Could someone give me a working definition of variance? From what I understand, standard deviation is the data's average amount of dispersal around the mean...so then we square that and get variance but I don't understand what variance tells me. Any help would be appreciated.
To keep this smooth we want to be even, then the simplest case is for which is the standard deviation.
(The average absolute deviation is used sometimes, but is not as generally useful as the standard deviation.)
Just to add to CaptainBlack's remarks, another reason why we like to work with variance is that variances "add up": that is, if X and Y are independent random variables, then the variance of X+Y is the variance of X plus the variance of Y:
This is not true of standard deviations.
Standard deviations, on the other hand, have a more natural dimension, the same as the original random variable. For example, if your sample data is measured in feet (people's heights, for example), then the standard deviation is also in feet. The variance, on the other hand, will be measured in square feet. So the standard deviation may have a more intuitive "feel" about it.