Let's say I have data on how much time (in hours) students spend doing homework per day at various schools across the United States. The values I have collected include: the average time spent on homework (aggregated at the school level), the average and maximum values for time spent on homework for each decile bin, and the standard deviations for both of the previous categories.
I have a hypothesis that the distribution of homework hours is a key predictor of some school-level outcome (let's say the school's rating). Specifically, I believe (counter-intuitively) that schools for which only a handful of students are working extremely long hours on homework are likely to do better than schools which have more symmetrical and even distributions of homework time.
So imagine this incorrect specification:
lm(school_rating ~ average_homework_hours + sd_homework_hours)
I'm concerned that:
You can't have
sd_homework_hoursin the regression, as it a function of the mean (which you can't therefore hold constant).
The coefficient on
sd_homework_hoursdoesn't actually capture the hypothesis of interest (it's more a measure of dispersion than the idea of the shape of the distribution). Perhaps what I'm trying to get at is something like skewness, or a Gini coefficient?
So I'm wondering what the correct approach is to including some measure of how asymmetrical the overall homework burden is. Are there other metrics that better capture the idea of the majority of a distribution being centered on a few individuals? Can you use skewness as an IV measure in a regression specification? Is this the right approach?
Just dawned on me that there's another way to think about this. Imagine lining up all the individuals who have completed a test at a given school and you rank-order them from least amount of homework time to most amount of homework time. You then plot a cumulative density plot of the total amount of homework time.
You can imagine schools where the addition of each student adds a fixed amount per student (so a perfectly increasing line). There are other schools where each additional kid barely adds anything to the total, until you get to the kids at the upper-end of the decile, at which point you see sharp increases.
What measure allows me to compare "sharp, peaked" increases in the plot versus slow and gradual increases?
Edit: Bumping this to promote visibility.