I work these days as a statistician and a lot of what I do is evaluating design of experiments; I started this job less than a year ago, after getting a PhD in mathematical statistics. I remember once trying to explain conditioning numbers and their use in statistics in terms of numerical stability and the spiel from numerical analysis, but I don't think I was understood. I was not actually using the conditioning number at the time to understand an experimental design, though, so I didn't worry too much about needing to explain it.
But a few months later I find myself looking at conditioning numbers in multiple contexts in order to understand why strange behavior could be emerging in some models being fit and why some experimental designs seem to be failing. (These designs, by the way, are space filling designs intended for computer experiments.) The conditioning number is becoming a go-to tool for myself and I think I need to be able to explain it to others. There are situations where correlation alone will not reveal potential problem in a design.
But I still remember how miserable it was to try to explain a quantity designed for numerical analysis to people who (while having PhDs) don't have training in numerical analysis and may not even be statisticians proper. Talking about numerical stability, sensitivity to small perturbations, singular values, etc. in a statistical context seemed to just go over people's heads.
So I hope someone has a good description about what the conditioning number is, in the context of statistics, that a non-statistician could understand.