Variance

Variance represents how spread out the data are. It is the average of the squared differences from the mean.

The distances from the mean are calculated by subtracting each x from the mean. These distances are squared and then averaged to arrive at the variance.

Because the differences are squared, the result is in squared units - for example, if the measurements are "miles," then the variance is "miles^2". Therefore, the variance value does not intuitively describe the data. To overcome this, the square-root of the variance is taken. The square-root of variance is called standard deviation.

Here is an example data set:

miles driven (x): 43, 70, 27, 36
n = 4
mean = 44 miles

differences
43 - 44 = -1
70 - 44 = 26
27 -44 = -17
36 - 44 = -8

differences^2
(-1)^2 = 1
(26)^2 = 676
(-17)^2 = 289
(-8)^2 = 64

The average of the differences^2:
(1+676+289+64)/4 = 257.5

The variance is:
257.5 miles^2

Standard Deviation

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.