You likely remember the spelling rule, “I before E, except after C.”

It’s been taught in English grammar textbooks at least as far back as 1866 and persists in modern school texts. In fact, it’s been called the “supreme, and for many people solitary, spelling rule.” The rule is supposed to help with the complexity of English.

Only, like many rules, it’s not very useful.

For starters, the version most of us remember is incomplete. The full rule is “I before E, except after C or when sounded as A, as in neighbor and weigh.” This expanded rule is much harder to remember and still not complete. The words science, forfeit, seize – and dozens of others – are exceptions.

The Merriam Webster dictionary amusingly tried to account for the myriad of exceptions with this whopper of a rule:

I before e, except after c
Or when sounded as ‘a’ as in ‘neighbor’ and ‘weigh’
Unless the ‘c’ is part of a ‘sh’ sound as in ‘glacier’
Or it appears in comparatives and superlatives like ‘fancier’
And also except when the vowels are sounded as ‘e’ as in ‘seize’
Or ‘i’ as in ‘height’
Or also in ‘-ing’ inflections ending in ‘-e’ as in ‘cueing’
Or in compound words as in ‘albeit’
Or occasionally in technical words with strong etymological links to their parent languages as in ‘cuneiform’
Or in other random exceptions such as ‘science’, ‘forfeit’, and ‘weird’.

It gets worse funnier.

A statistician named Nathan Cunningham tested the original I Before E rule on 350,000 English words. As expected by the first part of the rule, “ie” was about three times more prevalent than “ei.” However, despite the second part of the rule, “cie” words outnumber “cei” ones by about three to one. On average, I before E is just as common after C as it is after any other letter.

In fact, Cunningham showed a better rule would be I before E, except after W.

English is weird.

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You should always get the bigger pizza.

No, this isn’t some bigger-is-better American standard where size matters more than quality or consistency. It’s not because everyone loves pizza so you might as well get more of it (it’s the glutamate we find so irresistible).

And I’m definitely not using the number of slices rule that says the average adult will eat about three slices. That completely ignores the fact an extra large pizza has a slice which is much larger than a medium or small.

Instead, I’ll rely on math. Or maths for you Brits.

At most national pizza chains, a medium is typically 12” in diameter, a large is 14”, and an extra large is 16”. To compare the amount you’re getting with the options, we calculate the area of the pizza using the formula: Area = pi * radius squared. Using this formula, an extra large is 201.1 square inches, a large is 153.9, and a medium is 113.1 square inches.

To determine which one is the better value, we need to know the prices. As one example, a large Supreme Pizza recently was $23.29 while a medium was $21.39. The large pizza is 36% larger than the medium one but only 9% more expensive. The large is comparatively a bargain, making it an easy decision to get the bigger pizza.  

This trend seems to be nearly universal. NPR checked 74,476 prices from 3,678 pizza places across the U.S. and found you should almost always get the bigger pizza. And this doesn’t seem to have changed despite shrinkflation.

Not long ago, I mentioned this phenomenon to the GM of a Bay Area award-winning pizza chain. Grinning, he replied that Silicon Valley engineers had figured out this pizza arbitrage and, as a result, he priced based on square inches. For this pizza place, there is no price advantage to get the bigger size.

However, I do love their Big Sur special. So, I still get the bigger pizza.

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Whenever an expert makes a claim they cannot support with data, I’m reminded of the firewood fallacy.

Firewood fallacy is a term I coined earlier in my career after being frustrated by how many organizations suffer from group think. Decisions are often made based on institutional knowledge or hearsay, rather than independent research or data.

I borrowed the term from the following parable:

A group of settlers in a remote location needed to gather firewood to prepare for the upcoming winter. The group’s leader guessed it would be a relatively cold winter but was unsure how cold so wanted to check with experts. He journeyed into the next town where he was able to call the National Weather Service (NWS). The NWS confirmed his projection of a cold winter; as a result, the group’s leader instructed the settlers to gather more firewood.

A week later, the leader checked with NWS again but the forecast had changed: the prediction was now for a very cold winter. To be careful, the leader decided to collect more firewood. This cycle went on several more times, with increasingly dire forecasts from NWS followed by even more firewood.

Finally, the bewildered leader asked the NWS: “Why do you think the winter is going to be so ridiculously cold?” The NWS answered: “It’s because those settlers are gathering so much firewood.”

The firewood fallacy is a phenomenon that often happens in business. Decisions are made based on conjecture or poorly-researched facts. In extreme cases, the so-called experts merely repeat what they’ve heard from others.

What can be done to counteract the firewood fallacy?

In the extreme, follow the 1845 advice of Edgar Allan Poe:

Believe nothing you hear, and only one half that you see.

It’s likely simpler and easier to follow the example of the settlers’ leader and ask why more often. Over time, learn to ask better questions. Finally, embrace critical thinking.

When making decisions, it can be instructive to consult an expert. Just make sure you understand the basis of their expertise. Avoid the firewood fallacy.

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Shrinkflation happens when companies keep the price of an overall package the same while reducing the size or quantity of the items in the package. The word is a portmanteau of the words shrink and inflation. The first use of the term is often attributed to Pippa Malmgren.

Shrinkflation is a sneaky way to combat the effects of inflation. When ingredients and manufacturing are more expensive, companies have two primary choices – increase prices or make smaller products. People typically shop by price so raising prices reduces demand. “Consumers check the price every time they buy, but they don’t check the net weight,” said Edgar Dworsky, a consumer advocate and former assistant attorney general in Massachusetts.

As a result, most companies resort to reduced sizes. These size changes are often subtle, like making candy bars sold in multipacks smaller than the ones sold individually or changing the shape of products so you don’t notice the weight difference. Shrinkflation is another example of the numerosity heuristic.

Shrinkflation isn’t new – in 2003 Dannon shrunk its yogurt cups from 8 ounces to 6 ounces and in 2009 Häagen-Dazs reduced their ice cream pints from 16 ounces to 14 ounces. But it does seem much more prevalent. Buzzfeed provides 15 “sneaky changes” while BusinessInsider shows 13 “extreme examples.” But to really immerse yourself in shrinkflation examples, you have to check out this shrinkflation subreddit. Shrinkflation is so common it’s become endemic.

To combat shrinkflation, people have to pay attention to unit pricing not just package pricing. Unit pricing is the price per weight or price per item in the package. Many stores make the calculation for the consumer, although it’s usually in small print under the total price on the shelf. While math may not be your strong suit, doing a quick calculation can save you money

One upside to shrinkflation is it might save you some calories. If your candy bar is now 1.5 ounces instead of 2 ounces, you’re unlikely to eat a second one and therefore your calories reduce by 25%. Of course, you don’t save any money.

There’s lots of debate on when inflation will end but I’m willing to bet that shrinkflation never does. Shrinkflation has become endemic.

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For many years, my routine was a Sunday hike followed by a Sunday blog. For a variety of reasons, both my hiking and blogging have become erratic. After skipping yet another Sunday hike, I was surprised to find that I still had exceeded 10,000 steps per day for the past week. 10,000 is the number the World Health Organization recommends, fitness apps urge us to reach, and that has become conventional wisdom.

Surprisingly, there’s little to no science behind 10,000 steps.

Instead, it appears to be the result of good marketing. To capitalize on the popularity of the 1964 Tokyo Olympic Games, a Japanese company called Yamasa Clock created a personal-fitness pedometer called the Manpo-kei. The name derives from the Japanese words “man” meaning 10,000, “po” meaning steps, and “kei” meaning system. (Native speakers, please correct me if I’ve gotten this wrong.)

However, another theory – and one that appeals to me – is that the number was chosen because the Japanese character for 10,000 looks like a person walking. The easy-to-remember number and the visual reinforcement worked. The pedometer was hugely successful and the 10,000 number is still in use more than 50 years later.

While 10,000 steps is a memorable target, it can be counter-productive. A Duke University study found that people who tracked their steps enjoyed it less, reporting that walking felt like work. A University of Texas at Austin study showed that one hour of exercise was much less beneficial if the rest of the day was spent sitting and sleeping; 10,000 steps would better split up over multiple periods in the day.

In short, one size doesn’t fit all. There is no magic answer as to how many steps you should take every day. The best answer is whatever personally motivates you – and that might be to not even track how many steps you’re taking.

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Early in my career, my departmental VP hired a well-known management consulting firm to diagnose why an important project had failed. The management consultants used a technique called “5 Whys” to get past discussions of the failed outcomes and try to unearth the root causes. As the name implies, the technique is based on asking Why five times in a row – with the assumption that by the last Why you’ll reach the true cause of a problem.

This technique can work well (here’s a real-world example from a kitchen range manufacturer) but has its limitations. In a complex situation, you can suffer from tunnel vision which might lead to an incorrect conclusion. To counteract this, there are variants of the 5 Whys technique which intentionally create branches (alternative theories) and therefore explore new ground. In mathematics this is called avoiding a local maximum to search for a global maximum. For those interested in learning more, here’s a good article on variants of the 5 Whys technique.

The recent Elon Musk-fueled craze around first principle thinking reminded me that everything old is new again. First principle thinking, for those who haven’t seen the endless articles trumpeting this breakthrough, is a mode of thinking “designed to relentlessly pursue the foundations of any given problem from fundamental truths.” In science, theoretical work is considered to be from first principles if it starts directly with established science and doesn’t rely on assumptions. This approach isn’t really new: A case can be made it originated more than 2000 years ago with Aristotle.

Confused? Here’s a frequently-referenced video of Elon describing how first principles thinking challenged the traditional economics of making batteries:

5 Whys and First Principles are both examples of critical thinking but differ in how they approach the problem. 5 Whys is top down; starting with the observed result and trying to discover the underlying cause. On the other hand, the First Principles approach builds from basic truths to discover new solutions. Each has merit and is appropriate in different circumstances. When in doubt, try them both.

Regardless of which approach you use, the key to critical thinking is to be sensitive to your built-in biases and learn to challenge surface explanations. Be sure to ask more questions; just remember it’s hard to ask good questions.

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After writing about several examples of bias from insensitivity to sample size, a former colleague asked whether I thought performance in the workplace was subject to the same bias. She observed people were sometimes rewarded or even promoted for high performance, even if that performance was sporadic rather than sustained. She asked:

Shouldn’t the promotion of an employee be tied to a larger sample size (i.e., more time)?

The answer is heavily nuanced. In my own experience, performance at work is not solely based on your own skills. A variety of external factors come into play, including the difficulty of your tasks, the quality of your team, and even some luck. The more time you have to observe an employee, the more you can tell how much impact these external factors had.

James G. March, Professor Emeritus at Stanford University, describes this as the False Record Effect:

A group of managers of identical (moderate) ability will show considerable variation in their performance records in the short run. Some will be found at one end of the distribution and will be viewed as outstanding; others will be at the other end and will be viewed as ineffective. The longer a manager stays in a job, the less the probable difference between the observed record of performance and actual ability.

This doesn’t necessarily mean that seniority is a better way to determine promotions. Organizations that favor “time in seat” over short-term success lose the benefit of fresh perspectives and can become stale.

We are more susceptible to the False Record Effect early on in an employee’s career when there is less performance data available to make decisions. As Professor March cautions:

Within a group of managers of varying abilities, the faster the rate of promotion, the less likely it is to be justified.

In other words, the fast start might be an anomaly and the employee could have a regression toward the mean.

So, what should an organization do to avoid the false record effect? In business as in sports, recognize we likely will be biased by the hot hand fallacy. When deciding on a promotion, don’t just consider an employee’s most recent performance. Check whether this performance has been sustained over time, with different teams, in different situations, and with multiple success criteria. This increases the likelihood this performance is repeatable.

When in doubt, promote those who have a history of making others better, not just themselves.

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The law of small numbers (or hasty generalization) is the tendency to jump to a conclusion without enough evidence. In statistics, it’s called bias from insensitivity to sample size – generalizing from a limited number of events (a sample) selected from a much larger number of events (the population). For example, if a mutual fund manager has three above-average years in a row, you might conclude that the fund manager is better than average and will have a fourth above-average year. While it may be true, you cannot come to this conclusion from such a small amount of data.

This bias shows up frequently in the media. Imagine the headline summarizing a telephone survey of 500 rural inhabitants, in which 70% responded they didn’t have enough money for retirement. It would likely be something like this: People living in the country can’t comfortably retire. Most readers would ignore the details of the survey, including whether 500 people is sufficient to warrant the conclusion.

This is the challenge with small sample sizes. If the survey had only included 50 people, you’d be suspicious. And if it included 50,000 people, you probably wouldn’t be worried. But what about 500? Our intuition is good at the extremes but not in the middle. Without more information, it’s hard to know if the 500-person sample size is sufficient.

In the book Thinking, Fast and Slow, Daniel Kahneman elaborates:

The exaggerated faith in small samples is only one example of a more general illusion – we pay more attention to the content of messages than to information about their reliability, and as a result end up with a view of the world around us that is simpler and more coherent than the data justify. Jumping to conclusions is a safer sport in the world of our imagination than it is in reality.

Here’s a test of your sensitivity to sample size, courtesy of Max Bazerman in Judgment in Managerial Decision Making: A town has two hospitals; 45 babies are born each day in the larger one and 15 are born in the smaller one. Overall, about 51% of babies are boys but of course the exact percentage varies day to day. For one year, both hospitals tracked the number of days in which more than 60% of the babies born were boys. Which hospital had more of these days?

A. The larger hospital
B. The smaller hospital
C. About the same (within 5% of each other)

Did you guess C? Most people incorrectly choose C when the right answer is B. Having 60% boys in one day is a rare event and statistics tell us that we’re more likely to observe a rare event in a small sample than in a large one.

You can also see this effect in sports. In a sport with a long season like basketball, hockey or baseball, the standings 20 games into the season may not be representative of the final standings. But at the end of the season, the best team usually has the best record. This effect is why sports fans are often optimistic; on any given day, anything can happen – especially in the playoffs. As Michael Lewis writes in Moneyball: “In a five-game series, the worst team in baseball will beat the best about 15% of the time.”

Which brings us to advertising. The classic commercial which claims “4 out of 5 dentists recommend…” has been mimicked many times over the years. The law of small numbers teaches us this claim is meaningless unless we know the sample size.

My guess is that there were only 5 dentists. Which means they could be wrong.

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One rainy afternoon I was watching a re-run of a detective show when I began to think about the phrase “beyond a reasonable doubt.” How much doubt, I wondered, is reasonable?

If you have an analytics background, you might fall prey to the temptation of trying to quantify the doubt. Using the language of statistics, you establish a confidence interval that a specific defendant is guilty. Given our legal system’s underlying principle of “innocent until proven guilty,” we don’t want to mistakenly send someone to jail. Therefore, we might set the confidence level as high as 99.9%, rather than the typical 95%.

However, Professor Gary Wells has shown this isn’t really how legal decisions work in practice. People are reluctant to assign liability when the plaintiff’s evidence is only based on statistical evidence – even if this evidence is compelling mathematically. Consider this example:

An application for child support has been dismissed despite a blood test showing it is 99.8% probable that the man being sued is the father of a four-year-old girl.

In a series of experiments, Prof Wells compared peoples’ willingness to believe so-called naked statistical evidence versus other forms of evidence such as expert testimony. Even when the probabilities of each being accurate were identical, the subjects were more than six times more likely to convict when the statistical evidence was presented as the rationale for an expert decision. In the above example, the plaintiff’s attorney might have been more successful if the expert had testified “based on a blood test that is 99.8% accurate, I conclude that the defendant is the father” rather than simply stating “based on a blood test, there is a 99.8% probability that the defendant is the father.”

To a statistician, there is no mathematical difference between the two statements. But psychologically, the experiments show people react more favorably to statements of belief which are based on probability than they do to the probabilities themselves. This is especially true in complex or ill-defined court cases in which attorneys are highly unlikely to be able to establish an accurate assessment of the probability of innocence or guilt.

You shouldn’t be surprised. In legal thrillers, colorful and charismatic attorneys are more likely to win cases. Persuasive employees seem to get their way more often analytical ones.

I’m not suggesting that facts don’t matter but sometimes beliefs are stronger than statistics.

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These three laws, simple as they are, form much of the basis of probability theory. Properly applied, they can give us much insight into the workings of nature and the everyday world.
– Leonard Mlodinow

That quote is from Leonard Mlodinow’s book, The Drunkard’s Walk: How Randomness Rules Our Lives. The book contains examples as varied as politics, wine ratings, and school grades to show how a misunderstanding of probability causes people to misinterpret random events. Mlodinow’s three laws of probability are as follows:

1. The probability that two events will both occur can never be greater than the probability that each will occur individually.
2. If two possible events, A and B, are independent, then the probability that both A and B will occur is equal to the product of their individual probabilities.
3. If an event can have a number of different and distinct possible outcomes, A, B, C, and so on, then the probability that either A or B will occur is equal to the sum of the individual probabilities of A and B, and the sum of the probabilities of all the possible outcomes (A, B, C, and so on) is 1 (that is, 100%).

When we don’t understand probability, we fall prey to the conjunction fallacy. As I previously wrote,

we might hear separate rumors that corporate budgets will be cut soon and that the senior executive for our department is considering leaving the company. We judge each of these events on their own as unlikely – perhaps a 33% chance of budget cuts (the company is doing well) and a 25% chance of the executive leaving (she’s been here for 10+ years). But when we hear both rumors, our intuition that both events will happen is quite high – perhaps 50% or more. As a result, we spend more time than we should worrying about the funding for our project and maybe even update our résumé.

Assuming the executive is not leaving because of the budget cuts (i.e. the events are independent), the probability of both happening is 0.33*0.25 or only about 8% – not likely at all. Even if the events are related, by law 1 the probability of both happening cannot be more than 33%.

The Drunkard’s Walk provides another example based on empty airline seats which I’ve modified to strengthen the point. Imagine an airline has only one seat left on a flight and two passengers have yet to show up (they’ve overbooked the flight). From experience, the airline believes there is a 75% chance a passenger who books a seat shows up in time. Mathematically, overbooking makes sense if your goal is to fill the plane: the chance that neither will show up and the plane will fly with an empty seat is very low: 0.25 * 0.25 is 6%. On the other hand, it’s risky from a customer experience point of view: There is a 0.75 * 0.75 = 56% chance both show up and they must deal with an unhappy customer. From law 3, the probability that it all works out perfectly and one (and only one) person shows up is less than 38% (1 – 0.56 – 0.06). These are not great odds and yet airlines do this all the time.

Of course, the above assumes the passengers are independent. If they are traveling together, the situation is even worse. The chance both people will show up is 75% and that neither shows up is 25%. There’s literally no chance that exactly one person shows – the situation the airline is counting on. A combination of ignoring customer experience and not understanding probability might explain why we’ve had so many unfortunate airline incidents recently.

Yes, I realize this post had more math than you might be used to in my writing. The same is true with the book. But that’s sort of the point: we all need a little better understanding of probability if we want to make sense of our surroundings.

Or as Mlodinow writes, “probability is the very guide of life.”

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