[BruceZ]

The variance is the average squared distance from the average. Standard deviation is the square root of the variance. So in a sense you can think of it as kind of an average distance from the average, though it is not a simple arithmetic average because the square root of the average of squared distances is not the same as the average of the distances. That is, the average of squares is not the same as the square of the average. The square is taken so positive and negative distances are treated equally. The values will lie within +/- 1 standard deviation of the average 68% of the time.

So if x[n] are values of N measurments, and the average value is u, then:

variance = 1/N*sum[i=1 to N](x[n]-u)^2
standard deviation = sqrt(variance)

average distance from average would be:
1/N*sum[i=1 to N](|x[n]-u|)

If you're an electrical engineer, variance can be thought of as power, while standard deviation is rms amplitude. rms means "root mean square" or the square root of the mean of the square".