David R. Henderson  

Numeracy Watch: 5 is Less than 10

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Andrew Oxlade writes:

He [Ian Spreadbury] declined to predict the exact trigger but said it was more likely to happen in the next five years rather than 10.

There's some pretty serious innumeracy going on here. I'm not sure if it's Spreadbury or Oxlade who's innumerate. Oxlade might be misreporting what Spreadbury said.

It reminds me of one of my favorite examples that Richard Thaler and Cass Sunstein use in their book Nudge. I use it when I teach numeracy in class. They write:

Again, biases can creep in when similarity and frequency diverge. The most famous demonstration of such biases involves the case of a hypothetical woman named Linda. In this experiment, subjects were told the following: "Linda is thirty-one years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice and aIso participated in antinuclear demonstrations." Then people were asked to rank, in order of the probability of their occurrence, eight possible futures for Linda. The two crucial answers were "bank teller" and "bank teller and active in the feminist movement." Most people said that Linda was less likely to be a bank teller than to be a bank teller and active in the feminist movement.

This is an obvious logical mistake. It is, of course, not logically possible for any two events to be more likely than one of them alone. It just has to be the case that Linda is more likely to be a bank teller than a feminist bank teller, because all feminist bank tellers are bank tellers. The error stems from the use of the representativeness heuristic: Linda's description seems. to match "bank teller and active in the feminist movement" far better than "bank teller."


Just as all feminist bank tellers are bank tellers, the next ten years includes the next five years.


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COMMENTS (27 to date)
E. Harding writes:

Maybe if it doesn't happen in the next five years, it becomes much less likely to happen in the five years following. I consider such stuff to be either misphrasing or people reading in between the lines; not really worth paying attention to.

Norman Carton writes:

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David R. Henderson writes:

@E. Harding,
The key word in your comment above is “Maybe.” The fact is that you don’t know what is meant; nor do I. If we take it literally, it’s wrong. If we don’t take it literally, we don’t know how to take it.

Radford Neal writes:

It's not plausible that people really make as ridiculous a logical error as they supposedly do in the "Linda" story. More likely, they interpret the question in a way that makes it the sort of question someone might actually be interested in. For instance, they may take a description of a "possible future for Linda" as describing not just some randomly selected features of her future life, but ALL the important features of her future life. So "bank teller" means "bank teller, and doing nothing else worth mentioning". There is then no logical fallacy.

It's generally not safe to assume that psychological experimenters are smarter than their subjects.

Jim Dow writes:

I would take it to be more in line with the phrase "it will happen sooner than later". In other words, whenever it happens, that date is more likely to be closer to 5 years from now than 10 years from now.

Greg G writes:

I am more inclined to view this as an example of poor writing than poor numeracy. It is true, as David points out, that we can't be sure which it is.

It is more charitable to view it as poor writing because then it has the virtue of being more of a falsifiable prediction. On this reading it would mean something like: I expect it to happen in the next 10 years and I think it is even more likely to happen in years 1-5 than years 6-10.

RohanV writes:

I suppose it's technically not parsing the question right. But I suggest that when humans are presented with a list of options, we assume that the options are mutually exclusive.

And if the wording in the question is non-mutually exclusive, we assume it was an error on the drafter's part, and read in the extra words to make it mutually exclusive. People who don't intend mutually exclusive choices are generally trying to trick you.

So the Linda story becomes:

1. bank teller and active in the feminist movement.
2. bank teller and not active in the feminist movement

Of the two options, the first is more likely, and tracks with the responses.

Similarly, the first example is not innumeracy, but unclear writing. Most people will read it as more likely to happen in years 1-5, rather than in years 6-10. Which is a perfectly legitimate way of looking at it.

Jameson writes:

Using language with mathematical precision obscures, rather than helps, communication. This is exactly why people hate laywers.

If you honestly don't understand this reporter's phrase, I think you're surprisingly inept with the English language. The phrase obviously means "sooner rather than later" as one commenter already suggested.

Most of our "biases" are absolutely necessary for communication. I can't imagine any non-adversarial human relationship in which taking the other person absolutely literally would be advantageous. Unless you want to go around "lawyering" people all the time, and making no friends in the process, you need to read between the lines.

David R. Henderson writes:

@Jameson,
Using language with mathematical precision obscures, rather than helps, communication. This is exactly why people hate lawyers.
Interesting that you and I have such different takes. Cut the word “mathematical” and you have identified one of the things I most like about lawyers.

john goodman writes:

Nice post.

Mark Bahner writes:
I would take it to be more in line with the phrase "it will happen sooner than later". In other words, whenever it happens, that date is more likely to be closer to 5 years from now than 10 years from now.

Yes, absolutely. It's just a way of saying, "It will more likely happen in less than 5 years, than in 5-10 years."

Brian writes:

Redford,

The Linda example is a famous experiment done by Nobelist Daniel Kahneman, recounted in his book Thinking, Fast and Slow. 85 percent of business grad students at Stanford got it wrong. He was able to get 64 percent to give the right answer when he gave them only two choices. Kahneman is clear that most people use their System 1 thinking to answer it, which is based on how representative the description is. System 2 has to be used to get a better result.

Tom West writes:

Interesting that you and I have such different takes. Cut the word “mathematical” and you have identified one of the things I most like about lawyers.

Precision in language is one thing I really do appreciate. But over the years, I have learned that my enthusiasm is definitely not shared with most of the human race.

I can take being called pedantic when an ambiguity is parsed the same way by 90% of the listeners.

But it's still a bit hard when my demand for clarity annoys the audience when it turns out that half the audience had parsed it one way, and half parsed it the other.

Turns out correctly understanding what was said is often far less important than not interrupting the flow. And this is for technical discussions!

norman carton writes:

Kahneman and Tversky’s “Linda” problem is a semantic trick rather than a test of “logical” or “rational” thinking. This point has been elegantly made by Steven Poole at http://stevenpoole.net/articles/within-reason/

The following quote is taken from Poole’s blog:

If we adopt a wider sense of “rational”, some of our apparent cognitive hiccups don’t seem so silly after all. Take Kahneman and Tversky’s famous “Linda problem”. Imagine you are told the following about Linda:

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Now, which of these statements is more probable?

1. Linda is a bank teller.
2. Linda is a bank teller and is active in the feminist movement.

A majority of people say that 2 is more probable: that Linda is a bank teller and an active feminist. Well, as Kahneman points out, 2 cannot be more probable in the statistical sense, since there are many more bank tellers (the feminist ones plus all the rest) than there are feminist bank tellers. People who answer 2, he says, think it’s more probable that Linda belongs to a smaller population than to a larger, more inclusive one. Mathematically, this is just wrong. So Kahneman’s story is that we are primed by the irrelevant information about Linda’s personality to commit what he calls the “conjunction fallacy”, and so we make an “irrational” judgment.

But this does not take into account some important nuances. Consider what the philosopher Paul Grice would have called the “conversational implicature” of the puzzle as posed. According to Grice’s “maxim of relevance”, people will naturally assume that the information about Linda’s personality is being given to them because it is relevant. That leads them to infer a definition of “probability” that is different from the strict mathematical one, because giving the mathematical answer would render the personality sketch pointless. (After all, we could reasonably wonder, why did they tell me this?) Thus, respondents who give the “wrong” answer are interpreting “probability” as something more akin to narrative plausibility. (Tellingly, psychologists Ralph Hertwig and Gert Gigerenzer reported in 1999 that when people are given the same puzzle but asked to guess about relative “frequencies” instead of what is more “probable”, they give the mathematically correct answer much more often.) One might add that, if we are talking plausibility, then the notion that Linda is a bank teller and an active feminist fits the whole story better. All the available information is now consistent. Arguably, therefore, it is a perfectly rational inference.

[unclear quotation marks changed to indented blockquote--Econlib Ed.]

John T writes:

Calling either of these examples of innumeracy is pretty absurd. As others above have pointed out, the first can be explained by people assuming that 'bank teller' means that she is a teller, and nothing else is notable - so being a teller that is also a feminist is more likely. And the second can be explained by people assuming that it means 'sooner rather than later.'

So the most we can say is that Nobelist Daniel Kahneman and fund manager Ian Spreadbury aren't very precise with their words.

Shane L writes:

Despite studying higher level maths in school, after four years studying journalism at undergraduate level my numeracy was definitely quite poor. Our undergrad had very little to say about numbers, yet many of us would be quoting statistics in articles. I left the world of journalism and have a stats cert under my belt now, but one does read amazing statements in mainstream media by journalists who presumably also had little training in numbers. I read an Irish economist bemused at such mistakes just a few days ago:

"The piece... tells us that “adults whose highest educational attainment is the Leaving Cert earn 31 per cent less on average than those with a higher certificate or ordinary degree, and 100 per cent less than graduates with an honours degree”. I’m sure we all sympathise with how tough it must be to make ends meet on the latter salary."
http://www.irisheconomy.ie/index.php/2015/06/09/arithmetic-manoeuvres-in-the-dark-at-the-irish-times/

Ouch.

John Fembup writes:

It sure seems to me that the 5 years/10 years statement is ambiguous. Either it means something it does not exactly say, or it is logically wrong.

Several commenters assert the meaning is obvious, others disagree. Doesn't that establish the ambiguity of the statement?

I think a person who understands basic things about numbers would not intentionally write an ambiguous statement using numbers. Which, I think supports the original point about innumeracy.

Radford Neal writes:

Brian: If 85 percent of Stanford grad students get it "wrong", that's pretty strong evidence that the answer is not actually wrong at all. And an explanation that they're using "system 1" (whatever that is), as most people do in the circumstances, is actually further evidence of this. The meaning of a statement/question is what most people will take it to mean. It seems most people take the statements in this example to mean something other than what the experimenter claims it means (a claim that is therefore clearly incorrect).

Whether the experimenter has a "Nobel" prize is not very relevant.

Jim Dow writes:

@John Fembup

I would argue that the "gist" of the statement is obvious but that it's not "precise". That is, there are various interpretations that are consistent with the gist and so it is ambiguous in that sense, but it still carries meaning.

This was part of a conversation and conversations often work that way. Start with a general point to convey the basics of what you're trying to say and then add to that if people want more precision/clarification.

Since he's using round numbers (5 and 10) it's clear that this is just a rough statement anyway.

Mark Bahner writes:
The piece... tells us that “adults whose highest educational attainment is the Leaving Cert earn 31 per cent less on average than those with a higher certificate or ordinary degree, and 100 per cent less than graduates with an honours degree”. I’m sure we all sympathise with how tough it must be to make ends meet on the latter salary.

Yes, there's a real example of innumeracy, rather than a simple failure to speak clearly.

Mark Bahner writes:
I think a person who understands basic things about numbers would not intentionally write an ambiguous statement using numbers. Which, I think supports the original point about innumeracy.

I think the guy (Ian Spreadbury) David Henderson is accusing of "innumeracy" was speaking, not writing. Moreover, he was being interviewed by someone, so it wasn't a prepared speech.

It sure seems to me it must have been a "slow blog day" to accuse Ian Spreadbury of "innumeracy."

John Fembup writes:

Regardless whether whether speaking or writing, I think a person who understands basic things about numbers would not intentionally use numbers to utter an ambiguous statement. (I reluctantly excuse politicians and used-car salesmen.)

So I wonder, has innumeracy now become . . . normal? Acceptable, even?

Reasonable Q's to ask. At least reasonable because the blog comment has generated so much effort to defend a statement that either (1) does not say what it means, or (2) is logically wrong. Why all the effort to defend such a comment, I wonder, if understanding of numbers - pr aything else, for that matter - is still valued?

Hazel Meade writes:

I think most[ people infer from the context of the question that "bank teller" actually means "bank teller AND not active in the feminist movement".

So what people are really answering is whether P("bank teller and feminist") )(greater than) P("bank teller and not feminist").

Similarly, when someone says "more likely within 5 years than ten", what they are really saying is P(less than five years) (greater than) P(less than ten, AND greater than five).

Hazel Meade writes:

Credit to RohanV for making exactly the same point upthread.

Andrew M writes:

There's something else wrong with the usual presentation (as by Thaler and Sunstein) of the Linda case.

The diagnosis of the respondents' alleged error is that they fail to see that a conjunction can't be more probable than one of its conjuncts--which indeed it can't.

But that diagnosis assumes that respondents take the question to be which of the two options has the higher unconditional (or prior) probability, whereas they are almost bound to take the question to be which of the options has the higher conditional (or posterior) probability, i.e., probability conditional on the information presented about Linda.

And the answer to that question isn't settled merely by noting that the two options have unequal prior probabilities.

David R. Henderson writes:

@Andrew M,
But that diagnosis assumes that respondents take the question to be which of the two options has the higher unconditional (or prior) probability, whereas they are almost bound to take the question to be which of the options has the higher conditional (or posterior) probability, i.e., probability conditional on the information presented about Linda.
If I understand you correctly, then your last paragraph is wrong. Add an additional constraint, and the probability is still lower.

Andrew M writes:

I could easily be wrong, but I was thinking in terms of the use of Bayes' Theorem to compute posterior probabilities, things like P(h/e), the probability of h given some evidence e. We have two hypotheses with different prior probabilities and one body of evidence, the facts about Linda (so that the denominator is the same in both equations). That just leaves the so called likelihood (i.e., P(e/h)) in each equation. Here the hypothesis with the lower prior probability might conceivably gain an advantage, because the probability of the evidence given the conjunctive hypothesis might exceed the probability of the other hypothesis given the (same) evidence by a wide enough margin to more than compensate for the difference in prior probabilities between the two hypotheses.

That might not in fact be probabilistically coherent despite appearances (and I can't be bothered to figure out whether it is), but even if it isn't, my point stands, which is that standard presentations of the Linda case don't say enough about what the respondents do wrong. They need to be shown to have computed the conditional probabilities (of hypotheses given the facts about Linda) wrongly.

Sorry if this still isn't very clear!

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