## Gompertz Law of Human Mortality

Wednesday, August 5th, 2009

In 1825, a British actuary by the name of Benjamin Gompertz discovered what we now call Gompertz Law of human mortality:

Your probability of dying during a given year doubles every 8 years. For me, a 25-year-old American, the probability of dying during the next year is a fairly miniscule 0.03% — about 1 in 3,000. When I’m 33 it will be about 1 in 1,500, when I’m 42 it will be about 1 in 750, and so on. By the time I reach age 100 (and I do plan on it) the probability of living to 101 will only be about 50%. This is seriously fast growth — my mortality rate is increasing exponentially with age.

And if my mortality rate (the probability of dying during the next year, or during the next second, however you want to phrase it) is rising exponentially, that means that the probability of me surviving to a particular age is falling super-exponentially.

All those mortality statistics demonstrate that the “lightning bolt theory” of mortality can’t be right:

In this view, death is the result of a sudden and unexpected event over which you have no control. It’s sort of an ancient Greek perspective: there are angry gods carousing carelessly overhead, and every so often they hurl a lightning bolt toward Earth, which kills you if you happen to be in the wrong place at the wrong time. These are the “lightning bolts” of disease and cancer and car accidents, things that you can escape for a long time if you’re lucky but will eventually catch up to you.

The problem with this theory is that it would produce mortality rates that are nothing like what we see. Your probability of dying during a given year would be constant, and wouldn’t increase from one year to the next. Anyone who paid attention during introductory statistics will recognize that your probability of survival to age t would follow a Poisson distribution, which means exponential decay (and not super-exponential decay).

Just to make things concrete, imagine a world where every year a “lightning bolt” gets hurled in your general direction and has a 1 in 80 chance of hitting you. Your average life span will be 80 years, just like it is in the US today, but the distribution will be very different.

The accumulated lightning bolt theory comes slightly closer to fitting the data:

Maybe dying is a matter of accumulating a number of “lightning strikes”; none of them individually will do you in, but the accumulated effect leads to death. I think of it something like Monty Python’s Black Knight: the first four blows are just flesh wounds, but the fifth is the end of the line.

Shown above are the results from a simulated world where “lightning bolts” of misfortune hit people on average every 16 years, and death occurs at the fifth hit. This world also has an average lifespan of 80 years (16*5 = 80), and its distribution is a little less ridiculous than the previous case.

I was somewhat surprised when I read that Tyler Cowen called it one of the more interesting blog posts I’ve read in some time — not because it’s not interesting, but because survival analysis isn’t new.

(Gamers, by the way, might note that those two models are awfully similar to saving throws and hit points.)