THE VARIANCE AND STANDARD DEVIATION

The variance of a distribution is E((X_bar - X)**2), which is the mean of the squared difference from the mean. Hence, the variance measures the variability of the numbers from the average value of that same set of numbers. The standard deviation is the square root of the variance.

Another way to describe the variance is the sum of the squares of the difference between the numbers in X and the mean mu of the set X, divided by the number of values in X, as shown here:

variance = [SUM (xi - mu)**2 ] / n

For example, if the set X consists of {-10,35,75,100}, then the mean equals (-10 + 35 + 75 + 100)/4 = 50, and the variance is computed as follows:

variance = [(-10-50)**2 + (35-50)**2 + (75-50)**2 + (100-50)**2]/4
         = [60**2 + 15**2 + 25**2 + 50**2]/4
         = [3600 + 225 + 625 + 2500]/4
         = 6950/4 = 1,737

The standard deviation std is the square root of the variance, as shown here:

std = sqrt(1737) = 41.677

If the set X consists of...