0.08 | 0.40 | 2.58 | 6.66 | 0.53 |
The standard deviation is the square root of the variance.
