Square roots and sunscreen

Also, here's an interesting tidbit, from this New York Times piece on SPF. SPF, or "sun protection factor", is the number on the sunscreen bottle; if a properly applied sunscreen lets through a fraction p of the UV rays it's meant to protect against, then that sunscreen has SPF 1/p. (The numbers in the article talk about the proportion of the UV rays which are blocked; in this case, if a fraction q of the UV rays are blocked, the sunscreen has SPF 1/(1-q).)

Anyway, you're supposed to apply some ridiculous amount of sunscreen to your body, about an ounce. This seems like a lot to most people, because that stuff is expensive! So a lot of people underapply sunscreen. (I'll include myself here.) The article quotes Darrell Rigel, NYU dermatologist, as saying that if you apply half the sunscreen you're "supposed" to, you have to take the square root of the SPF.

That sounds obvious once you think about it -- but I'll admit I'd never thought about it. Say I have a sunscreen that allows through one-sixteenth of the light which hits it when applied properly. Now imagine splitting it up into two coats, each of which allows through the same proportion of the light that hits it. One-fourth of the light makes it through the outer coat; one-fourth of that light makes it through to the skin.

Of course there are issues with this analysis, but according to this paper in the British Journal of Dermatology it appears to hold up. And applying twice the usual amount of sunscreen apparently squares the SPF. (The effect is actually a bit less than this, because sunscreens don't block all wavelengths equally, nor does the sun's spectrum contain all wavelengths equally.)

This all implies that if you want to compare prices of sunscreens, you should divide the cost of the sunscreen by the product of the bottle's volume and the logarithm of the SPF. Do sunscreen prices actually work this way?