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July 8, 2009



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.  Below are some statistics for mortality rates in the United States in 2005, as reported by the US Census Bureau (and displayed by Wolfram Alpha):

USA-death_ratesThis data fits the Gompertz law almost perfectly, with death rates doubling every 8 years.  The graph on the right also agrees with the Gompertz law, and you can see the precipitous fall in survival rates starting at age 80 or so.  That decline is no joke; the sharp fall in survival rates can be expressed mathematically as an exponential within an exponential:

P(t) \approx e^{-0.003 e^{(t-25)/10}}

Exponential decay is sharp, but an exponential within an exponential is so sharp that I can say with 99.999999% certainty that no human will ever live to the age of 130.  (Ignoring, of course, the upward shift in the lifetime distribution that will result from future medical advances)

Surprisingly enough, the Gompertz law holds across a large number of countries, time periods, and even different species.  While the actual average lifespan changes quite a bit from country to country and from animal to animal, the same general rule that “your probability of dying doubles every X years” holds true.  It’s an amazing fact, and no one understands why it’s true.

There is one important lesson, however, to be learned from Benjamin Gompertz’s mysterious observation.  By looking at theories of human mortality that are clearly wrong, we can deduce that our fast-rising mortality is not the result of a dangerous environment, but of a body that has a built-in expiration date.


The lightning bolt theory

If you had never seen any mortality statistics (or known very many old people), you might subscribe to what I call the “lightning bolt theory” of mortality.  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:

Your probability of survival according to the "Lightning Bolt Theory"

Your probability of survival according to the “Lightning Bolt Theory”

What a crazy world!  The average lifespan would be the same, but out of every 100 people 31 would die before age 30 and 2 of them would live to be more than 300 years old.  Clearly we do not live in a world where mortality is governed by “lightning bolts”.


The accumulated lightning bolt theory

I think most people will see pretty quickly why the “lightning bolt theory” is flawed.  Our bodies accumulate damage as they get older.  With each misfortune our defenses are weakened — a car accident might leave me paralyzed, or a knee injury could give me arthritis, or a childhood bout with pneumonia could leave me with a compromised immune system.  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.


Your probability of survival according to the “accumulated lightning bolt” theory

Fortunately, this theory is also completely testable.  And, as it turns out, completely wrong.  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.  Still, it’s no Gompertz Law: look at all those 160-year-olds!  You can try playing around with different “lightning strike rates” and different number of hits required for death, but nothing will reproduce the Gompertz Law.  No explanation based on careless gods, no matter how plentiful or how strong their blows are, will reproduce the strong upper limit to human lifespan that we actually observe.


The cops and criminals inside your body

Like I said before, no one knows why our lifespans follow the Gompertz law.  But it isn’t impossible to come up with a theoretical world that follows the same law.  The following argument comes from this short paper, produced by the Theoretical Physics Institute at the University of Minnesota [update: also published here in the journal Theory in Biosciences].



But what happens if your internal police force starts to dwindle?  Suppose that as you age the police force suffers a slight reduction, so that they can only cover every spot 12 times a day.  Then the probability of them missing a criminal for an entire day increases to e^{-12} \approx 6 \times 10^{-6}.  The difference between 14 and 12 doesn’t seem like a big deal, but the result was that your chance of dying during a given day jumped by more than 10 7 times.  And if the strength of your police force drops linearly in time, your mortality rate will rise exponentially.



More questions than answers






UPDATE: G&L reader Michael has made a cool-looking (if slightly morbid) web calculator to evaluate the Gompertz law prediction for different ages.  If you want to know what the law implies for you in particular, and are not terribly handy with a calculator, then you might want to check it out.

147 Comments leave one →
  1. bellisaurius permalink
    August 2, 2009 12:04 am

    Interesting. I can definitely buy the model as old age approaches. My only curiosity would be that some of the things that kill us in the 18-35 range are concerning “bolt from heaven” events like gunshots and poisonings (although, perhaps poisoning can fit withing the model, but I’d think car accidents and homicide don’t)

    • gravityandlevity permalink*
      August 2, 2009 12:19 am

      You’re right. It would be interesting to isolate mortality statistics for things like homicide or car accidents and see if they are consistent with a “lightning bolt theory”.
      It may be that the Gompertz law survives only because the causes which dominate our mortality rates (like cancer, heart disease, and stroke) are a result of our bodies breaking down and not a random environment.

  2. August 2, 2009 10:01 am

    It seems there is a deviation from Gompertz law for older people in such a way a 100 years old has more life expectancy than an 80 years old person. Reliability
    theory for machines applied to live expectancy is an interesting possibility. You have a good book chapter here
    and the wikipedia entry

  3. wally permalink
    August 2, 2009 12:18 pm

    Why do people die? Why do individuals in ANY species die? More important yet: why do individuals in ALL species die?
    The answer is not difficult: if individuals did not die, the species itself would die. Life is species-centric, not individual-centric. It doesn’t take much thought to see why a species needs to keep casting out and turning over new, learning, adaptable individuals… if it does not, any environmental change dooms the species. It is disastrous in natural life – just as in human society – if any individual hangs on too long or acquires too entrenched a position of dominance.
    Doctors and researcher who work to extend the life span of individuals really should look at the philosophical underpinnings of their work; much of what they do can be harmful rather than helpful to society at large.

    • gravityandlevity permalink*
      August 2, 2009 2:09 pm

      An important idea: that individual mortality is essential for the survival of the species. But I can’t believe that a doctor working to prolong the life of his patient, under any circumstance, is doing something morally questionable.

    • gwern permalink
      August 2, 2009 5:34 pm

      Wally: individual death is unnecessary for evolution; imagine a growing population in which ‘death’ is simply ceasing to reproduce. The percentage of alleles in the gene pool can still change, and evolution can still occur – despite not a single individual dying.

      Aside from that, natural selection inherently operates on *individuals* and not *groups* (which is why group selection keeps getting proposed, and keeps getting slapped down; see ).


      Tell that to various species, of varying complexities, whose individuals do not age, and only die by accident or lose individual identities. They seem to get along fine. The bacteria, for example, outweigh and outnumber us many times over, yet they don’t really age…


      I’d be *very* careful about this line of thought. Crude analogies about the ‘health of the species’ and ‘protecting the weak’ lead straight to Social Darwinism and eugenics.

      • gravityandlevity permalink*
        August 2, 2009 5:43 pm

        You’re right that individual death is not necessary for evolution. But if there is going to be some balanced consumption of resources, a species needs to either die off or know when to stop breeding. For most animals, the population remains in check because a balance is reached between “rate of having offspring” and “rate of dying off”. For something like bacteria, I imagine that chemical signals can trigger/prevent reproduction when intra-species competition for resources gets too intense.

        • August 17, 2009 1:24 pm

          This assumes a fixed quantity of resources. In the case of human beings, the situation is more dynamic. While there may ultimately be some need to limit resource consumption, we will do well to remember that humanity’s greatest resource is human ingenuity. Unlike most animals, we have the ability to discover and develop new resources. That’s why many people think (with horror or hope depending on the thinker) that humanity has transcended biological evolution and will now chart its own course.

          Anyway, suppose there is intra-species competition for scarce resources. Won’t that just cause the weaker members of the species to die off? Far from harming the species, wouldn’t this just enhance evolutionary progress in the long run?

    • Richard permalink
      August 6, 2009 8:06 am

      Take a few days to read Richard Dawkin’s “The Selfish Gene”.

  4. smartpeopleiknow permalink
    August 2, 2009 5:45 pm

    I’d be interested in knowing if this distribution holds for all deaths, or just natural causes. Also, if this distribution holds for certain types of death (e.g. cancer, or forms of cancer) vs other types of death (e.g., heart attack, respiratory diseases). I’d like to understand how much of a “Law” it is. (I am always suspicious of laws based of correlation).

    • gravityandlevity permalink*
      August 2, 2009 5:50 pm

      You’re right to be suspicious, and to point out that certainly not every type of death can follow this same distribution.
      Cancer incidence rates, as a whole, seem to follow the Gompertz Law. But not every type of cancer does. Incidence of testicular cancer, for example, peaks in your late 20s.

      Nonetheless, it seems that the causes of death which dominate our mortality rates follow the Gompertz Law. The reason why remains a mystery.

  5. ahier permalink
    August 2, 2009 6:26 pm

    Excellent essay! I think that there has been more progress in research relating telomerase and aging than we might think. Have you seen the Association Between Telomere Length, Specific Causes of Death, and Years of Healthy Life in Health, Aging, and Body Composition, a Population-Based Cohort Study? I think that telomoerase shotening and fused chromosomes are good suspects for the ‘criminals’ in aging.

    • gravityandlevity permalink*
      August 3, 2009 9:49 am

      Thanks for the link! I have looked at statistics like this before, and tried to put together some kind of theory of “shortened telomeres as criminals.” It became too hard to justify after a while, though. The specifics of “when does a short telomere lead to cancer” were too daunting for me.

  6. August 2, 2009 7:09 pm

    Makes one think of Nobel Prize winning chemist Linus Pauling. He believed large doses of Vitamin C on daily basis re-armed the immune system’s ability to defend against molecular level disease attacks. In a sense the Vitamin C’s stereo-chemistry effect was to fill in or act to enhance the “communication” ability of cells against dis-ease attacks. True or not, it is worth noting that Pauling contracted prostate cancer at age 91 and died as a result of its spread. Not sure that proves anything about his life being extended or not.
    There’s also the Hayflick Limit theory about a kind of clock in our cells that determines the number of times our cells will rebuild; supposedly some people’s Hayflick Limit is greater than others. I wonder if that setting can be adjusted by use of vitamin therapies, or some other chemistry of immune enhancement? I have read that scientists have looked into the idea of “resetting” the Hayflick Limit; one imagines a kind of DNA type clock setting deep in the helix of molecules. Not sure about accuracy of all this but it wouldn’t surprise me that one day we learn how to push physical chemistry of human body to its so-called “natural maximum”….Perhaps at that point we’ll be seeing Old Testament like people who live hundreds of years. Poor bastards! That’s a lot of long afternoons staring into the Void.

    • Dave Orel permalink
      October 7, 2009 11:25 am

      I would gladly take another lifetime or two. It’s been so much fun so far, but it’s like drinking from a fire hose. Too much to see and do to squeeze it into one lifetime.

  7. August 2, 2009 7:56 pm

    your post brought to mind the last stanza of Audre Lorde’s poem “The Black Unicorn”:

    “So it is better to speak
    we were never meant to survive.”

  8. matt permalink
    August 2, 2009 10:11 pm

    Shouldn’t our survivability rate be lower in the age 0-1 yrs range. This doesn’t seem to capture the real world statistics of infant mortality rates.

    • gravityandlevity permalink*
      August 2, 2009 11:00 pm

      You’re right about infant mortality. The Gompertz law is only really valid for ages 25+, after the serious risk of childhood diseases.

      • August 9, 2009 12:19 pm

        It wouldn’t surprise me if from 0 to 25 years the rate will behave in an inverse Gompertz law way (every X amount of years you would have half the risk of dying).

    • gwern permalink
      August 3, 2009 4:58 am

      If you read the linked PDF book chapter about aging, I think a number of the charts do show a high infant mortality rate (relatively speaking), with mortality rates declining to ~10 years old, and then beginning their endless climb back upwards (forming the ‘bathtub’ shape).

    • Darren permalink
      August 18, 2009 9:22 pm

      Thats an excellent point. The chance of dying during the first year of life isnt near zero nor lower than life between 2 to 5 years. The chart should be able to indicate that life is more fragile in the beginning than in the decades to follow. Additionally, this would be more noticeable when comparing survivability rates between industrial and developing nations. Earlier someone pointed out an interesting question that it hasn’t been specified if this Law pertains only to the natural causes of death and if its inclusive of all diseases. I would suspect that it only pertains to the decrease of survivability as a result of the aging process (natural only no diseases).

  9. Amit permalink
    August 3, 2009 12:04 am

    Its not just infant mortality that the Gompertz law fails to describe, as pedroj says, its also life expectancy once a person reaches around 80.

    That said, and considering that incidence of cancer follow the Gompertz law as you say, I think it appears more that the correlation is centered on the incidence and death from age related diseases, which have the time and opportunity to affect us from roughly 30-80.

    Once we hit 80, our survivability seems to increase since we’ve shown a particular resilience (perhaps genetic, perhaps lifestyle, both) against age related diseases. Our bodies continue to break down, but there’s no particular reason it seems we can’t live into the 100s, as some people do. While many doctors and scientists think there is a natural age barrier beyond which the body cannot reach, some believe that we may be able to rejuvenate the body so that we technically will never “reach” that age (limit), regardless of how long we live.

    As far as what is being suggested by the statistics, I think we should likely remain cautious and skeptical, and probably not draw broad conclusions from this data.

  10. August 3, 2009 1:14 am

    “I can say with 99.999999% certainty that no human will ever live to the age of 130. ”

    This is preposterous. In hunter-gatherer, pre-agricultural times, the human life expectancy was about 20 to 30 years. That’s what it was for Egyptians as well as Western Europe in Late Roman and in Medieval times. It didn’t rise to 40 years until around the year 1870. It reached 50 in 1915, 60 in 1930, 70 in 1955, and today it continues on the rise over 80 (a little more for women and a little less for men, – i.e see info: The rest of the world is retracing the European increment in longevity. What is the cause of this stunning, unprecedented, humanitarian transition? The germ theory of disease, public health measures, medicines, medical technology.

    From this perspective we can see that with the adoption of proper hygine, the germ theory of disease and the advent of modern medicine, human life expectancy has gone up rapidly in a very short span of time.

    In the past two centuries, life expectancy doubled, with most of the increase occurring in the last century. Today with the rapid advances in Biotechnology, Information Technology and the Nanotechnology revolution, this trend of increased life-expectancy continues. Judging from similar articles, this future shock will surpass most expectations. It seem to me that it is a grave mistake to attempt to predict human life expectancy without taking human technological progress into account, which the author clearly made a note to disregard purposefully.

    Last year, Scientists made a genetic breakthrough and carried out experiments on mice which made them live 45 per cent longer and left them free from tumours. And just last month Rapamycin, an immunosuppressant that enables elderly mice to live longer (“First Drug Shown to Extend Life Span in Mammals”) was unveiled. There is no telling whether this specific drug will play a role, however, I am very certain of the likely hood that humanity will overcome the 130 year life-expectancy barrier that author of this blog so readily disregards as extremely improbable.

    • gravityandlevity permalink*
      August 3, 2009 5:45 am

      Sorry andres, I only meant to say that the current distribution of lifetimes is extremely sharp, not that it will never improve in the future.

    • January 11, 2012 7:04 pm

      This is incorrect, based on a mistaken understanding of what “average” means for a dataset that’s not a normal distribution.

      Back in the day, the average life expectancy was rather low, yes. But this was because a *whole lot* of people died in childhood. Those dying children pulled down the average for everyone else.

      Another significant fraction of deaths were from wars, which killed *many* more people historically than they do now. Tribal societies tend to devote a relatively massive percentage of their population to warriors, even today.

      Childhood death and war killed about 50% of the population up to a few centuries ago, which is why families had to be so large – a population, on average, had to have 4 kids per couple to maintain their numbers.

      When you omit all the childhood deaths and war deaths, the average life expectancy rises *significantly*. People regularly made it to 60 back in prehistory, and it was not particularly rare to hit 70. Modern medicine, especially in the last century, has certainly raised the average, but it’s nowhere near as significant as the numbers naively make it appear.

      • Grammaticist permalink
        January 5, 2015 5:32 pm

        “This is incorrect, based on a mistaken understanding of what “average” means”

        Actually, it is 100% correct, based on precisely what we all know the word “average” to mean. Add up each person’s lifespan, divide by the number of people, and voila – an average. When somebody says that the average lifespan centuries ago was 30 or 40, we all know that this is precisely what it means. You seem to take issue with the inclusion of infant mortality into such calculations – why? Statistically, it would make no sense not to include them.

    • Calorta permalink
      March 27, 2012 8:53 am

      you seem to be confused with 2 different concepts: median life span VS. maximum lifespan. Median life span has increased a lot, as you said yourself, but not maximum life span. Humans have reached 120 yrs old a century ago, and despite all technology we still are not able to cross this mark. There are many interventions that can expand median life span, but none that can extend maximum, at least not for humans.

  11. Sebastian permalink
    August 3, 2009 6:58 am

    Dear Author,

    I very much greatly enjoyed this piece. I learned quite a bit, and it got the gears in my head turning. I’m quite interested in longevity so it really hit the spot for me.

    One piece of feedback, a kind of crazy one:

    I’ve written a number of pieces myself. If you come in dense in the beginning, you’ll lose *a lot* of your audience. A whole lot. I try to be not dumb, but even I saw this second paragraph and almost clicked the back button to find some pictures of kittens playing or whatever. Take a look:


    That’s heavy! And I try to be a pretty well-read guy, and try to learn a decent vocabulary, and don’t mind handling numbers and thinking. Most people don’t really do any of those four! Paragraph two, I reckon, might’ve lost a lot of really decent people for whom your piece seemed too dense.

    But that’s a shame! It was a brilliant piece! And I think anyone that comes here is probably smart enough to go through the numbers and words and thinking if they figure it’s worthwhile. Just most of us have ADD in the internet age.

    If you’d started a little slower, simpler, shorter, easier, then got more technical as you went on, it’d probably catch a few more people. Maybe the cops and criminals analogy right away? Or just allude to it, with a, “If I told you that your immune system worked like cops chasing criminals…” or some such. Just something easy to wade people in, with the numbers and words and terms and exponential increases later.

    I say this, because I am in admiration of the piece, and I should like if everyone got hooked and read it, and continued reading your works thereafter. Thanks for writing it, and best wishes.

    • gravityandlevity permalink*
      August 3, 2009 9:47 am

      Thanks for the tip, Sebastian. I wish I could somehow get statistics for what percentage of initial readers made it through a given number of paragraphs. Maybe that would follow something like the Gompertz law, with an average lifetime of about 2 paragraphs. : )

      • Dr Sam permalink
        August 3, 2009 12:22 pm

        your website stats will show you how long a particular visit lasted which will give you an estimate of if the readers read the article in full or not

  12. August 3, 2009 5:19 pm


    Can you explain how you calculated this? Was there some underlying
    assumption about the total number of humans who will ever be born?

    • gravityandlevity permalink*
      August 3, 2009 8:42 pm

      Sorry, I was pretty sloppy with that estimate, and then even sloppier with my explanation.
      If you assume that the Earth will hold about 50 billion people for the next 4.5 billions years (the age of the Earth so far), then there will be about 2.8 * 10^18 people. The probability that any one of them will live to age 130 (assuming, again, a static lifetime distribution) is e^(-0.003 e^((130 – 25)/10) ) = 4.8 * 10^-48. So the probability that anyone will live to age 130 is
      (2.8 * 10^18) * (4.8 * 10^(-48)) = 1 * 10^-29.

      If I took that seriously I would have had to put 21 more 9’s in the statement above. But I don’t take myself that seriously (and, as others have pointed out, there is a deviation from the Gompertz law for very old age), so I just put a bunch of 9’s to make a point. Sorry to be misleading.

      • MTarlton permalink
        August 4, 2009 1:15 pm

        March 27, 2009

        “Officials in Kazakhstan say they have a found a woman who will this week celebrate her 130th birthday, making her 16 years older than the oldest known human currently living.
        Sakhan Dosova – a mother of ten – says she has never visited a doctor nor eaten sweets. She is addicted to cottage cheese and puts her longevity down to her sense of humour.

        Her remarkable age came to light during a census in Karaganda in northern Kazakhstan. Demographers were astonished to find that she was also on Stalin’s first census of the region in 1926 when her age was given as 47.”

        Read more:

        Follow up:
        “woman thought to be the world’s oldest person at 130 has died after slipping on the bathroom floor of her new flat.
        Sakhan Dosova broke her hip in a fall last month and never recovered.
        She had been given the flat by officials in Kazakhstan who were embarrassed she was living in overcrowded conditions with her impoverished family. ”

        Read more:–slipping-bathroom-new-flat-given-celebrate-age.html#ixzz0NEtaJwP9

        • May 17, 2011 4:22 pm

          Extraordinary claims require extraordinary evidence. Old people lose track of their age very easily, especially in poorly literate or turbulent areas.

          Historians actually estimate literacy rates in places like Roman Alexandria based on how biased ages are from the correct distribution – the more tombs or memorials recordings ages which end in 0s or 5s, the lower the literacy rate. (This is similar to statistical checks on voting fraud – looking for # of totals votes which are too regular and insufficiently random. Humans are bad at making up random numbers.)

          And people lie about much smaller things for much less gain than an entire apartment.

  13. August 5, 2009 2:31 am

    I find this fascinating. This formula would suggest that to extend our lives we should be figuring out how to reduce the risk of death in 25 year olds from 1/3000 to 1/12,000 or more. That way, with the doubling risk of death at 8 year intervals, by the time you hit 100 years old, your risk of dying the next year would only be 12.5% instead of 50%.

    I find this formula fascinating. You could make the old folks live longer, by doing nothing more than making the young folks die less often. I often think of dying young as a lightening bolt phenomenon. But this formula would suggest that dying young is the gradual loss of cops as well.

    Perhaps our bodies come into this world with their maximum number of cops and we start losing them immediately and continuously our entire lives. And all the money we spend on end of life is delaying what our clocks set in to motion on the day we were born.

  14. August 5, 2009 5:56 pm

    No matter what they numbers we are fighting a battle we can not win. The goal in life is not to live the longest life but to live the best life we can in the time we have. Anyone can die at any time. And that may be a good or bad thing. Because who really knows what happens next?

  15. August 6, 2009 9:19 am

    Blanding’s turtles do not age as most other animals do ( ), and almost always die of either accidents or infections. Does Gompertz law hold for them, too? If so, we can assume that the primary reason for it is an immune system that declines over time. If not, we can assume that there is something (probably genetic) that drives an “expiration clock.”

    • gravityandlevity permalink*
      August 6, 2009 9:47 am

      Wow, thanks for the link. I looked up this paper, written by the biologist who is the protagonist of the story you linked to: (sorry, I couldn’t find a free version online). He seems to claim that measurements of mortality rates among his turtles are very much consistent with a Gompertz law, at least for ages 13+. The remarkable thing is that instead of their mortality rate doubling every 8 years, it seems to double only every 16 years.

  16. yann permalink
    August 7, 2009 7:54 am

    Dumb question: how did you calculate the survival probability for the “accumulated bolt” model?

    • gravityandlevity permalink*
      August 7, 2009 8:56 am

      That is very much not a dumb question. In fact, I’m not even sure I could calculate the probabilities exactly. I got the “accumulated lightning bolt” graph above by writing a simple numerical simulation, with 100,000 simulated lives. For each year of a simulated life, a random number was drawn that represented a 1/16 chance of being “hit by a lightning bolt”. Once a person got hit 5 times it was counted as a death.

      Mortality rates for a “single lightning bolt strike” model are pretty easy to derive, and a “double lightning bolt strike” is also possible although more tedious to work out. I imagine it’s possible to give an exact analytical expression for any number of “accumulated lightning bolts”, but it was more work than I wanted to do. : )

      • yann permalink
        August 7, 2009 3:45 pm

        ah ok, thanks for the reply.
        I had tried to work your model out using combinations but ended up with a formula that excel refuses to compute. I’ll try to re-view it, was hoping you had the answer:)

        • D.O. permalink
          August 16, 2009 9:05 pm

          Must follow a simple Poisson. Probability to receive less then N lightning bolts (an therefore survive) is P(t) = (1+ (lt) +(lt)^2/2+…+ (lt)^(N-1)/(N-1)! )e-lt. t is time, l is rate in inverse t units.
          [Author’s edit:]

      • gravityandlevity permalink*
        August 16, 2009 10:49 pm

        Thanks, D.O. You’re right, this is the cumulative distribution function for the Poisson distribution.

        • August 19, 2009 7:28 pm

          Actually, what you’ve written is the Erlangian distribution for k = 5. The Poisson distribution is a discrete distribution; the cumulative distribution function would give you the probability that there are no more than k events in an interval of length t. What you’ve given is instead the probability density function for the Erlangian distribution–the probability density that it takes time t to accumulate k events in a Poisson stream.

          The reason this yields people living up to 160 years in substantial numbers is that k = 5 is a relatively low index for the Erlangian distribution. You can increase k (and also increase the rate of events from 1/16 to 80/k), and get narrower and narrower distributions. In the limit as k grows without bound, you approach a normal distribution. However, the standard deviation goes as 80/k3/2, meaning that when k gets very large, essentially everyone dies at exactly 80.

          I think you are right that it doesn’t mimic Gompertz’s Law, but the reason is not just a matter of spread. The shape is probably not right, although I haven’t checked it out yet.

  17. Matt Powell permalink
    August 9, 2009 2:27 am

    I think a useful metaphor is to consider is the difference between how error behaves in digital systems. As a digital signal is exposed to noise, that error accumulates as a probability of corruption.

    Shannon’s Theorem says that you can reduce that probability of corruption nearly to zero, but to do that you have to encode your signal. Basically you’re trading chance of data loss for time and energy spent encoding and decoding.

    The little bacterium doesn’t put much energy into encoding his signal. He doesn’t age because his signal is basically a single bit and when he encounters the slightest bit of noise he’s toast.

    We humans have substantially more redundancy in our signal so we can take a lot more damage before it becomes unrecognizable. However, that damage is always occurring at a non-zero rate and it’s going to be increasingly expensive in time and energy to reduce it further.

    • May 17, 2011 4:26 pm


      The extremophiles might disagree with you ( especially the radiophiles. in particular can have its entire genome shattered into countless pieces by 30,000 grays (5 is enough to kill a human) and still put itself back together and live happily.

  18. Alonso Alvarez permalink
    August 11, 2009 7:33 am

    Great article!
    If you don’t mind, I will comment it in Spanish in my blog.

    • gravityandlevity permalink*
      August 11, 2009 7:40 am

      Por supuesto! Me alegro que lo hayas disfrutado. :)

  19. D.O. permalink
    August 16, 2009 8:50 pm

    Another possible model for Gompertz law. The life is a battle between attackers and defenders. Defenders are able to defend the castle of life up to some point in the severity of attacks. If the severity of attacks has an exponential tail (maybe as a result of binomial distribution of the large number of small “attack agents”) and the defence level is linearly falling (say, as a result of previous attacks killing off some of the defenders) you will have the Gompertz law. Now, it might be possible that 2 scenarios map on each other (that is different only by metaphorics), but thinking of it for the better part of 5 minutes i don’t see how.


  20. jimmyc permalink
    September 15, 2009 2:40 am

    I like the idea of lifetimes and was struck that the death rate doubling every 8 years is close to the 7 years it takes for the human body to replace most of its cells. There are a number of avenues to pursue here:
    1. We have some cells (weak without care), which when not cared for by simple means can lead to much more rapid demise. Thus we invent medicine to counter the simpler things such as diseases, while eating good foods in moderation to further promote health.
    2. The most complex cells (strong and long lived) are susceptible to accumulated damage or more simply time. Radiation, the atmosphere, and time provide a flux of chance encounters which promote cell decay or replication errors, thus capping our mortality.
    3. the issue of the “weakest” cells (strong but overexposed) or those most susceptible to damage.
    4. Then there are the weakest cells with the shortest lifetimes that are easily replaced and offer little in accumulated damage.

  21. June 9, 2010 2:56 am

    the reason it follows exponential decay only after age 30 because depletion of stem cell in the body.

    stem cells create stem cells, before age 29

    stem cells create progenitor cells age 29~42

    progenitor cells create differentiated cells age 42-60

    differentiated cells divide (3 more cycles) 60~85

    differentiated cells into senescence (2 more cycles)

    ===> death

  22. Nasdaq7 permalink
    August 5, 2010 11:33 pm

    When I was younger, I never gave these things much thought. I now realise how important this knowledge is.

  23. July 27, 2011 1:08 pm

    Personally, I’m glad people don’t live so long. I just can’t fathom having to listen to some of the talking heads on TV and Radio for another 50 or more years.

    • July 27, 2011 10:44 pm

      That’s a really terrible reason to condemn billions of people to decades of suffering and then death; I hope you are only making a poor jest and that isn’t a real reason.

      • September 24, 2011 11:03 pm

        And how many people would you be condemning to suffering and poverty by hoping people live longer? JohnathonPDX has not said he hopes people die young, just not longer.. Food and many necessities of life are a limited resource. Its simplistic to believe that extending life to say 200 would be a GOOD thing for humanity and well being. Maybe for the privileged few, but the majority of the world is already condemned to suffrage and death.

        • Anonymous permalink
          January 9, 2012 11:27 pm

          That falls under “problems we’d love to have”. Given a world where everyone will die and we don’t know how to fix that, and a world where some people will die and we just need to extend the fix to more people, I’ll take the latter every time.

  24. Daniel permalink
    January 11, 2012 11:26 pm

    Just to clarify, under the lightning bolt theory your lifespan would follow an exponential and not a Poisson distribution. The main point is still correct though, and your probability of survival past age t would be e^(-c*t) for some c > 0.

  25. January 12, 2012 5:38 pm

    Great analysis of the mathematical laws describing human mortality rates. I think the “cops and criminals” notion is fairly plausible at the immune system level, where random mutations are an environmentally given hazard and cop-style repair works well while we are young, but diminishes until the odds tip dramatically against survival. Nonlinear processes of growth and decay abound in biology. Super-exponential processes can bring about even sharper growths or decay, of which death beyond a certain age (~80) is a good example.

    Inspired by this analysis I have written a post on my own Blog about different types of growth and decay processes:

  26. July 31, 2012 1:43 pm

    The fact that people are not biologically immortal keeps the heat on infectious diseases in that they must constantly infect new people just to be able to continue to exist, since each infected person will die in due time even if not killed by that disease.

    There was also the point that, if I remember right, Steven Pinker made in one of his books. Conceptually, there’s no reason we can’t be biologically immortal (i.e. death would only come from accidents or similar events, no such thing as death from “old age”). However, even if we were, we would still be susceptible to lightning bolts, car crashes, poisoning, lethal disease, and the like. Therefore, in this situation, putting infant mortality aside, then all else being equal, you’re more likely to be younger. Given that, it’s in the interest of the species or family group for more resources to be diverted away from the old and towards the young. So it makes sense from an evolutionary perspective that your caloric requirements decrease as you age, eventually coming to a point where they are zero, since you’re dead.

  27. July 6, 2013 3:34 pm

    Is the formula correct in the printed article? Seems like it should be e to the -.0003 instead of -0.003. All stats work out if that’s the case. I couldn’t find another listing of the formula anywhere in my subsequent searches to confirm my suspicions.

    • Brian permalink*
      July 8, 2013 7:07 pm

      No, I think it’s correct as written. Your version of the formula would put the average lifespan at around 105 years, rather than 80.

      • Marco permalink
        December 30, 2013 2:34 pm

        – 0.003 is correct as you say, Brian, the confusion comes from another typo. The probability of dying during the next year for a 25-year-old American is 0.3%, not 0.03%. 10 times more likely to die, sorry for the bad news😦

  28. suzanne permalink
    October 14, 2013 3:17 pm

    I don’t trust a writer who misspells “minuscule.”

    • Brian permalink*
      October 14, 2013 3:40 pm

      Now that I corrected the typo, do I become more trustworthy?

  29. December 30, 2013 11:21 am

    All the more reason why Aubrey De Grey’s talk at TEDxSalford seems relevant about maintaining your body..

  30. Greg Shubert permalink
    December 30, 2013 12:18 pm

    Small correction: In the 4th paragraph of the “Cops and Criminals” section, you write, “The difference between 14 and 12 doesn’t seem like a big deal, but the result was that your chance of dying during a given day jumped by more than 10 times.” I think it should read “more than 7 times”. The ratio of exp(-12)/exp(-14) = exp(2) which is 7.4; matched by the ratio of the two mortality rates you calculated (8E-7 to 6E-6) which is 7.5.

    • Brian permalink*
      December 30, 2013 10:02 pm

      Thanks for the correction!

  31. jim permalink
    January 1, 2014 1:14 pm

    Thanks for this great article. Your cops vs criminals analogy is a predator-prey system (i.e., Lotka-Volterra). This system is robust enough to describe not just one species preying on another, but both preying on each other. It can also describe competition for the same resources or symbiosis. It has also been generalized to n species.

    This may seem like it’s getting too complicated. Perhaps it is, but my suspicion is that some simple properties might be found. Anyway, my point is that in this case the population dynamics (of “cops and criminals”) is described by logistic functions. The Gompertz distribution is a specific case of the generalized logistic function which also makes an appearance in physics, e.g., the Fermi-Dirac distribution and Woods-Saxon distribution.

  32. Jason permalink
    March 17, 2014 7:00 pm


    This is one of the largest leaps in logic I’ve ever seen.

    Please don’t leave academia. We can’t have you making decisions that will affect the quality of life of anyone.

  33. José Antonio permalink
    September 1, 2014 11:16 am

    An interesting article Brian. I think that there is little hope of ever “explaining” Gompertz’s law in a strong sense, I mean deriving it from some simple stochastic mechanism. It seems to me that the situation here is rather like trying to “explain” why IQ’s in the population follow fairly well the nomal distribution, I don’t expect here either a simple explanation.
    Another thing that occurs to me is that this law refers to how the survival rates of plenty of people behave, not how the survival rate of any give person behaves. These are different things. Only if we suppose that the populations that satify the law are made up of very similar individulas could we go from the whole to the one (typical). Perhaps Gompertz’s law also reflects important genetic differences among individuals.
    And finally just for clarification the “accumulated lightning bolt” model follows a gamma distribution with shape parameter 5 and scale parameter 16. Your plot can be reproduced with the Mathematica sentence Plot[1 – CDF[GammaDistribution[5, 16], x], {x, 0., 200}]

    • Brian permalink*
      September 1, 2014 11:25 am

      Thanks Jose Antonio. I agree with everything you said, except that I hold out more hope for a satisfying rationalization, if not a full “explanation”, of the Gompertz law. In particular, there are satisfying derivations of the normal distribution (as arising naturally from the sum of many independent things, which is not hard to imagine happening for IQ), and it would be nice to have at least a comparable level of understanding of the Gompertz Law.

      • José Antonio permalink
        September 1, 2014 11:38 am

        I’m perfectly happy with “satisfying rationalization” to pretty much express our hopes, it was only that I felt “explanation” was too strong here. For instance I do not think that the relation between the normal distribution and the sum of (pretty) indendent factors model really explains the IQ’s distribution, it really expresses our vast ignorance about the factors and their possible interactions. But it is true that Gompertz’s law is much less universal than the normal law, so perhaps it could be hiding much more.

    • September 1, 2014 11:54 am

      I don’t think there’s any mystery about IQ these days: the heritable half is caused by thousands of genetic variants of small effect (, so you’ll get a normal distribution; and likely the nonshared environment half is also a lot of small random effects too.

      • José Antonio permalink
        September 2, 2014 3:55 am

        Well, I’m not a biologist, but more in the mathematics-physics area, so that may be perfectly right. But from the fact that we can partition the factors affecting IQ in two broad classes, and saying that there are a lot of them in both classes, and that their effects sum (this in fact this we deduce from the normal distribution of IQ, not the reverse, as far as I know), and that there are not stong non-linear interactions among them (again this I suppose we infer from the normal distribution we observe), I do not see that there is not any mystery as you say. Don’t forget that the normal distribution is the first-guess distribution to model errors for example, where we ignore almost everything.

  34. Daniel permalink
    November 28, 2014 12:45 pm

    I’m surprised that no one has looked at this issue from the opposite end of the spectrum. If one accepts that the entropy of the universe will lead to an ultimate state of temperature equilibrium (a so called heat death) then it follows that either two things must be true–either the human species must die out long before that happens or they must evolve into something that we humans living today would not recognize as human. What is true for humans must be true for anything else we would recognize as life today.

    Thus, there is an entropic limit to life which can be defined as the eventual age of the universe minus some iota of time. Once that is accepted as true then the actual life span of any individual life is determined by their entropic rate–how fast they decay relative to the entropic limit of life itself. A 1000 year old rock has a slower entropic rather than a 400 year old tree, which in turn has a slower entropic rate than a human, which in turn has a slower rate than a bee.

    This would imply that what we call disease is more accurately conceptualized as entropic differentials between life forms.Viruses and parasites have high entropic rates relative to their hosts, causing disease. Lifestyle and/or genes at the margin effect the entropic rates of cell death, leading some people to live longer than others. Viewed like that, Gompertz Law measures the ratio between various entropic rates that leads to the death of an organism. If this perspective is valid then it would mean that Gompertz Law implies two things.The first is that iota of time frame which separates the entropic limit to life from the eventual age of the universe is to a large part defined by the entropic ratios that lead to the death of individual life. More intriguing, it would suggest that a real increase in the actual range of the human life span would slow the entropic rate of the universe down. This is because in order for Gompertz Law to function as a fixed constant as life spans increase the decrease in entropic rate must cut across the board. That is to say that as the entropic rate of the human slows down the entropic rate of viruses and other organisms must evolve into a slower rate as well.

    If one looks at chemistry–as one should–as bound by the laws of physics then one doesn’t need to turn to immunology and cops and robbers to explain Gompertz Law. Basic physics does nicely, thank you very much.

  35. August 28, 2015 9:41 pm

    Then the probability of them missing a criminal for an entire day decreases to…
    should be:
    Then the probability of them missing a criminal for an entire day INCREASES to…

    • Brian permalink*
      August 28, 2015 10:13 pm

      Thanks! Someone caught that one before, and it should be corrected now.

  36. December 9, 2015 4:28 am

    Opened my closing eyes..

  37. May 23, 2016 11:21 am

    Great article! I just didn’t understand one part: “The likelihood of them missing a particular spot for an entire day is given (as you’ve learned by now) by the Poisson distribution: it is a mere e^{-14} \approx 8 \times 10^{-7}.” How did you come to this number?


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