Monday, May 16, 2016

Variation within and between

Say, you weigh some fifty-year olds from A-ville and some from B-ville.  They don't all weigh the same amount.  We look at how the people from A-ville vary.  We look at how the average or central tendency of the whole A-ville group varies from that of B-ville. If the A group includes widely different weights but the A average differs only a little bit from the B average, we say that the villages don't differ in an important way.

That approach is comparing variation (in weight) between villages with the variation within a village.  This is the central concept of the "analysis of variance", a.k.a. "anova", a central part of modern statistical analysis of experimental data.  Many variables of interest in human affairs are scattered among people in a way that does not split them very reliably or emphatically into separate groups.  For instance, most of the time, a man is stronger than a woman but depending on the groups, we might find that the average man differs from the average woman less than the men differ among themselves.  

Another way of thinking about variation within and between is to think of "overlap".  If the heaviest A-ville man is lighter than the lightest man from B-ville, then weight does cleave the men: A's are light and B's are heavy.  Unless we fudge the samples or groups, natural groups often overlap and are not cleanly separated.  If we take the A high school football team and compare body weights with the B elementary school football team, we may find that there is a clear grouping but only because we purposely chose groups that don't overlap.

We can think about variation in connection with group labels.  If we know a man is from A-ville, how likely is it that he is heavier than average.  Does group membership tell weight?  Does gender tell strength?

Much research today is a long, tough search for variables that matter.  Even if they only matter a little, that may be enough to get research and development started.  Say we do find that the A-villagers tend to be lighter, not all the time but somewhat lighter on average.  That finding might launch a search for an explanation of the lighter weight and lead to better body weights for both villages.

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