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

Dec 10, 2017

See 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:

$\overline{x} = \frac{1 + 2 + 2 + 2 + 3}{5} = \frac{10}{5} = 2$

The standard deviation is:

$\sigma = \sqrt{\frac{{\left(1 - 2\right)}^{2} + 3 \cdot {\left(2 - 2\right)}^{2} + {\left(3 - 2\right)}^{2}}{5}}$

$\sigma = \sqrt{\frac{1 + 1}{5}} = \sqrt{\frac{2}{5}} \approx 0.632$

Example 2

Let the set be:

${S}_{2} = \left\{0 , 1 , 2 , 3 , 4\right\}$

The mean is:

$\overline{x} = \frac{0 + 1 + 2 + 3 + 4}{5} = \frac{10}{5} = 2$

The standard deviation is:

$\sigma = \sqrt{\frac{{\left(0 - 2\right)}^{2} + {\left(1 - 2\right)}^{2} + {\left(2 - 2\right)}^{2} + {\left(3 - 2\right)}^{2} + {\left(4 - 2\right)}^{2}}{5}}$

$\sigma = \sqrt{\frac{4 + 1 + 1 + 4}{5}} = \sqrt{\frac{10}{5}} = \sqrt{2} \approx 1.414$

As you can see from the examples the means are the same, but the standard deviation of the second set is bigger because the elements differ more than in the first example.