# Can a data set with two or three numbers have a standard deviation?

Oct 29, 2015

Yes it can.

#### Explanation:

Yes. It is possible to calculate the standard deviation of any non empty set of numbers. However it would be difficult to interpret such value for such a small set of data.

Examples:

1. A sigle number would have a deviation of zero:

Let $A = \left\{x\right\}$ be a data set with only a single value.

The mean $\overline{x} = \frac{x}{1} = x$

The standard deviation: $\sigma = \sqrt{\frac{{\Sigma}_{i = 1}^{i = n} \left({x}_{i} - \overline{x}\right)}{n}}$

In this case $n = 1$ so the formula is reduced to: $\sigma = \sqrt{\frac{x - \overline{x}}{1}} = 0$.

1. Let $A$ be a set of 4 numbers: $A = \left\{2 , 3 , 5 , 8\right\}$

The mean $\overline{x} = \frac{2 + 3 + 5 + 8}{4} = \frac{18}{4} = 4.5$

The deviation would be:

$\sigma = \sqrt{\frac{{\left(2 - 4.5\right)}^{2} + {\left(3 - 4.5\right)}^{2} + {\left(5 - 4.5\right)}^{2} + {\left(8 - 4.5\right)}^{2}}{4}}$

$\sigma = \sqrt{\frac{{2.5}^{2} + {1.5}^{2} + {0.5}^{2} + {3.5}^{2}}{4}}$

$\sigma = \sqrt{\frac{6.25 + 2.25 + 0.25 + 12.25}{4}}$

$\sigma = \sqrt{\frac{21}{4}} = \sqrt{5.25} \approx 2.29$