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Notes -
That’s what I was looking at. Number of lives at 83 over number at 79 should be percent surviving.
How’d you figure?
I was not as clever as you and simply took the probability of surviving each year from ages 78-81 and multiplied them. That gave me a combined probability of survival to age 82 of 77.58%.
I guess the entire difference is down to if you should index by 78 or 79 then. The table is on "Exact age," so I guess @netstack was right to use 79. He even rounded down from there to "about 24%" from 24.6%, so it probably is about right interpolating. As the comments above point out, additional factors probably are more important at that level of accuracy already though.
I actually have to deal with these tables for work, and you'd calculate by 78 or their stated age. At least that's how the professional economists do it for expert reports. If you graph it it's easy to see why — the probability doesn't follow a set function but wanders based on extrinsic factors and random variation. For example, a newborn's chances of dying in the next year are the equivalent of a 50 year old man's. But it drops sharply after one year and continues dropping until age 8, when it starts permanently rising. There's then a jump around age 16, probably due to driving (and poorly at that), etc. In other words, it's derived from actual data. And the actual data can't be granular down to the day because it would be a nightmare to calculate and would probably end up wonky because of limited sample size (how many people aged 17 and 301 days die in a given year?) So they base the data on anyone who is a given age, even if they may be nearly a year apart. So if you're 78 and 240 days then your probability is what it is for 78, full stop, no rounding up. On your 79th birthday you use the higher number.
A probability density function does not have to be transcendental to be able to integrate for a cumulative probability. I have no way of knowing what convention you use for work, and there might be good reasons to use a left hand rule numerical integration for your application, but there is nothing magically more correct about a left hand rule integration. Probability of death is strictly increasing by the time you reach 78. If you use the left hand rule to integrate over a region where a function is monotonically increasing, you will systematically underestimate the area under the curve.
With respect to the Social Security Actuarial Life Tables, it is in fact meaningful to talk about regions between nodes. The numbers in the table are not raw population deaths. In fact part of the methodology for producing the table in the first place is reconciling five year central death rates and exact age one year probabilities. Once you have meaningful nodes 365 days apart you are not dealing with a sampling problem if you want to estimate a value at 301 days—you are dealing with an interpolation problem. They do anticipate people using the table for intra-node calculation. From the methodology notes:
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I was also surprised the number was that large, but also got 24.6% both taking the ratio of "Number of lives" and the complement of the product of the complement "Death probabilities."
Interestingly, in the notes they include cause-specific ultimate rates of reduction, so you could exclude the violence category if you are only considering health related causes.
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