Is it ok to use statistical methods on summary statistics, as they're random variables?
I ran into an interesting problem while working with a client, but will try to keep it very general as the specific instance is a lot to explain.
Suppose you're looking at the color distribution of something like meat and you have 50 samples of meat. You have standard measures of color used in your industry, say A*, B*, and C* which relate to the pixel values of certain colors in the meat. Essentially, each piece of meat has a distribution of A* values that can be thought of as a distribution of color values in the meat.
If you grouped your meat into three kinds (normal, semi-splotchy, splotchy) would you be able to run an ANOVA on the mean of A* values across the three kinds (levels) of meat.
We've summarized it visually, but the client really wants to talk about a statistically significant difference, which I'm not sure is necessary or possible. But it seems plausible to me, and my intuition says it should be valid.