melannen: Commander Valentine of Alpha Squad Seven, a red-haired female Nick Fury in space, smoking contemplatively (0)
melannen ([personal profile] melannen) wrote 2016-04-10 11:28 pm (UTC)

It depends. With a replacement rate of 2 (I'm doing that just 'cause the math comes out cleaner, but it doesn't make a huge difference) you could have a society where everybody has two kids, or a society where half the population has 2 kids, and 1/4 have 0 kids, and 1/4 have 4 kids. Or a society where 3/4 the population has 1 kid, and 1/4 has five kids.

But your average still has to be 2, and so for every family with less than 2 kids, there has to be a family with more than 2 kids. And the families with more than 2 kids will have more kids in them, so it actually comes out that the more families you have that are below the average birthrate, the more kids are from families with more than the average number of kids.

This may seem paradoxical, so here's an example with numbers:
.25% of women have 1 kid.
.5% of women have 2 kids.
.25% of women have 3 kids.

This comes out right - .25 + .25 + .5 = 1 family, and .25 * 1 + .5 * 2 + .25 * 3 = 2 kids.

But because the large families have more kids per family, when you look at number of siblings per kid instead of kid per mom, you get:
.125 of kids have no siblings
.5 of kids have 1 sibling
.375 of kids have 2 siblings.
--so the average number of siblings per kid is actually 1.25, even though the average kids per family is still 2.

If your % of hypothetically-childbearing people with less than two kids is greater than 50%, then over 75% of your children need to come from families with 3+ kids in order to make an average of two kids per family. In the example where 3/4 of families have 1 kid and 1/4 have five, the average siblings per kid is actually 2, and about 2/3 of kids are from large families.

This is why fantasy worlds with mostly only children and no large families worry me.


Basically, birthrate = average number of kids per childbearing person, so for every family with less than the average, there needs to be more kids born in a family with more than average, to balance out to the average.

But the larger-than-average families have more kids per family, so siblings from those families get counted more than once in the sibling-per-kids count, since they are siblings to more than one other kid, so the more kids in a large families, the more siblings-per kids. And the more small families, the more kids in large families are needed to balance them, and the larger the imbalance in average kids per family vs. average siblings per kid. This generalizes to any birthrate or distribution of family sizes.

...I would write out the general equations but I don't have the math for it today, and yet would rather try to work out the math from scratch than try to remember enough to google it, because it's that kind of day...

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