Something I HATE about Averages

If you had to summarize a statistical distribution with one number…well, I guess you could do worse than taking an average.

Even so, there are many things to hate about averages.

But I’ve been trying to articulate something very particular that I hate about averages, and what better place to take a fresh stab at it than a Friday blog post?

You’re probably already familiar with the idea that averages are extremely sensitive to outliers: The average net worth of folks at the bar increases 1,000x when Bill Gates walks in.

But before moving on from that (hopefully?) familiar point, let’s get our feet wet with some back-of-the-napkin math.

I promise it won’t get too involved.

Say there’s 100 people at the bar, each with the average American net worth of right around 1 million bucks—way higher than I would have guessed as a recent grad student.

So far:

Average net worth: $1 million Total net worth: $100 million

All good? Just checking.

Now Bill Gates walks in for a pint. But even after decades of philanthropy, his net worth is still a cool $137.8 billion. (That’s 137 billion, 800 million dollars.)

Let’s update the bar’s figures.

Total net worth: $137.9 billion

(Thanks for pitching in $0.1 billion, everyone else.)

To find our new average, we divide $137.9 billion by 101 people, and we get:

Average net worth: $1.365 billion

To be clear, that’s 1365x the $1m average we started with.

And it all came from one guy.

I’ve forgotten what folks without stats degrees know about averages, but I certainly hope this extreme sensitivity to outliers is a pretty familiar point. (In fact, it should probably make us a bit suspicious of this “average American net worth” figure we started with!)

But it points to a larger problem about how averages work.

I think you’ve read enough for today, great article, Ricky!

Every average is really just an efficiency ratio.

But that's a huge claim, so let's back off and try to build up to it.

So first, how would you try to increase an average?

We can come up with a few different strategies. Maybe we could start by looking for inputs that give us a lot of output.

STRATEGY 1: Include more high performers.

Wanna increase average net worth?

Include more billionaires.

Wanna increase miles per hour?

Include more open highway time.

Wanna increase yards per reception?

Include more long catches.

Wanna increase screen time per day?

You probably shouldn’t.

But uh…include more sick days at home?

“I can’t wait to get on my cell phone.”

High performers drag everyone up. But low performers drag everyone down. So what if we could get rid of inputs that don’t give us much output?

STRATEGY 2: Exclude low performers.

Wanna increase average net worth?

Exclude grad students.

Wanna increase miles per hour?

Exclude rest stops.

Wanna increase yards per reception?

Exclude catches for a loss.

Wanna increase screen time per day?

I don’t know why you’re being like this.

But uh…exclude vacation days spent sight-seeing?

Now that’s more like it.

But what if we can’t change which inputs we’re working with? Well, we can still try to work with what we’ve got.

STRATEGY 3: Improve everyone’s level.

Wanna increase average net worth?

Give everyone $5.

Wanna increase miles per hour?

Always drive 1 mph faster.

Wanna increase yards per reception?

Always dive helmet-first for 1 more yard.

Wanna increase screen time per day?

Fine, stare at your phone for an extra 15 minutes before you fall asleep.

But you don’t have to improve everyone’s level. (That’s probably hard to impossible anyway!) It’s probably much easier to improve a handful of inputs instead.

STRATEGY 4: Make selective improvements.

Wanna increase average net worth?

Give one guy a zillion bucks.

Wanna increase miles per hour?

Haul ass on that one backroad that’s always empty.

Wanna increase yards per reception?

Juke the last defender to turn a small gain into a long TD.

Wanna increase screen time per day?

Nuke one day a week where you don’t even leave bed.

But wait a minute. I can increase the average net worth of my ten friends by giving everyone five dollars, or by giving one friend fifty bucks.

Averages don’t care. After all, my friends’ total net worth increases by $50 either way. And I’m still dividing by the same ten friends. So the average goes up by $5 either way.

Let’s take a sec to reconsider how averages are calculated:

1. We sum up the total output.

2. We sum up the total input.

3. We divide total output by total input.

And that’s it.

An average is just a ratio of two sums.

That means I can double the average by doubling the total output, or by halving the total input, or I guess by messing with both. Total output, meet total input. As far as averages are concerned, nothing else matters!

To an average, the shape or context of a distribution is invisible. An average begins with two summary statistics (total output & total input) and applies exactly one function to them: division. Individuals are just contributors to these larger sums, nothing more.

So averages aren’t subtle or context-sensitive creatures. They never really see individual data points for what they are. And that means that when we argue using averages, the numbers we’re appealing to simply don’t embed many of our values, beyond how efficiently total input is converted into total output.

Let’s listen in on a bit of sample dialogue. (If you happen to be one of my students reading this, hi! These averages and comments are fictionalized.)

“The average grade on this assignment was a C+.” “That’s pretty low!” “Well, 3 students didn’t turn it in at all. Excluding them, the average was a B+.” “Hey, that’s pretty good!” “But it’s buoyed by 3 really strong students who all got an A+. Excluding them, the average was a B.” “Well, that’s about what we expected.”

Obviously, grades are their own can of worms. But how did students do overall? Well, we took an average, an efficiency ratio of points per student:

average grade = total points / total students

So I have no idea how any particular student did! But if I think of my class as a single superstudent repeating the same assignment over and over, well I guess they got a C+.

Things started making a bit more sense once we took multiple averages…but not really. Now my class consists of:

• 3 students who failed

• 3 students who aced it, and

• one superstudent (going n-6 times) who did about what we expected.

Phew, I sure am glad I know how my class did on the assignment!

Anyway, that’s why I write out the entire distribution of grades whenever I’m norming grades with other instructors. Okay, I just gave out:

• 3 A+’s

• 5 A’s

• 7 A-’s

• etc.

Am I being too harsh or too easy on the high end? How about the low end? How symmetrical is my grading distribution, and is that even a problem?

Why don’t you take a look at the shape of this distribution, and then we can dive in to particular papers if you want? Oh, you’re surprised I gave 3 A+’s. Yeah, but take a look at em!

Averages summarize distributions a bit too neatly, which often keeps us from exploring them more deeply.

And that’s something I really hate about averages.