Standard deviation (represented by the symbol σ) shows how much variation or "dispersion" exists from the average (mean, or expected value).
σ – standard deviation, E(X) – expected value of X, X – random variable
Standard deviation gives us information how dispersed data (like age, inflation, market, share prices, ...) is. A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data points are spread out over a large range of values.
Example of two sample populations with the same mean and different standard deviations (red SD = 10; blue SD = 50).
In order to better understand what is standard deviation and why it is important we prepared a short example.
Consider, that we examined two groups regarding their age. Age distribution in group A is 11, 15, 18, 20, 21, 24, 60, 71, 78, 82 and in group B is 36, 36, 37, 38, 39, 40, 41, 42, 44, 47. Average (mean) in both cases is equal 40. Does it mean that both groups are identical? Is average enough to describe and compare both groups? The answer is NO.
Using only average one can get wrong opinion that two groups are identical. In fact, differences are significant. The first group have big age differences. The second group is more homogenous and age of different people is similar.
Differences of each group can be more precisely described with standard deviation. In case of group A standard deviation is equal 27.5 and in case of group B 3.4.
Standard deviation gives us information if observations in the group are similar or different. The smaller standard deviation the more similar observation and bigger concentration around the average (mean). The bigger standard deviation the bigger dispersion (differences between observations).