Are a majority of people an oldest or only child?

This is equivalent to asking are the majority of people a youngest or only child since there are as many oldest children as there are youngest children.

So, let’s be more precise. Are a majority of people first children?

I personally am a first, and I know quite a few first children. But, are a majority of the people I meet also first children?

Human fertility dictates that in a large enough population at least a plurality of people are first children since any mother that has children for sure has a first child but may or may not have a second, third, fourth, etc… But in order to determine if first children actually represent a majority, we need to know the average number of children per mother.

According to an article published in 2015 by the Pew Research Center, “moms have 2.4 children on average.” Pretty sure that refers to mothers in the U.S.

Since the number of mothers is equal to the number of first children we can take the reciprocal of the average number of children to find what percentage of children are first children.

1/2.4 = 46.7%

So at least among children of the mothers included in the data set, the majority of children are not first children. As long as the average number of children per mother is > 2, first children won’t represent a majority.

Are a majority of second, third, etc… children youngest children?

My wife is her mother’s fifth child, but she isn’t her mother’s youngest child. If my wife was to meet another fifth child, what is the likelihood that person is also not their mother’s youngest child?

Or more formally, for what numbers n is it accurate to say a majority of nth children are youngest children?

We are going to need more than the average number of children per mother to answer this question. But what statistics do we need? One possibility is the following.

a = number of mothers with only n children
b = number of mothers with n or more children
m = a/b = % of nth children who are also the youngest in their family

The Pew Research article referenced above doesn’t provide a or b, but it does provide a few other valuable statistics. Namely the % of mothers with 1 child, the % of mothers with 2 children, the % of mothers with 3 children, and the % of mothers with 4 or more children.

Given those %’s we can figure out m for n = 2 using the %’s from 2014.

c = total number of mothers
a/c = % of mothers with 2 children = 41%
b/c = % of mothers with 2 or more children = 79%
(a/c)/(b/c) = a/b = m
m = (a/c)/(b/c) = 41%/79% = 52%

So at least among the children of the mothers included in the 2014 data, a majority of the second children were their mother’s youngest child. This was also true for the 1994 children, but it wasn’t true for the 1976 children. In fact, because we only have a % for mothers with 4 or more children, we can’t say at what n the majority of nth children were the youngest in their family, but it’s > 3.

Are a majority of people middle children?

We’ve already determined that at least for children in the available data set first children are not the majority. And since there are as many youngest or last children as there are first children, we can also conclude that youngest children alone are not a majority. But if we combine first and last children, are there enough of them to out represent middle children?

A straightforward approach would be the following. (Please excuse me for repurposing symbols.)

a = number of mothers with only 1 child
b = number of mothers with more than 1 child
c = total number of children
m = (a + 2b)/c = % of children who are first or last children

Unfortunately the Pew Research article doesn’t provide a, b, or c outright, but we can use the statistics we have to still reach m. The following uses the 2014 statistics.

d = total number of mothers
a/d = % of mothers with only 1 child = 22%
b/d = % of mothers with more than 1 child = 78%
c/d = avg number of children per mother = 2.4
((a/d) + 2(b/d))/(c/d) = (a + 2b)/c = m
m = ((a/d) + 2(b/d))/(c/d) = (22% + 2 * 78%)/2.4 = 74%

So, of the children included in the 2014 data, 74% are first or last children, meaning only 26% are middle children.

The Pew Research article doesn’t provide a precise average number of children per mother for 1976, but instead states, “In the late 1970s, the average mother at the end of her childbearing years had given birth to more than three children.” Assuming 3 is the average number of children per mother in 1976, the percentage of first and last children from that set is calculated as follows.

a/d = % of mothers with only 1 child = 11%
b/d = % of mothers with more than 1 child = 89%
c/d = avg number of children per mother = 3
m = ((a/d) + 2(b/d))/(c/d) = (11% + 2 * 89%)/3 = 63%

So even with a larger average family size, in the 1976 data set middle children are still a minority.

What average number of children per mother and % of mothers with only 1 child would result in middle children being a majority?

Technically b/d can be defined in terms of a/d (i.e. b/d = (1 — a/d)). That along with some variable renaming results in a simplified equation for m.

x = a/d = % of mothers with only 1 child
1 - x = b/d = % of mothers with more than 1 child
y = c/d = avg number of children per mother
m = % children who are first or last children
m = (x + 2(1 - x))/y = (2 - x)/y

Set m = 50%, the value at which middle children represent half of all children, and you end up with a nice linear equation.

m = (2 - x)/y = 50%
y = (2 - x)/50% = 4 - 2x, x >= 0, x <= 1, y >= 1

If we graph the equation, the area above the line represents points at which middle children would be a majority of all children, and the area below the line represents points at which middle children would be the minority.

Graph of the function 4 - 2x = y. x is the % of mothers with only 1 child. y is the average number of children per mother. The line represents the boundary between middle children being a minority (points below the line) and middle children being a majority (points above the line).

Not represented in the graph is the fact that a population where x = 1 necessitates y = 1. But for any other point with x ≥ 0, x < 1, y > 1, we should be able to manufacture a population to match those values, however unlikely it would be to find that population in the real world.