Question

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

Answer 1

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={x}` be a data set with only a single value.

The mean `bar{x}=x/1=x`

The standard deviation: `sigma=sqrt((Sigma_{i=1}^{i=n}(x_i-bar(x)))/n)`

In this case `n=1` so the formula is reduced to: `sigma=sqrt((x-bar(x))/1)=0`.

1. Let `A` be a set of 4 numbers: `A={2,3,5,8}`

The mean `bar(x)=(2+3+5+8)/4=18/4=4.5`

The deviation would be:

`sigma=sqrt(((2-4.5)^2+(3-4.5)^2+(5-4.5)^2+(8-4.5)^2)/4)`

`sigma=sqrt((2.5^2+1.5^2+0.5^2+3.5^2)/4)`

`sigma=sqrt((6.25+2.25+0.25+12.25)/4)`

`sigma=sqrt(21/4)=sqrt(5.25)~~2.29`