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Using data from US vital statistics I created a simulation that was based on the number of children a woman has over her lifetime as follows (out of 10000):

0   4100
1   5800
2   8200
3   9380
4   9780
5   9900
6   9960
7   9985
8   9995
9   9998
10  10000

So for example, out of 10000 women 4100 will have no children, 1700 will have 1 child, 2400 will have 2 children and so on. However, when I run the simulation, the average number of children per woman is 1.28, a lot lower than expected. According to real demographic data, women in the US have an average of 1.87 children over their lifetime.

To make the value come out as it should (around 1.87) I had to use radically different values:

{ 3600, 4800, 6400, 7680, 8780, 9900, 9960, 9985, 9995, 9998, 10000 }

Where am I going wrong?

enter image description here

In this table why is the 15-50 rate 1.29, but when broken out, the aggregated rate is 1.9?

Tyler Durden
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    Please explain how you obtained $1.87$ when the mean of the distribution you present is $1.29.$ What's the source of your data? They disagree somewhat with [recent US Census data](http://www.census.gov/hhes/fertility/data/cps/2010.html). – whuber Aug 12 '16 at 01:37
  • @whuber 1.87 is the actual measured value from vital statistics. You kind of answered the question. If the mean of my distribution is 1.29, then those values must be wrong somehow. – Tyler Durden Aug 12 '16 at 01:42
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    I agree with @whuber the mean of the presented distribution is 1.29. Quickly checking some WHO data on fertility rate, the U.S. total births per woman is approx 1.9. What's the link for the U.S. Vital Statistics data? (You might be misreading it.) Another thought: I recall from demographic stuff I've done that an important distinction is cohort vs. period. Eg. period life expectancy for 2016 would assume 2016 mortality rates over his/her entire lifetime. Cohort life expectancy for someone born in 2000 would use 2001 mortality rates for year 1, 2002 rates for year 2 etc... quite different! – Matthew Gunn Aug 12 '16 at 01:44
  • The issue centers around exactly what your data mean. I suspect the tabulation you present is for women of all ages. Certainly women 15 to 19 years old are averaging far less than 1.87 children. Their statistics are going to skew this table, suggesting you might be misinterpreting it. – whuber Aug 12 '16 at 01:45
  • 1.29 is implausibly low, it would imply huge population decline over the long run, and my strong priors are that the U.S. is not THAT far below the replacement rate of approx 2.1 births per woman. – Matthew Gunn Aug 12 '16 at 01:46
  • @Matthew The value 1.29 nevertheless is correct when you average in *all* women of childbearing age, regardless of age. The reason is that a great many of them have not finished childbearing. It's a censored dataset. It is quite clear that the data presented in this post are not "lifetime" figures in the sense of a *full, completed* lifetime. – whuber Aug 12 '16 at 01:48
  • @whuber That could be, but then it wouldn't be "...number of children a woman has over her *lifetime*" as quoted in the question. The OP may need to look up or link to the data he's using and be *very* careful about definitions. – Matthew Gunn Aug 12 '16 at 01:50
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    @Matt It comes down to whether "lifetime" means "for women who have reached menopause" or "for all women, even teenagers." The figures in this question are for the latter, but the simulation needs the former. (+1 for a stimulating question, BTW.) – whuber Aug 12 '16 at 01:51

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