Here are some calculations building on the standard deviations and Bomb#20's argument.
200 lb is not the average weight, so let's be fair. The average weight of a young man with the given densities is about 80 kg or 175 lbs.
For an 80 kg black man, his volume is:
80 kg/(1.075 g/mL)= 74.4 L
For an 80 kg white man, his volume is:
80 kg/(1.065 g/mL)= 75.1 L
It is a difference of 0.7 L, or 0.7 L*(1.07 g/mL) = 750 g = 1.65 lb of extra buoyancy force for whites than for blacks.
So, the average black man in a swimming pool is like the average white man but with a very dense 1.65 lb weight in his pockets. Doesn't seem like so much, but a little extra weight can count for a lot when trying to stay afloat.
The inequality is greater at the right tails of the overlapping normal distributions. I found the standard deviation of body densities in
this study. It is 0.012 g/mL for whites and 0.015 for blacks. The two distributions have different means and different standard deviations, so the calculations are a test of my ability to think straight, but I will give it a shot.
Given a racial density difference of 0.01 g/mL, this means the average body densities of whites and blacks are about 0.83 white standard deviations apart and about 0.66 black standard deviations apart.
Using a
z-score calculator, assuming an extra weight of 1.65 lb, with z=0.66 black standard deviations, Q is 0.25, and it means that 75% of blacks are like the average white but with at least an extra 1.65 lb weight in their pockets. With z=0.83 white standard deviations, Q is 0.20, so only 20% of whites are like the average white with at least an extra 1.65 lb weight in their pockets.
What if it is a body density equal to an extra
5-pound weight in their pockets? For whites, this is 5 lb*(0.83 SD/1.65 lb)= 2.52 standard deviations above the white mean. This means Q is 0.005868 or 1 in 170. One in 170 whites have a body density equal to an extra 5-pound weight in their pockets. But, for blacks, this is 5 lb*(0.66 SD/1.65 lb)= 2 black standard deviations above the white mean and equal to 2 minus 0.66 black standard deviations = 1.33 black standard deviations above the black mean. Another way to calculate this is that 5 pounds of extra weight for the average white is just 5-1.65=3.35 pounds of extra weight for the average black, and 3.35 lb*(0.66 bSD/1.65 lb) = 1.34 black standard deviations above the black mean. For z=1.34, this means Q is 0.090123 or 1 in 11.
One in 11 black men is like the average white man but with an extra five-pound weight in his pockets, or 15 times as many blacks as whites.