Occasional reflections on Life, the World, and Mathematics


Someone has set up a macabre “death clock“, a web site where individuals can enter a few personal statistics — birthdate, sex, smoking status, and general level of optimism, and it will calculate a “personal date of death”, together with an ominous clock ticking down the seconds remaining in your life. (For Americans, ethnic group is a hugely significant predictor, but I’m not surprised that they leave this out. Ditto for family income.) It’s supposed to be a sharp dose of reality, I suppose, except that it’s nonsense.

Not because no one knows the day or the hour, though that is true, but because the author has built into the calculator a common but elementary misconception about life expectancy, namely, that we lose a year of expected remaining life for every year that we live. Thus, when I enter my data the clock tells me that I am expected to die on August 6 2042. If I move my birthdate back* by 10 years — making myself 10 years older — my date of death moves back by the same amount, to August 6 2032. If I tell it I was born in 1936 it tells me that my time has already run out, which is obviously absurd.

In fact, every year that you live, you lose 1 year, but gain a proportion a remainder equivalent to the probability that you might have died. Thus, a 46-year-old US man has expected remaining lifespan 33.21 years. He has probability 0.00365 of dying in the next year; if he makes it through that year and reaches his 47th birthday, his expected remaining lifespan is (33.21-1)+.00365 x 32.21 = 32.33 years.** So he’s only lost 0.88 years off his remaining lifespan. In this way, it’s actually possible to have more expected remaining lifespan at an older age than at a younger, if the mortality rate is high enough. Thus, if we go back to 1933 mortality rates, the expected lifespan at birth was 59.2 years. But a 1-year-old, having made it through the 6.5% infant mortality, now has 62.3 years remaining on average.

This is another way of expressing the well-known but still often not-sufficiently-appreciated impact of infant mortality on life expectancy. The life-expectancy at birth for US males is 76.4 years. But that obviously doesn’t mean that everyone keels over 5 months into their 77th year. 60% of the newborn males are expected to live past this age, and a 77-year-old man has 10 remaining years on average.

Of course, these are all what demographers call “period” life expectancies, based on the mortality rates experienced in the current year, and pretending that these mortality rates will continue into the future. Based on the experience of the past two centuries we expect the mortality rates to continue to fall, in which case the true average lifespans for people currently alive — the “cohort life expectancies” will exceed these period calculations — but there is no way to know. If an asteroid hits the earth tomorrow and wipes out all life on earth, this period calculation will be rendered nugatory (but there will be no one left to point that out. Hah!) The true average lifespan of the infants born this year will not be known until well into the 22nd century. Or, if Aubrey de Grey is right, not until the 32nd century.

* Or is it moving my birthdate forward by 10 years when I make it 10 years earlier? Reasonable people disagree on this point! And there’s interesting research on the habits of mind that lead one to choose the metaphor of the stationary self with time streaming past me, or the self moving like a river through a background of time.

** Actually, it’s (33.21-1)/(1-.00365)

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