Middle children

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!

How many children are middle children?

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.

How about a room of 22 people?

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.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


2 comments on “Middle children

  • 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!

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