Discretization of normal distribution over a finite range

Question

If I only have data about the mean and standard deviation of a distribution over a finite discrete range (e.g. integers 1 to 5). How do I properly reconstruct the distribution (= a distribution that has the same mean and standard deviation, assuming it is close to normal)?

Answer 1

If $p_i$ is the probability of outcome $i$ and you want to achieve a certain $\mu$ and $\sigma$, you need to obey the constraints $$ \begin{align}\sum p_i&=1\\ \sum ip_i &= \mu\\ \sum i^2p_i &= \sigma^2+\mu^2\end{align}$$ These are three equations in five unknowns, so there are many solutions (let us ignore the possibility that the solutions conflict with the conditions $p_i\ge 0$) and we can impose additional conditions "at will". Since you want "close to normal", I suggest we take the conditions from the next few non-central moments of the normal distribution: $$ \begin{align}\sum i^3p_i &= \mu^3+3\mu\sigma^2\\ \sum i^4p_i &= \mu^4 + 6\mu^2\sigma^2 + 3\sigma^4.\end{align}$$ So now you have five linear equations in five unknowns - and I hope it turns out that all $p_i$ are $\ge0$ in your specific case.

Answer 2

If there are just five values (possible), I fail to see the point of trying to fit to some "standard distribution" or even a continuous one (like normal). The (relative) frequency table says it all, in a simpler way, and it's even easy to visualize (e.g. like a histogram).

I know the thread is very old; I just couldn't resist.