Written by Colin+ in hypothesis testing, statistics.

At an academic conference; 22 people in the room. Speaker asks who is a middle child. There is only one in the entire group - him. Striking (if anecdotal) confirmation of stereotypes about birth order.

— Leigh Caldwell (@leighblue) December 14, 2018

As a loyal listener to *More or Less*, my first thought here is, “is that a big number?” And as a proud geek, my second thought is, let’s model it!

Let’s suppose that the cliche of 2.4 children is reasonable, and that the number of children in a famliy follows a Poisson distribution: $C \sim Po(2.4)$.

I want to know - en route to the final answer - the probability that a randomly-selected child is a middle child. That is to say, the second of three, or the third of five; I suppose an only child is *technically* a middle child, but they’re also special in their own way.

So, running the numbers, about 9% of families in this model have no children. 22% have one child, 26% two, about one-in-five have three children, one in eight have four, 6% have five, 4.5% the field (and to be super-generous, let’s assume they’re all seven-child families.)

But that’s *families*, not *children*. Assuming 100 families, 22 are only children; 52 have one sibling; 60 are in a three-child family, 50 in a four-child family, 30 have four siblings and 32 (in our generous model) are one-in-seven.

How many of these are middle children? Twenty of the three-family children, six of the five-family and four or five of the sevens - altogether, around 30.

The total number of children is 246, so it’s reasonable to say we’d expect one child in eight to be a middle child.

Given there are 22 people in the room, we’d probably expect around three middle children. Is finding just one an anomaly? This requires a binomial expansion - 22 people, each with (under a null hypothesis yadda yadda) a 1-in-8 chance of being middle would give us $M \sim B \br{22, \frac{1}{8}}$.

And we can easily work out that the probability of having no middle children in the room is about 5%, and the probability of only one about one-in-six. So the probability of having one or fewer middle children is somewhere north of 20%, and not especially remarkable.

Even taking into account the fact that the person asking was a middle child himself, and presumably wouldn’t have asked if he weren’t, with $n=21$ we still have $P(M = 0) \approx 0.06$ - more unusual, certainly, but still not statistically significant.

## Alison

Hmmm. I would interpret “middle child” differently, so that in a family of four there is an eldest, a youngest and two middle children. (In my case, it’s complicated too by having half-siblings… I am a youngest child to one of my parents, and one of the middle children to the other!)

## Colin

I suppose real families are more complicated than mathematical ones!