Why men aren't 5'11 (and other statistical ghosts)
Phantom gaps in the normal distribution
This graph shows US male height distribution, both “measured” and “reported.”
The “measured” data follows a normal distribution. But when men are asked to self-report their own height, there is a suspicious dip in the number of guys who describe themselves as 5’11” (and a correspondingly large number of guys who are exactly 6 foot).1
These sorts of “holes in the distribution” show up in lots of places besides self-reported height:
Missing IMDB ratings: “the 9 gap”
My fellow Inkhaven resident David Gros has an excellent post about why some movies have a hole in their IMDB ratings. To recap the most salient part: most film ratings on IMDB have what looks like a “normal distribution” of ratings. For example, here’s the ratings for Toy Story 4:
It has an average rating of 7.6, so as you would expect, most of the ratings are 7 or 8, with the frequency of ratings dropping off from there in either direction. But for certain movies -- what you might call “cult hits” -- the distribution looks more like this:
It’s tempting to chalk this one up to “activist fans” (this is what you’d expect to see see if a bunch of sock puppet accounts were voting 10), but if you look at other films that have a big “9 gap,” most of them don’t have massive “fandoms.”
There’s certainly a sense in which “dog lovers” and “horse lovers” have a sort of “cult fandom” quality to them, but I don’t think that dog lovers are creating sock puppet accounts on IMDB to give A Dog’s Purpose a 10/10 rating. These votes express actual human preferences.
So what are those preferences? I think some movies hit in a way that invites you to consider them more emotionally and less quantitatively.
When a movie emotionally breaks you and you immediately hug your dog, that’s not a 9/10 experience; the only rating that feels emotionally honest is a 10. Nobody says “this movie made me cry about my dog, but in a rigorously calibrated way that deserves a 9.”
Ditto for when you have a slumber party that turns into a KPop singalong night, or when you dress up, drive to the local independent theater, and get ready to do the Time Warp again. That’s not a 9/10 experience. You can’t even really quantitatively compare it to other films.
The “9-gap” in IMDB ratings doesn’t require a hardcore fandom or some conscious campaign to boost the number of 10 ratings. It just needs a movie whose appeal is “bimodal” for a certain audience.
Retail discounts
Retail discounts also follow a bimodal distribution: items are either shallowly discounted, or deeply discounted, but seldom moderately discounted.
PC game market Steam is currently doing a “Black Friday” sale where the top-selling discounted software titles demonstrate this:
Scroll through the Steam bestsellers list and you’ll see practically no games with a 25-35% discount. Either the game is getting propelled to the top of the list by a steep 40-70% discount, or it’s popular enough to make the top sellers list without a substantial discount.
Income
In the same way that certain IMDB titles have a “9 gap” at the high end, the lower end of the income distribution has a very small number of households with a reported income of $5,000-10,000: Either you make >$10k, or you’re ~broke, with very few people in between, as you can see by looking a the orange bars on the left:
You can also see something more directly analogous to a “9 gap” if we zoom in on the middle of the graph: Notice how there’s a slight dip for $95-100k, and a corresponding bump for $100k-105k. What explains this?
It’s a bit like the 5’11 guys calling themselves 6-foot: this census data showing self-reported survey income. There are probably some people making $98k or $99k who “round up” to $100k.
But if we set aside the census survey and look at IRS income data, we discover something different:
The tax reporting incentives
Mortenson and Whitten (2020) find evidence of “bunching” at what they describe as “kinks” in the income scale, points where the Child Tax Credit and Earned Income Tax Credit peak.
For example, many taxpayers, particularly taxpayers who are self-employed, cluster around the first income level “kink” ($13,650) where the Earned Income Tax Credit hits its maximum:
They find that 89% of the taxpayers who are at the “bunching” points are self-employed, meaning that they have finer control over their income than people working W-2 jobs.
Likewise, this is what the distribution of taxpayers looks like for at the point at which the Child Tax Credit is maxed out and stops increasing with additional earnings ($16,333):
The tax code creates incentives, and people respond to them. Who knew!
This post was largely inspired by David Gros’s post The Missing 9: Why Some Movies Have a Hole in Their IMDb Ratings.
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Interestingly, this graph also shows a related phenomenon in which men under-report being 6’1” and over-report being 6’2”. What is it about being 6’1” that makes these guys want to exaggerate their height by 1 inch? Don’t they know that 6 foot 1 is tons fun when you’re dressed to a tee?
Really well written! To continue on with this theme, there are barely any “moderately” priced SaaS products - you’re either a dirt cheap 20 usd a month sub or a bespoke 100k solution with not much in between, mostly because salesman have a lot of associated fixed costs and thus there’s an implicit floor on prices for products that require salesmen to sell.
I am reminded of "Churov's Saw", a fascinating little phenomenon in Russian elections where for some reason, a weirdly large number of districts report turnout numbers ending in 5 or 0.