I hope this question fits the forum. I'm not a statistician, but have received decent training in statistics and use statistical modeling in my daily work. I've been tasked to give a very condensed "info-package-plus-practice-workshop" about how to run and interpret statistical analyses in social sciences to a group of researchers who have little to no statistical training or experience.
There will be a lot of practical exercises but first I'd like to say something about why we use statistical analyses in the first place. I was thinking of going about it by focusing on a design with two groups and a continuous outcome/response variable and presenting and answering the question of "why can't we just look at the raw group means and see which group has a higher mean?"
I was then going to explain that if we just look at the raw group difference, we ignore within-group variability, and statistical tests help us with that by adjusting the between-group difference to within-group variability1.
However, I realized I can't quite find the words to express, in plain, common sense language and/or through a practical example, WHY large within-group variability undermines between-group differences. i.e. why do we need to adjust for within-group variability. If we have a large between-group difference, why can't we just go by that and ignore the within-group variability?
I'd be grateful if someone could help me find the right words (or tell me this is not a good way to teach about this).
1I will mention other reasons for conducting statistical tests too, but I already have an idea how to explain those.