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March 9th, 2017, 12:56 PM   #1
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Median age or Life expectancy? Population that ever lived

Hey guys, first post here

Say you have a graph that gives the population of the Earth (or any country) over time. The total population ever lived (in an era) is actually easy to calculate: all the deaths in the era plus current population (and plus all the emigrants if you are going to factor them in as well). Some say, you have another way of figuring this out, by getting the primitive and calculate the area beneath the graph (I will shortly come to my question, this is not a lecture I am just giving some helpful background). This you can do with for example paint net, with the magic wand.

So I made an equation:
All deaths + current population = area of graph
However, one will notice the area of graph is way higher. First it is crazy, but then you notice the obvious thing: it means that for the next t a total new population would have happened, which is wrong ofcourse as some live on.

So, I made the equation further:
All deaths + current population = area of graph / x
And the thing is, I want to know what x is. (average) Median age, or (average) life expectancy? This may not be really a mathematical question, but I am sure some people here are good statistians (I spelt that wrong) who know how to look at this.

I may be unclear, if so you can ask me to clarify!

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March 9th, 2017, 01:59 PM   #2
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It is unclear to me at least.

All deaths + current population = area of graph????

What are the labels on the axes of the graph. Probably time and number of people. Then the area represents people years. Is that meaningful?

What is x? You do not say.

In fact it is not clear what you even want to measure.

Is it population alive at a given time?

Then your equation should be

Starting population + births + immigration - deaths - emigration = ending population.

Is it the current population alive plus those now dead, a sort of cumulative population (not that I see what utility such a number might have).
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March 10th, 2017, 05:56 AM   #3
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Yeah I have been unclear. The point I want to find is: can I find the average life expectancy or median age, with these numbers? Why do you say? Well, let's say the average life expectancy is infinity, then we would have the total population that had ever lived. However, that is not the case. What I am saying is that, or atleast what my thoughts are, is that it should be dependent on the average life expectancy how many of the total population that has ever lived would still live today.

This is something which has sprung to my mind once, this is not for homework or study or whatever, this is something I would like to discuss with other interested people.

I think you are right about the people * years, but I think it does not really matter, because: propose we have another graph, that of velocity in the y ax and time in the x as. Well, the distance would be the area. But what actually happened is that you calculate the sum of the velocities of every point along the graph, multiplied by time frame (so every second). If we now go back to my graph, with indeed the population on the y ax and time (in years) on the x as. But if we divide the number of people * years by an number in years (for example median age or life expectancy), we get the population.

That was something that I was thinking about.

Let's make an example:
Let's say we have a population of 10 million, that grows with 1% every year (birth rates 20 per 1000, death rates 10 per 1000, emigration and immigration convienantly zero per 1000).

If we go measure the total population that ever lived in the upcoming 10 years, we take the average population (which is one year = 5), which is then 10.51 million people. Total deaths in 10 years is death rate/1000*average population*number of years =
0.01*10.51*10^6*10=1.05million deaths. Current population is 10*1.01^10 = 11.05 million. Total population ever lived is then around 12.10 million.

If we make the primitive of the formula 10*1.01^t we get (1/ln(1.01))*10*1.01^t. At t=10 this would give 1110 million people (*years). Now let's make the original equation I had:

total population ever lived (deaths plus current population) = primitive /x
x = primitive / total population ever lived = 1110/12.10 = 91.8 years.

And now I am wondering: what is x? Is it meaningless? Is it the life expectancy?

I had a few days back made a formula, for the life expectancy:
k = (total population ever lived in era/median population of era)-1
average life expectancy of era = number of years in era / k

Median population is the population at the year where there have been living as much people before ever lived as there will be people ever living in the upcoming years.

This formula has been suprisingly good, I have come to get pretty believable numbers, like I calculated the life expectancy of Iraq with this formula and it was about 61 years while the life expectancy in Iraq is actually 67 years. However, I find it a weird formula, like I can't really explain it why this variable, why that variable. I was just like one night making this formula, like I knew totally what I was doing but now I don't. I just fill in the formula.
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