What is the difference between controlling for mean versus median income in analyses of income inequality?

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Abstract
It is routine to control for “average” income when assessing the independent effect of income inequality on health, but authors have used different measures, for example, percentage poverty,1 per capita or mean income,2 and median income.3 4 However, as the distribution of income in a population is always positively skewed (that is, a long thin tail for the few with high incomes), the median income and percentage poverty are necessarily correlated with any measure of income inequality. For example, for a given total income of $1 billion in population of 100 000 people, the average income is the same ($10 000) regardless of how that income is distributed. Assume the distribution of income is log-normal (that is, the proportion of people at each income level (y axis) has a normal distribution plotted against the log of that income (x axis)). Figure 1plots the cumulative proportion of the population with up to the given income on the x axis, for three scenarios: low income inequality (SD log income = 0.4); medium income inequality (SD 0.6) and high income inequality (SD 0.8). As the total income is fixed (and hence the average income is fixed), the median income must differ under each scenario and is given by the intercept of each curve with the gridline for the 0.5 cumulative proportion: $8777, $7942, and $6904 for the …