Using stat methods on summary statistics

Question

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.

Answer 1

I wouldn't say that what you're interested in testing actually involves any "summary statistics". You're asking whether you can group your data in some disjoint way (that is to say no particular meat sample falls into a combination of two A* categories) and then test whether the means of some particular variable (A*) are equal across the grouping levels (normal, semi-splotchy, splotchy). This seems to be to a perfectly reasonable time to use ANOVA. Of course, as usual, you need to check whether the assumptions of ANOVA are met. Those assumptions are that your errors are normally distributed, have constant variance and all independent. By virtue of your problem statement you should be all but guaranteed independence of meat samples (that is, unless the meat came in separate batches and certain batches were influenced in their "splotchey-ness" which you should certainly consider) but you will definitely have to check the variance and normality assumptions.

Now to a hammer everything looks like a nail but I would be careful about drawing conclusions solely from visual displays of data. While you didn't claim that is something you've done in the past, you seemed to indicate you were inclined to do so. Visual displays are certainly helpful for gaining intuition into statistical problems but I would recommend using a quantitative method to make decisions on the quality of a product.