The Standard Deviation

Despite its usefulness, the variance has an unfortunate hitch. When we squared the deviations, by definition we also squared the units in which those deviations were measured. The variance of the juvenile homicide arrest data, then, is 44.86 arrests squared, and the variance for prosecutors’ offices is 150.62 trials squared. This obviously makes no sense. The variance produces oddities such as crimes squared and years squared. The variance is also difficult to interpret—the concept of the “average squared deviation score” is not intuitive and does not provide a readily understandable description of the data. Something needs to be done to correct this. Luckily, a solution is at hand. Since the problem was created by squaring, it can be solved by doing the opposite—taking the square root. The square root of the variance (symbolized s, or sometimes sd) is the standard deviation:

Standard deviation: Computed as the square root of the variance, a measure of dispersion that is the mean of the deviation scores. Symbolized as s or sd.

The standard deviation of the juvenile homicide arrest data is

And s for the prosecutors’ jury trials is

The square root transformation restores the original units; we are back to arrests and trials now and have solved the problem of impossible and nonsensical squares. Note that the standard deviation—just like the squared deviation scores and the variance—can never be negative.

Substantively interpreted, the standard deviation is the mean of the deviation scores. In other words, it is the average distance between the individual raw scores and the distribution mean. It indicates the general spread of the data by conveying information as to whether the raw values cluster close to the mean (thereby producing a relatively small standard deviation) or are more dispersed (producing a relatively large standard deviation). The standard deviation is generally presented in conjunction with the mean in the description of a continuous variable. We can say, then, that in 2015, the eight states considered here had a mean of 4.50 juvenile arrests for homicides, with a standard deviation of 6.70. Similarly, this sample of prosecutors’ offices handled a mean of 14.57 felony jury trials, with a standard deviation of 12.27.