# No One Asked Me: Height Differences

Do you think the height difference is too much if a girl is 4’11 and a guy is 5’8?
Just wondering what other people’s thoughts are… My friend thinks it’s too much of a difference.
-Anonymous

Okay Anonymous, let’s talk this out.  There’s a few different ways of looking at this type of problem, and figuring out how much of a height difference is “too much”.  The first thing you should know is that height for males and females follow two different, but similar, normal (gaussian) distributions.   They look like this:

Aw cute, they’re holding hands…er, tails!

That’s what it looks like when two normal distributions are similar in shape but have different averages.  For the most part they stay on their own sides, but there’s some overlap.  Now, some people1 will tell you this is a good example of a bimodal distribution, but there’s some controversy about that2, so tread lightly, Anonymous. Those same people will also tell you this means it looks like a camel….

This camel would like to point out that comparisons to his humps are ALSO a potentially inaccurate analogy and yet no one’s writing papers about that issue. He has feelings too you know.

….maybe you should avoid this analogy too. Regardless of what we call it or how we describe it, you’ve noticed this before and you know what it means – most women will wind up with men who are taller than them.  Quite handily for your question, people love to track these distributions as much as Stats 101 profs love to use them as examples, so we have a nice data set from 2007-2008 here.  We’ll work off of that.  I like this data set because it’s kind of cute, and height is measured rather than self reported, which means it’s likely more accurate than a lot of the other more shifty looking data sets out there.

Now Anon (can I call you Anon?), I’m going to assume from your question that you’re on the young side.  I don’t know if you’re pre-pubescent or post, but just know that if either of you are younger than 15 or 16, there’s still a shot one of you could grow a bit more.  For the purposes of this exercise though, let’s assume you’re both in the 20-29 range.  While technically adult height ranges could be anywhere from 21 inches to 8’3”, we know that in reality over 99%3 of adult  men you meet will be between 5’3” and 6’4”, and 99% of adult women will be between 4’10” and 6′ tall4.

Since you didn’t give your gender, we’re going to look at this from two different angles.

If you are the girl:

If you’re our 4’11” girl, you may not want to go ruling out a 9 inch height difference so quickly.  Only 33.1% of men are shorter than this, so it’s actually more like than not that you’ll meet a guy whose even taller than 5’8”….and thus more than 9” taller than you.

If you are the guy:

Well now the story kind of changes.  If you’re our guy here, your chances of meeting a girl shorter than 4’11” are actually pretty small….only 2.6% of women are shorter than this.  Thus nearly every woman you meet will be less than 9 inches taller than you.  You’ll actually meet women taller than you 5 times as often as you’ll meet a woman shorter than 4’11”.

If you just want to fit in:

Alright, so now that we told you your chances, lets take this from another angle.  What happens if you are not concerned about you personally, so much as you’re concerned about everyone else.  How often do people, in general, wind up with someone 9 inches taller than them or more?  Well, for that we have to use some slightly different data.  When I mentioned the bimodal controversy up there a while back, I linked to another nifty data set that gave me some more information:  mean and standard deviation of male and female heights.  Those are cool because now I can blatantly exploit their good nature to calculate how likely it is that we will find a woman 9″ shorter than her male partner.  To do this we….you know what?  You are not going to be interested.  If you want to see the calculations, go here.  I want to show you on of the equation I used though, because it’s initially kind of funny looking but when you see it in action you find yourself oddly attracted to it, like a Yugo or Benedict Cumberbatch:

The Yugo literally looks like a bad drawing of a car came to life

Take that baby out for a spin and it tells you that there’s a 17% chance that he’s more than 9 inches taller than her.

BUT YOU’VE BEEN LIED TO, BY ME AND POSSIBLY BY EVERYONE:

Alright Anon, I gotta come clean.  I’ve been lying to you, and I’m sorry.  In my haste to impress you, I totally made a few things up that I shouldn’t have.  It’s hard for me to say this, but here we go: I never should have presumed that any of this was random. I was trying to make life easier on myself by making some assumptions, but I can’t do that to you now that we’re friends and all.  All the calculations I just did presume that men and women just get thrown together without anyone ever thinking anything or having preferences.  That’s completely not true and you and I both know it.  People like what they like,  and what they like frequently includes height preferences, at least enough to mess with my calculations.  As a little bit bonus of life advice, any time anyone shows you a statistic, it’s a good idea to try and figure out what assumptions they made on their way to getting it.  Everyone makes assumptions to get their math to be a little easier5, but sometimes those assumptions make our results way less useful.

So what’s the real story if we don’t presume everything’s random? Well, Fivethirtyeight.com did some good math on this a few months ago while answering a question about how often in real life a man would be shorter than his partner here, and they linked to an interesting study that suggested in the real world, 9 inch (or close) height differences would occur in 30% of couples. That’s even more than the 17% we came up with up earlier, and a few more calculat.  So you’re not alone Anon, not even close.  Everything else being equal, the height difference shouldn’t be a problem, and it definitely won’t be that unusual.  In fact, given how common it is, you may want to consider that one of your friends has a crush on whoever you’ve got your eye on, and is trying to talk you out of it so they can have him/her to themselves.  I’m unimpressed with their advice here, and the math agrees with me.  Good luck and god speed you crazy kids you.

1. Up to and including every stats 101 prof ever.
2. This paper ends with one of the most amusing conclusions of all time. Essentially they conclude that a. male/female height distributions are not really technically bimodal but b. there’s no other good quick classroom demonstration they can think of to illustrate the concept and c. readers should think of one and tell them so we can stop using this
3. 99% sounds like a lot, but keep in mind that at least in the USA this leaves nearly 2.5 million adults outside of these ranges. 1% of a large number is a LOT.
4. Tangentially related bonus fact:Right around 5’7” we hit a kind of magical crossover place, where it’s equally likely that men and women will be that height.  Put another way, if I were to say something like “my cousin is 5’7””, and you were trying to guess my cousin’s gender based only on that statement, you’d have to guess completely at random.
5. If you think I’m bad, don’t even get me started on physicists.