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A measure of spread for a distribution of a random variable that determines the degree to which the values differ from the expected value.
The standard deviation of random variable X is often written as σ or σX.
For a discrete random variable the standard deviation is calculated by summing the product of the square of the difference between the value of the random variable and the expected value, and the associated probability of the value of the random variable, taken over all of the values of the random variable, and finally taking the square root.
In symbols, σ = 
An equivalent formula is, σ = 
The square of the standard deviation is equal to the variance, Var(X) = σ2.
Example
Random variable X has the following probability function:
| x | 0 | 1 | 2 | 3 |
| P(X = x) | 0.1 | 0.2 | 0.4 | 0.3 |
A bar graph of the probability function, with the mean and standard deviation labelled, is shown below.
See: population standard deviation, standard deviation
Curriculum achievement objectives reference
Probability: Level 8
