# Small data sets with integral sample standard deviations

I'm looking for small data sets ($N\approx10$) that have integral (or even rational) sample standard deviations. Given a list of observations $\{x_1,\ldots,x_N\}$ with mean $\bar{x}$, the sample standard deviation is given by

$$s=\sqrt{\frac{\displaystyle\sum_{i=1}^N (x_i-\bar{x})^2}{N-1}}$$

Notice the $N-1$ in the denominator here. With a simple google search I've found an article by Frank J. Dudek giving small data sets with integral population standard deviations, but the population standard deviation is computed with the formula above except the denominator of the fraction is $N$.

I'm aware of data sets that have the following form:

$$\{a,\ldots,a,\frac{a+b}{2},b,\ldots,b\}$$

where the number of $a$'s appearing is equal to the number of $b$'s appearing. The sample standard deviation for this set is

$$s=\left|\frac{b-a}{2}\right|$$

However, I'd like an example that is a little more interesting than this one. Thanks in advance for any help.

• Curious why you are focusing on standard deviation. Maybe this is for computer generated homework questions and checking? If you focus on variance instead, then it will always be rational if the data are. – user52817 Feb 8 '17 at 2:53
• Not a full answer, but you and others may find this useful: the Simple Wikipedia page for Standard Deviation has a handful of examples: simple.wikipedia.org/wiki/Standard_deviation#Basic_example In particular, the "More Examples" section has 3 small data sets (n=4) all with mean 7 but with stdevs 7, 5, and 1. This can be an instructive example for students to motivate the need for stdev and why the formula captures what it should. – Brendan W. Sullivan Feb 19 '18 at 15:37

Sure, try this data set: $-2, -2, -1, -1, -1, 0, 0, 0, 3, 4$.

Unless I've fudged things up, it has $\overline{x} = 0$, $\sum\left(\overline{x}-x_i\right)^2=36$, $N-1=9$.

• (Sorry about the original formatting--everything should be in order now.) – Jered Wasburn-Moses Feb 8 '17 at 13:40

These are easy to just make as needed:

 1 2 3                   has sd 1  (n=3)
6 8 8 8 8 9 9           has sd 1  (n=7)
1 2 3 4 4 5 6 7         has sd 2  (n=8)
2 3 3 4 5 5 6 7 7 8     has sd 2  (n=10)
1 1 3 3 4 5 5 5 6 7     has sd 2  (n=10)
2 3 5 7 7 7 11          has sd 3  (n=7)
1 3 5 5 6 6 6 7 12      has sd 3  (n=9)
1 1 1 1 1 1 2 3 5 8 9   has sd 3  (n=11)
0 2 3 3 3 5 5 6 9 14    has sd 4  (n=10)
3 3 3 4 4 5 5 7 11 15   has sd 4  (n=10)


Of course you can shift these up or down as desired.