GrareeCowui

2022-01-30

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

### Answer & Explanation

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: ${S}_{1}=\left\{1,2,2,2,3\right\}$ The mean is: $\stackrel{―}{x}=\frac{1+2+2+2+3}{5}=\frac{10}{5}=2$ The standard deviation is: $\sigma =\sqrt{\frac{\left(1-2{\right)}^{2}+3\ast \left(2-2{\right)}^{2}+\left(3-2{\right)}^{2}}{5}}$ $\sigma =\sqrt{\frac{1+1}{5}}=\sqrt{\frac{2}{5}}\approx 0.632$ Example 2 Let the set be: ${S}_{2}=\left\{0,1,2,3,4\right\}$ The mean is: $\stackrel{―}{x}=\frac{0+1+2+3+4}{5}=\frac{10}{5}=2$ The standard deviation is: $\sigma =\sqrt{\frac{\left(0-2{\right)}^{2}+\left(1-2{\right)}^{2}+\left(2-2{\right)}^{2}+\left(3-2{\right)}^{2}+\left(4-2{\right)}^{2}}{5}}$ $\sigma =\sqrt{\frac{4+1+1+4}{5}}=\sqrt{\frac{10}{5}}=\sqrt{2}\approx 1.414$

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