An hour wait at Disney

You are at Disney. You planned ahead. You open Touring Plans (theme park visit optimizing software). Here’s how the creator thinks about what you see.

“Let’s say we were trying to predict what (wait time) Disney is posting for Rock ‘n’ Roller Coaster at Disney’s Hollywood Studios. We have to do two things. We have to predict the number Disney is going to show on the wait time sign in front of the ride, and then we have to predict the actual wait time. We have to predict both because if we showed only our expected wait time and you walk up to the ride and see the posted wait time is sixty minutes and we are predicting five minutes, then you won’t believe our number.” – Len Testa, Causal Inference, October 2021

Like weather reports or repair times, Testa and his team generate more value by being less accurate.


In the fall of 2021 Disney released a similar feature to what Testa created: Genie. It will be an interesting TiVo problem .

$1 Toronto real estate

Average is like my reciprocating saw: never as useful as I expect. Part of the reason average sticks around is economics, It’s cheap to produce.. Average is a crude tool, like with student loan debt, and often hides the heterogeneity of a situation.

We’re entering an era of precision. One covid lesson has been the effect size of heterogeneity. At the macro level, the impact of covid depends on time and place. At the micro level, the impact depends on age, immunity, and social network. Covid was (is?) difficult to judge because there are a lot of factors that need fitted together.

If we need precision we should probably think about distributions at least as often as we think about averages. An example is the periodic one dollar real estate listing. Yes, this generates attention, spins up the market mechanism, and might be the marketing magic an owner needs. But it also changes the distribution of offers without changing the average offer.

“When you give people a listing price they ask if it’s worth more or less and by how much, so they anchor at the listing. If you don’t have an anchor people build a valuation from first principles. The average (offer) doesn’t change but the distribution does. For a one dollar listing you get some really high rates and some really low ones. In the listing price you get distributions around what the asking price was. This is a world where the seller doesn’t care about the average, they only care about the top end of the distribution.” – Dilip Soman, The Decision Corner, October 2021

Maybe this is being too hard. Average, like the saw, has its uses. The aim here is to combine numeracy with psychology to get by in the world. That means presenting the ‘best’ wait times or predicting rain more often. Being numerate is understanding that the average age is 78, but if you make it to 65 you’ll probably live well past eighty.


“The average looks like 10-12 years lost due to Covid – but that’s an average of a distribution with a very odd shape, a highly skewed distribution, some people have lost forty years of life. The peak of the distribution is people who lost less than a year of life.” – David Spiegelhalter, Risky Talk October 2021

How to board an airplane?

Everyone knows how to board an airplane, back to front. It’s logical. Back to front means that if someone is taking their time in seat 21C the person in 20A can still seat themselves.

Physicist Jason Steffen built a computer model to see how much faster back to front was relative to front to back. He was shocked. The difference was minimal. Hmm, Steffen scowled at his code, is there a better way?

What if instead of 30 to 1 or 1 to 30, a plane boarded everyone in row 30, then rows 1, 2, 3…. That might work right? The folks in row 30 would have time to stow and seat without holding anyone up.

That kinda works. Steffen’s code continued and compared the 30, 1, 2…29 option to the 1, 2,…30 option. The code noted the faster sequence, and switched around two more numbers. Again it kept the faster option and computed another switcheroo. The fastest boarding process turned out to be boarding every other row.

“It turns the boarding process from a serial process, where one person gets to their place, puts their luggage away, and sits down into a parallel process where you send in fifteen people, say in all the even rows, and they all put their luggage away and sit down at the same time. And then you send in the next group of people.” – Jason Steffen, August 2021

Steffen’s code was a Markov Chain Monte Carlo, a way to solve problems through computer code and exploration. But like wet bias or wait times, ideal solutions may not be the best.

One problem with Steffen’s method is when people travel in groups, especially families. Another obstacle is the culture of air travel, there’s some established norms. Further confounding the case is an airline’s incentives. Faster turns do matter, but relative to upgrades how much does saving time save the company?

A landmark numbers walk

We tend to remember more when things are connected. We tend to remember more when there is a story.

For instance, about 1,000 ants equal the length of one beetle. Rather, one Beetle, 1,000 of which lined end to end equal about the length of Central Park. Imagine that. One-thousand VW bugs lined up along the length of Central Park with one-thousand ants lined up along each bug.

Now another step, 1,000 Central Parks is about the width of Australia, ten of which is about the length of the equator.

That’s a nice story and it gives us some Landmark Numbers.

“What I mean by landmark numbers is something that sticks in your brain, you don’t set out to memorize it but it sticks in your brain as a reference point. It becomes a ready reference for you to relate to. These are numbers where, if you have them ready, they help you make yardstick comparisons of the things you hear about. It’s something where you hear a number on the news you go: ‘Hold on.That’s not a big number because it compares to this landmark number in some way.'” – Andrew Elliott, Talks at Google, December 2018

Another landmark number is 4 million. That’s about the number of Americans of any single age. That’s via Tim Harford.

Landmark numbers fit nicely with Maxims for thinking analytically. Richard Zeckhauser suggests that simple cases, extremes, and everyday analogues help us think better. A basic understanding needs context though, and landmark numbers provide just that.


Similar to Landmark Numbers is thinking like Fermi.

Fermi questions, answers, and landmarks

A Fermi question is something like, How many rolls of toilet paper do the residents of Columbus Ohio use in a week? Fermi questions are silly but embody some serious thought. Namely, how do we think about the world?

There’s some fun little math behind a Fermi question but the hardest part is often the start. For instance, how many people live in Columbus Ohio? Tim Harford knows. Rather, Tim Harford has a suggestion.

“Andrew Elliott—an entrepreneur who likes the question so much he published a book with the title Is That a Big Number?—suggests that we should all carry a few ‘landmark numbers’.”

Landmark numbers are figures we can use to guide our thoughts about the world. For instance, there are about four million Americans at any age under sixty. New York City has a population of about nine million. Columbus Ohio has a population of about one million. This is actually quite helpful just for a start.

Using a Zeckhauser maxim, “when you are having trouble getting your thinking straight, go to a simple case.” If every resident of Columbus Ohio used half a roll a week, how many rolls of toilet paper would they use? That’s easy! We have a million people, each uses half a roll, and that’s 500,000 rolls per week. No wonder we had a shortage.

Tsk tsk, Enrico Fermi would scold us. You can do better. And indeed we can. That is the point of thinking about Fermi questions. We can do better and even if we make a mistake, even if we make a few mistakes, we can still very likely be right. The reason is because of the random walk nature of our guesses. Some of our guesses will be too high (Columbus actually has 898,000 residents) and some will be too low, but overall these kinda-sorta balance out and that puts us in the right ballpark. Not only that, but making additional steps doesn’t necessarily mean additional steps in error. That, and more examples are here.

Good decision making takes nouns and verbs. We’ve got good verbs like inversion, mean reversion, extreme examples, and such. Landmark numbers give us a few nouns to work with too.


The very good Fermi book inspired this post: Fermi Knowledge.

Numeracy at Best Buy

We note that numeracy is important but it is hard to box in what numeracy is and how to use it well. Generally it is this idea that numbers explain some parts of the world well and we should use those numbers in a world full of people.

Yes?

Maybe an example will help:

“The Best Buy Geek Squad was reporting the mean (repair time) to their customers. A customer walks in to get their computer fixed, the part is on backorder, and the Geek Squad would quote the mean time to repair the computer. Of course, that means plenty of people’s repairs were not the average amount of time. So they changed and reported the 95th percentile time. They ranked the past times and now quote that to the customer – which is what people want!” – Elea Feit, March 2018

Like wet bias, maybe we can’t handle the truth. Or rather, maybe the way we see the world makes more sense one way rather than another.

One thing people are pretty bad at is randomness. We use stories to connect actions to events. Another thing we tend to miss is thinking that what did happen was the only thing that could have happened. It’s not.

We work around this through design. For instance, we know innovation is important but without separate metrics and incentives it’s less likely to happen. Put another way, it’s the framing stupid.

Wet bias makes sense. Being less honest than possible also makes sense. Quoting average waits may be more accurate but it’s less valuable.


Design and framing were two of my favorite ideas. For the ideas vitamin-style in a daily email drip, buy the email-drip on Gumroad. Find it on Amazon too.

A confusing life expectancy calculation

“Statisticians are sometimes dismissed as bean counters. The sneering term is misleading as well as unfair. Most of the concepts that matter in policy are not like beans; they are not merely difficult to count, but difficult to define…the truth is more subtle yet in some ways easier: our confusion often lies less in numbers than in words.” – Tim Harford, The Data Detective, 2021

One of Harford’s goals is to help people understand the world more as it is and less as they wish it. Harford kindly covers ideas like base rates, sampling bias, and algorithm associations.

That last one has some quite funny anecdotes. For instance, one AI system was trained to distinguish healthy skin from cancerous skin. Crunching and comparing over and over are two things computers do really well, so this seemed a good fit. And it was! The AI categorized correctly. But computer code is like a mango slicer – it has a singular use. In the case of the skin cancer, what the AI “learned” was that if a ruler was present it was cancer.

That’s funny.

But also not. One economic principle that’s going to affect (is affect_ing_) work is the idea that as something gets cheaper it’s used more. LEDs and cameras are two recent examples, name an electronic product that does not have one of those. Data too, is going to be part of our lives more, and Harford wants us to think about the numbers a bit more. For instance, what does “life expectancy” mean?

“They take the relative risk at every age and they integrate it. They ask, if the relative risk this year stayed constant forever, how long would someone born today live? That’s where we lost a year, but that’s assuming Covid stays and the year we just had gets repeated .” – Adi Wyner, Wharton Moneyball, July 2021

This isn’t the only way to calculate life expectancy, but it was the way that lead to headlines like, “US Life Expectancy in 2020 Saw Biggest Drop Since WWII, With Virus Mostly to Blame”. That’s true, but is that how most people understood it?

Most of what happens, and Harford starts his book on this idea, is that we think fast. “Biggest drop”, “WWII”, and “Virus” are all oh-boy-this-is-bad bits of information. But we dig in to what the words really mean and things look a little better.

Our tendency to think fast doesn’t have to be a hinderance. We can use this tendency to be more numerate. Books like Harford’s bump up (be Bayesian baby) these ideas. Riddles like: most British men live past the average age help too. A steady dose of numeracy uses the availability heuristic for our own good.


Not into the book thing? Harford has great podcast that cover these ideas. Wharton Moneyball is another with more of a sport’s bent. Gambling podcasts too cover these ideas. As Tyler Cowen said, it’s not that these things are VERY IMPORTANT but that if we see them more we update our mental toolboxes so they are marginally more important.

A confusing life expectancy calculation

“Statisticians are sometimes dismissed as bean counters. The sneering term is misleading as well as unfair. Most of the concepts that matter in policy are not like beans; they are not merely difficult to count, but difficult to define…the truth is more subtle yet in some ways easier: our confusion often lies less in numbers than in words.” – Tim Harford, The Data Detective, 2021

One of Harford’s goals is to help people understand the world more as it is and less as they wish it. Harford kindly covers ideas like base rates, sampling bias, and algorithm associations.

That last one has some quite funny anecdotes. For instance, one AI system was trained to distinguish healthy skin from cancerous skin. Crunching and comparing are two things computers do really well, so this seemed a good fit. And it was! The AI (read: computer code) categorized correctly. But computer code is like a mango slicer – it has a singular use. In the case of the skin cancer, what the AI “learned” was that if a ruler was present it was cancer.

That’s funny.

But also not. One economic principle that’s going to affect (is affect_ing_) work is the idea that as something gets cheaper it’s used more. LEDs and cameras are two recent examples. Data too, is going to be part of our lives more, and Harford wants us to think about the numbers a bit more. For instance, what does “life expectancy” mean?

“They take the relative risk at every age and they integrate it. They ask, if the relative risk this year stayed constant forever, how long would someone born today live? That’s where we lost a year, but that’s assuming Covid stays and the year we just had gets repeated .” – Adi Wyner, Wharton Moneyball, July 2021

This isn’t the only way to calculate life expectancy, but it was the way that lead to headlines like, “US Life Expectancy in 2020 Saw Biggest Drop Since WWII, With Virus Mostly to Blame”. That’s true, but is that how most people understood it? Does if this previous year repeated forever seem like a good conditional?

Most of what happens, and Harford starts his book on this idea, is that we think fast. But we can use this tendency to being more numerate. Books like Harford’s bump up (Bayesian baby) these ideas. Riddles like: most British men live past the average age help too. A steady dose of numeracy uses the availability heuristic for our own good.


Not into the book thing? Harford has great podcast that cover these ideas. Wharton Moneyball is another with more of a sport’s bent. Gambling podcasts too cover these ideas. As Tyler Cowen said, it’s not that these things are VERY IMPORTANT but that if we see them more we update our mental toolboxes so they are marginally more important.

Numeracy + Psychology

One of the consistent behavioral psychology findings is the framing effect. People judge what is pointed out and consider the number attached to it. Two out of every three dentists approve chewing no-sugar gum. Sure, but do they caveat that with increased flossing? Heck, no one cares. The thinking goes that if it was flossing that was important someone would have mentioned it.

This effect is most often seen in medical communication and Matt Yglesias captures it perfectly here:

But that headline is good. It’s salient – 4 people. It’s got friction. The real surprise is that they didn’t say ‘warn’ rather than ‘said’.

This kind of psycho-logic-magic needs countered with another kind of psycho-logic-magic.

We can assume two things work: (1) that people pay attention to and value what someone points out to them. This is normal, helpful, and completely understandable. It works. Most things that most people say are relevant to our lives. (2) that new news works. Different is interesting. This is also, normally helpful and understandable.

Here’s the pitch. This is the angle, the message. Here’s the psycho-logic-magic for vaccine interventions: opportunity cost.

If you’re pro-vaccine point out all the things that will be back to normal once people get it. Grandparents will visit grandchildren. Sports will resume. Christmas won’t be cancelled. Freedom and fellowship. Dining out and date nights. Cruise ships and college trips. Find whatever people value and point it out. People do not consider the opportunity cost unless it is explicit.

Closing note: if SkininTheGame is the ultimate signal, my wife had her second dose last weekend.

What the mean age means.

The Math of Life and Death is a good addition to the when are we ever going to use this collection of popular science books. With numeracy being so important in life, a regular diet of these ideas keeps someone mentally fit. Consider the story of student loans as one example.

Often the mathematical manuscripts show mean and median differing in network systems like in the case of income or the social graph. Or, how much Bill Gates skews average wealth but not legs.

Kip Yates reminds us of other instances.

“However, ecological fallacies can be more subtle than this. Perhaps it would surprise you to know that despite having a mean life expectancy of just 78.8 years, the majority of British males will live longer than the overall population life expectancy of eighty-one years. At first this statement seems contradictory, but it is due to a discrepancy in the statistics we use to summarize the data. The small, but significant, number of people who die young brings down the mean age of death (the typically quoted life expectancy in which everyone’s age at death is added together and then divided by the total number of people). Surprisingly, these early deaths take the mean well below the median (the age that falls exactly in the middle: as many people die before this age as after). The median age of death for UK males is eighty-two, meaning that half of them will be at least this age when they die.”

Kip Yates

Numeracy is becoming more important because we are generating more data. Luckily, we don’t have to become mathematicians but we do have to see if ideas pass the sniff test. We have to think about how survivor explains sampling, and consider gambling parlays. We have to be mathematically minded.