What does a small standard deviation signify? What does a large standard deviation signify?

Explanation: Standard deviation indicates how different are the elements in the set. If the elements are close to each other, then the standard deviation is close to zero (it is zero only if the elements are identical). The bigger standard deviation, the more different elements in the set Example 1 Let the set be: $S_1=\{1,2,2,2,3\}$ The mean is: $\bar{x}=\frac{1+2+2+2+3}{5}=\frac{10}{5}=2$ The standard deviation is: $\sigma =\sqrt{\frac{(1-2{)}^2+3\ast (2-2{)}^2+(3-2{)}^2}{5}}$ $\sigma =\sqrt{\frac{1+1}{5}}=\sqrt{\frac{2}{5}}\approx 0.632$ Example 2 Let the set be: $S_2=\{0,1,2,3,4\}$ The mean is: $\bar{x}=\frac{0+1+2+3+4}{5}=\frac{10}{5}=2$ The standard deviation is: $\sigma =\sqrt{\frac{(0-2{)}^2+(1-2{)}^2+(2-2{)}^2+(3-2{)}^2+(4-2{)}^2}{5}}$ $\sigma =\sqrt{\frac{4+1+1+4}{5}}=\sqrt{\frac{10}{5}}=\sqrt{2}\approx 1.414$

It signifies that your data are closely distributed (instead of widely spread) around the mean value. This generally implies that your data dont