# Exact probability and asymptotic approximation

From Wikipedia

A large amount of computing resources is required to compute exact probabilities for the Kruskal-Wallis test. Existing software only provides exact probabilities for sample sizes less than about 30 participants. These software programs rely on asymptotic approximation for larger sample sizes. Exact probability values for larger sample sizes are available. Spurrier (2003) published exact probability tables for samples as large as 45 participants. Meyer and Seaman (2006) produced exact probability distributions for samples as large as 105 participants. Critical value tables and exact probabilities from Meyer and Seaman are available for download at http://faculty.virginia.edu/kruskal-wallis/. A paper describing their work may also be found there.

I think exact probabilities are the accurate probabilities, but why does it say that "Existing software only provides exact probabilities for sample sizes less than about 30 participants. These software programs rely on asymptotic approximation for larger sample sizes"? Is it wrong?

Thanks and regards!

It's clearly true of many software programs, and for the reason explicitly stated in the first sentence you quote - after all, unless you very carefully implement efficient algorithms for exact calculations of tail probabilities, the number of calculations grows extremely rapidly (well in fact it also grows very rapidly even if you do implement such calculations, it just lets you push the achievable $n$ up some distance).
Since you give the reason why the asymptotic distribution is used for large $n$ right in your question, I'm puzzled what else you're after here, except perhaps an explanation of why more software hasn't implemented the most efficient known algorithm.