Small data sets with integral sample standard deviations

Question

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.

Answer 1

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

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

Answer 2

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.