The standard deviation of a set of values is a measure of how widely the values differ from each other. Specifically, the standard deviation is the square root of the average of the squares of the differences between the data values and their mean. If the distribution of the values is normal then about 68% of the values will fall within one standard deviation of the mean.
Examples help illustrate this concept. Learning the median height of basketball players tell us that half are above that height and half are below. Learning the standard deviation of their heights tells us how varied their heights are.
Mathematically, the standard deviation of a random variable X is:
where the expected value of X is E(X).