The Problem of Innumeracy

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Podcast Transcript

Over the last several centuries, there has been a concerted effort to raise literacy rates around the world. 

For the most part, although there is still work to be done, we’ve done a pretty good job. The vast majority of people on the planet know how to read and write.

While literacy has improved, despite our world becoming ever more dependent on technology, overall mathematical literacy has not improved.

Learn more about numeracy, or mathematical literacy, on this episode of Everything Everywhere Daily.

In the 1980s, the fast food chain A&W was competing with the much larger fast food chain McDonald’s.

One of the most popular McDonald’s hamburgers was, both then and now, the quarter pounder. 

A&W thought they came up with a great idea to compete with the McDonald’s quarter pounder. They introduced a hamburger with a third-pound hamburger patty instead of a quarter-pound patty for the same price, and they beat the quarter-pounder in blind taste tests. 

A&W thought they had a winner on their hands. 

However, the A&W third-pound burger wasn’t the success that they had hoped. 

A&W conducted customer surveys to find out why people didn’t like the new burger. The owner of A&W, Alfred Taubman, wrote about what they found in his book Threshold Resistance

More than half of the participants in the Yankelovich focus groups questioned the price of our burger. “Why,” they asked, “should we pay the same amount for a third of a pound of meat as we do for a quarter-pound of meat at McDonald’s? You’re overcharging us.” Honestly. People thought a third of a pound was less than a quarter of a pound. After all, three is less than four!

This case is the poster child for what has been dubbed innumeracy, the mathematical equivalent of illiteracy. 

Innumeracy isn’t necessarily the inability to do basic arithmetic or multiplication; rather, it is an inability to understand and reason using basic mathematical concepts and logic.  It doesn’t really have anything to do with how well you did in math classes or the highest level of mathematics you took.

What I want to do in this episode is go over some of the biggest mistakes people make with basic math. Some of these you might have seen before, and some you might be guilty of yourself.

The first has to do with very large numbers. Most people don’t have a problem with numbers that are on a human scale. Things that are counted in the dozens, hundreds, or even thousands are numbers that we can grasp. Most of us can intuitively understand the difference between one thousand dollars and ten thousand dollars. 

However, there are many things in science or economics where we encounter numbers in the billions, trillions, or even greater. Most people don’t even know what septillions, quadrillions, or octillions even are.

Numbers at this scale are difficult for people to grasp intuitively. 

One helpful way to grasp this scale is to think about it is how long a million, billion, and trillion seconds are.

One million seconds will pass in 12 days. One billion seconds will pass in 32 years. One trillion seconds will take 31,688 years. 

Scientists and mathematicians have developed an easy way to express these numbers using what is known as scientific notation. Scientific notation is based on ten raised to the power of something. The number that it is raised to, aka the exponent, is just the number of zeros.

For example, one million is 106, as one million has six zeros. A billion is 109.

If you wanted to express the number of humans on Earth, it would be 8 x 109, or 8 times a billion, or eight billion. 

With this, we can easily express huge numbers without making our eyes bleed, looking at a ton of zeros. 

The diameter of the earth in meters is 12.7 x 106. Or 12.7 million meters, or 12,700 kilometers.

The diameter of the observable universe is 8.8×1026 meters. I don’t even know the name of a number that big, and I don’t need to know it because it doesn’t matter. 

You can also express incredibly small numbers the same way by making the exponent negative. One hundredth is 10-2. The diameter of an atom is approximately 10-10, or 1/10 of a nanometer. 

As handy as scientific notation is for really big and really small numbers, most people don’t understand it, so you will very seldom see it used in public discourse.  

The inability to conceptualize big numbers also leads to problems when it comes to statistics and probability. 

There is a dating company called eHarmony. They claim that their service makes a match with members every 14 minutes.  They phrase it this way because it makes it seem that you might only be 14 minutes away from making a match. 

Assuming their number is correct, it really doesn’t tell us anything because we don’t know how many people are signed up for the eHarmony service. 

The most recent public number I could find was that there were 10 million people who were using their service. 

With one match every 14 minutes, that would mean that the system only makes approximately 103 matches per day and about 37,542 per year….out of a population of 10,000,000. 

So, in an average year, only about 0.4% of the members of eHarmoy will find a match. 

Not nearly as exciting as one every 14 minutes. 

Another example of making numbers look good is with pharmaceutical products. 

Let’s say a certain pharmaceutical product claimed that if you took their drug, your risk of dying from a certain type of cancer would be reduced by 10%.  10% isn’t something to sneeze at, so that sounds like a pretty good thing. 

However, those numbers cited almost always cite relative risk, not absolute risk. That 10% reduction doesn’t tell you the absolute risk before the drug was taken. 

If the odds of dying from a particular cancer is 1/100,000, then even assuming the data on the drug is correct, the odds would now be 1/110,000.  Miniscule to still pretty minuscule. 

Almost all pharmaceutical efficacy percentages are reported as relative risks, but they never say that because most people will think that it is really absolute risks. 

Speaking of odds, I’d like to propose a hypothetical game. 

In this game, we’ll flip a coin. If the coin comes up heads, you give me $1. If it comes up tails, I’ll give you 98 cents. 

Given the way I proposed the game, you’d be smart if you declined to play. You clearly are going to lose in the long run.

Yet, that game I proposed is basically every game at a casino…..except they usually pay much less than 98 cents.

The cards, dice, and spinning wheels are designed to obfuscate the fact that the house has an edge. The more you play, the greater the odds are that the house will come out ahead. 

Casinos will feed this irrationality when they can. For example, many roulette tables will have a digital display showing what the last several numbers and colors to come up were.

They make no claims regarding what this information means, but it is really encouraging what is known as the Gambler’s fallacy. If you see that red has come up five times in a row, then surely, black is due. 

In reality, it doesn’t matter what happened in the past. If red comes up five times in a row, then the odds of it coming up again are exactly the same as if black had come up five times in a row previously. 

Odds and probability come up in a lot more situations than in casinos.

Let’s suppose the weather forecast for the weekend calls for a 50% chance of rain on Saturday and a 50% chance of rain on Sunday. What are the odds that it will rain over the weekend?

If you are really innumerate, you would say the odds are 100% because 50% and 50% equals 100%, but this is obviously wrong. 

If each day were independent of each other, the odds of it raining on either of the two days would be 75%. That would be the odds of getting at least one head to come up on two consecutive coin flips. 

However, the real answer is…..we don’t know. The reason why we don’t know is that the weather isn’t like a coin flip. The odds of rain on two consecutive days are not independent of each other. 

One storm cloud might pass over near midnight on Sunday morning, in which case the forecast could be for the same event split into two days. Or, it could be two separate events separated by almost 48 hours. 

Another area where people are confused by probability is elections. 

Let’s say there are two candidates. Candidate A is polling at 52%, and candidate B is polling at 48%. 

Many people think that because the polling results are published in percentages, they are the same as the odds of winning. They are not.

If the polling is consistent and there is little variance, someone polling at just 52% might have an 80 or 90 percent chance of winning. 

Also, when political polling is conducted, it is always given with a margin of error. Most people ignore the error margin and take the numbers at face value. However, the errors are there for a reason. If there is a 3% margin of error and two candidates are within 3%, then it is basically a statistical dead heat. 

Risk assessment is also an area where people don’t think mathematically. People tend to inflate the risk of dramatic, low-probability events which appear on the news and underestimate the risks of more probably yet commonplace events. 

A good example of this is the risk assessment people have with terrorism. Thousands if not millions of people have canceled or changed their travel plans because of the threat of terrorism. Yet the odds of being a victim of a  terrorist attack are extremely low, and are dwarfed by many other things.

One thing that few people worry about is actually the single biggest killer of travelers: automobile accidents.  You are hundreds of times more likely to be involved in an automobile accident than you are in a terrorist attack.

I’ll end by noting something that many people also completely misjudge the odds of, coincidences. 

A coincidence, by definition, is a low probability event. However, certain coincidences aren’t that improbable.

For example, if you have a group of 25 people, the odds are better than 50% that at least two people will share the same birthday. It doesn’t seem like it because we think of someone having the same birthday as us, but with any two people, it is more probably than not.

Coincidences happen all the time. Predicting any particular coincidence before it happens is improbable, but pointing out one after the fact isn’t that big of a deal because the universe of possible coincidences is extremely large. 

Humans like to find patterns, so there is a tendency to try to find reasons why coincidences occurred, even if there wasn’t anything that was the cause.  Not accepting coincidences can lead to conspiracy theories and pseudoscience, trying to find underlying reasons to explain something that doesn’t actually require explanations.

Innumeracy isn’t as obvious as illiteracy, but it is far more prevalent. It isn’t just something that affects regular people. You can often see evidence of it in news reports and even in academic research papers. 

Almost forty years after A&W failed with the launch of their ? pound burger, they reintroduced it in 2021. This time, however, they changed their marketing. Instead of calling it the ? pound burger, they changed the name to the 3/9 pound burger.

Everything Everywhere Daily is an Airwave Media Podcast. 

The executive producer is Darcy Adams.

The associate producers are Thor Thomsen and Peter Bennett.

Today’s review comes from listener Dave from Kenai AK, over at Apple Podcasts in the United States. He writes:

Easily one of the best podcasts I’ve found

I’ve listened to every episode, and they are ALL GREAT! It is rare to find a podcast where every episode is interesting, but this is that podcast. Thank you for the great show!

Thanks, Dave!  And welcome to the completionist club! I’ve been to a part of your neck of the woods. I’ve been to the Kenai Peninsula and Kenai Fjords National Park, but never all the way to the town of Kenai itself. Glad to know I have people listening in the 49th state. 

Remember, if you leave a review or send me a boostagram, you too can have it read the show.