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~~Take a life table, check the column that says lx, that means how many people of the model are alive with x years, normally l_0 = 10000. Then go down until you found lx = 5000 or lx = 1/2 l_0, that x is your life expectancy at birth. If you want to know the life expectancy at any age, look for the value of lx at that age, then look in how many years that value half, that's the expectancy. Source: I'm an actuarie and study that in college, but don't ask me to go deeper because I don't use that at my job.~~
I was wrong.
That marks the median age of death, but I don’t think that’s the definition of life expectancy (though maybe the term is used loosely or imprecisely to mean the median age of death or the average (mean) age of death in different situations...). The definitions of life expectancy I found claim it’s the mean age of death, which would make sense because the expected value of a random variable is the arithmetic mean. That said, the median on the life tables that I have found seem to correlate much more closely to the age of 77 versus the life expectancy at birth which is much lower (78-79 for the median and 74 for the life expectancy at birth for 2020 data)… but the actual paper is behind a pay wall so I have no idea what they’re actually computing for 77 years of life expectancy… my guess is that it’s the median and not the mean, but maybe they’re considering people over a certain age or something… either way, the mean / median getting confused is an issue and I wish people were more clear about what metric is actually being communicated.
Okay, I couldn't look at this table when I responded last night (I thought you were referring to the zip files, not the PDFs at the bottom). Got a chance to look at them on my computer today!
Would you call the point where I_x = 1/2 I_0 the life expectancy at birth? In the life tables you link to (direct link to the 2020 table) there's an "expectation of life at age x" column which differs! My understanding is that in official metrics of "life expectancy" they usually mean the "life expectancy at birth", which is calculated in the "expectation of life at age x" column in this data set, do actuaries use a different definition?
In this table "life expectancy at birth" is estimated at 74.2 for men in the USA in 2020. This is calculated in this table by computing T_0 / I_0, which is the arithmetic mean for the ages of death in this period. The estimate for the median age of death in this table is between the ages of 79 and 80. There's about a 5 year difference between these two numbers, and furthermore only about 40% of the population of men has died by the ages of 74-75 in this table, which is quite different from 50% if we assume "life expectancy" is this arithmetic mean. These are pretty big differences, and I really wish people / articles would be more clear about how the number they're quoting was actually calculated and what it means! The estimated median age of death from the point I_x = 1/2 I_0 is a useful measure too, but I have no idea what a random person or article intends when they say "life expectancy" :|. I've grown to deeply distrust any aggregate measure that people discuss informally or in news articles... It's often very unclear how that number was derived, what that number actually means in a mathematical sense, and if it even means anything at all.
I think im going crazy because I was sure that the ex was the easily calculated as I told before, but I just checked a couple of different mortality tables and the math doesn't check out. I have no worked with life tables since college, but I'm going to ask my boss, that used to work with life insurance, about it tomorrow and follow up with his answer.
I just talked to my boss and he didn't knew where the fuck I learned that because it had no sense. Maybe I saw some wicked up life tables where for some reason it worked like that. The formula is sum(i=x,w) l_i /l_x, that means (l_x + l_x+1 + ... + l_w) / l_x, where w is the maximum life of the table.