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I am trying to find out what the world distribution of life expectancy looks like.

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    $\begingroup$ Statistician here. The answer would depend on what, exactly, you're asking about. The answer you have now wasn't explicit about what precise question it was answering but seems to be answering "if I randomly chose a country, each with equal probability, and looked up its life expectancy, what would the distribution of that life expectancy look like?". I expect you want something different to that (e.g. perhaps something closer to "if I randomly chose a person from the world, what would their life expectancy at birth have been?"). The most interesting/useful questions may be harder to answer. $\endgroup$ – Glen_b Nov 3 '15 at 6:35
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Let's see!

I took the most recent WHO data from here and did a quick an dirty analysis in R. Here is the histogram as well as a normal distribution with the same mean and standard deviation as the actual data:

enter image description here

Does not look very normally distributed. In fact, the shapiro test confirms this impression:

Shapiro-Wilk normality test

data:  df$life_expectancy
W = 0.94141, p-value = 4.338e-07

Edit

Commenters requested to weigh the life expectancy by population size.Well...the result is pretty ugly:

enter image description here

Apparently, the large populations of China and India produce disproportionally high bars. However, keep in mind that we are using averaged values which means that all the variation within countries is not represented by the histogram. We would actually have to have data on individual ages of death for a random sample of the world population to finally settle the question. :/

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    $\begingroup$ Have you tried weighing by the number of inhabitants? It's true that there's no reason to expect a gaussian. But the distribution you show also seems rather weird :) $\endgroup$ – cel Nov 3 '15 at 5:21
  • $\begingroup$ Who kills all those people just before they turn 80? $\endgroup$ – BlindKungFuMaster Nov 3 '15 at 8:55
  • $\begingroup$ It would definitely be interesting to see how the distribution looks when weighted for population. $\endgroup$ – 123 Nov 3 '15 at 15:15
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    $\begingroup$ I'll see what I can do $\endgroup$ – Hav0k Nov 3 '15 at 18:51
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    $\begingroup$ Both graphs can't answer the question, since they're done by the mean in each country. So, all people from India and China fall in a single column each, when with real data they would be more spread, and, who knows, make something closer to normal distribution. Though it's probably hard to get the real data. $\endgroup$ – Rodrigo Nov 6 '15 at 21:53
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A few people thought it would be interesting to see what the distribution looked like if we plotted the number of people dying at each age, so I took data from the SSA (which admittedly isn't global data, but it's probably fairly reflective of the world overall) and plotted the number of deaths per 100,000 at each age.

This looks like it makes sense - the high death rate at birth, the plateau during the 20-40 range, then the rapid escalation past middle age, and eventual decline shortly after reaching the average life expectancy range.

However, while it looks a lot smoother than plotting the averages of individual countries, it's definitely not normally distributed.

EDIT:

If you extrapolate the above death rates using the global life expectancy of 71.0 instead of US life expectancy, the chart looks roughly the same, but is adjusted to reflect what the death rates of the planet most likely look like.

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