The standard deviation is a measure of dispersement, or, how spread out the data are. Each value in the data lies a distance from the sample mean (x minus x-bar). These distances are averaged in order to give a general sense of how the values tend to vary.
The sample mean is the center of the data, where there are an equal amount of negatives as positives. So, the sum of the differences will equal zero. Therefore, the differences must be squared before they are averaged. Squaring cancels the negatives. This average of the squared differences is called variance.
Variance is then square-rooted. This result is the standard deviation.
