Interpreting the Standard Deviation

A large standard deviation, which is the square root of the variance, indicates that the data points are far from the mean, and a small standard deviation indicates that they are clustered closely around the mean. For example, each of the three populations $\{0, 0, 14, 14\}$, $\{0, 6, 8, 14\}$, and $\{6, 6, 8, 8\}$ has a mean of 7. Their standard deviations are 7, 5, and 1, respectively. The third population has a much smaller standard deviation than the other two because its values are all close to 7.

Standard deviation may serve as a measure of uncertainty. In physical science, for example, the reported standard deviation of a group of repeated measurements gives the precision of those measurements. When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements is of crucial importance. If the mean of the measurements is too far away from the prediction (with the distance measured in standard deviations), then the theory being tested probably needs to be revised. This makes sense since they fall outside the range of values that could reasonably be expected to occur, if the prediction were correct and the standard deviation appropriately quantified.

Application of the Standard Deviation

The practical value of understanding the standard deviation of a set of values is in appreciating how much variation there is from the average (mean).

Climate

As a simple example, consider the average daily maximum temperatures for two cities, one inland and one on the coast. It is helpful to understand that the range of daily maximum temperatures for cities near the coast is smaller than for cities inland. Thus, while these two cities may each have the same average maximum temperature, the standard deviation of the daily maximum temperature for the coastal city will be less than that of the inland city as, on any particular day, the actual maximum temperature is more likely to be farther from the average maximum temperature for the inland city than for the coastal one.

Sports

Another way of seeing it is to consider sports teams. In any set of categories, there will be teams that rate highly at some things and poorly at others. Chances are, the teams that lead in the standings will not show such disparity but will perform well in most categories. The lower the standard deviation of their ratings in each category, the more balanced and consistent they will tend to be. Teams with a higher standard deviation, however, will be more unpredictable.

Comparison of Standard Deviations

Example of two samples with the same mean and different standard deviations. The red sample has a mean of 100 and a SD of 10; the blue sample has a mean of 100 and a SD of 50. Each sample has 1,000 values drawn at random from a Gaussian distribution with the specified parameters.