# Is this an example of extrapolation?

I saw a joke on facebook today where professor cat states:

There are two types of people in this world: Those who can extrapolate from incomplete data

My brother thinks that this really isn't extrapolation. M-W defines extrapolate as:

to infer (values of a variable in an unobserved interval) from values in an already observed interval.

The joke seems to fit this definition, but I'm not convinced one way or the other. What do you think? Is there a better word for this kind of reasoning?

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But this isn't "a kind of reasoning". It's a joke, based on the unexpected twist of not defining the other type of people (who can't extrapolate?). Similar to "There are three types of people: those who understand binary, and those who don't." I think it's Not a Real Question. – FumbleFingers Aug 29 '12 at 0:56
But the joke is based on the definition of extrapolation. If this isn't extrapolation, then it takes steam out of the joke. Hence, my question. – DCookie Aug 29 '12 at 1:49
And I disagree, BTW. This is a type of reasoning. If it weren't you wouldn't be able to draw a definitive conclusion. Just because it's a joke does not mean there is no reasoning involved. – DCookie Aug 29 '12 at 1:59
@FumbleFingers- I thought the joke was "There are 10 kinds of people in the world..." – Jim Aug 29 '12 at 2:18
You say you're 'not convinced either way'. Can you elaborate on reasons for not being convinced? That would give something to actually work with, because as it is, it works fine for me. Hold on..your brother is the one with the problem? So you want us to justify someone else's position in an argument where we can only guess the issues? You need to elaborate quite a bit more. – Mitch Aug 29 '12 at 13:46

The straightforward answer to your question is that the joke is an example of extrapolation.

A discussion of the mathematics of extrapolation misses the point. The word extrapolate is not limited in meaning to a mathematical sense. The word has a more general primary sense¹ which is the sense of the word used in the joke:

to say what is likely to happen or be true by using information that you already have (Macmillan Dictionary)

When you work from information that you already have (the text of the joke) to discover what is likely to be (the implied punchline: “and those who can’t”), you are extrapolating.

The general sense of the word is about as long-lived as the mathematical one. Online Etymology Dictionary says examples of the word extrapolation date back only to 1867. The original meaning of extrapolation was mathematical, but the “[t]ransferred sense of ‘drawing a conclusion about the future based on present tendencies’” was already in use by 1889. The verb (extrapolate) began to be seen in 1874.

(I recommend Macmillan and Oxford over Merriam-Webster.)

## Notes

1. To check this fact, I consulted all online dictionaries on the first page of the Google search results for `[ define extrapolate ]`. Except specialized dictionaries, most report the general sense of extrapolate as the primary sense. All general-purpose dictionaries but Merriam-Webster qualify the mathematical sense as only applicable to mathematical, scientific, or statistical contexts:
– American Heritage Dictionary of the English Language, Fourth Edition
– Bing Dictionary
– Collins English Dictionary – Complete and Unabridged – technical sense first
– Dictionary.com
– Macmillan Dictionary
– Merriam-Webster – technical sense first
– Oxford Dictionaries
– V2 Vocabulary Building Dictionary
Specialized dictionaries (technical sense only):
– American Heritage Science Dictionary (science dictionary)
– Mosby’s Dental Dictionary (medical dictionary)
– Oxford Dictionary of Biochemistry (science dictionary)
– Saunders Veterinary Dictionary (medical dictionary)

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How does simply stating that it is extrapolation or restating a definition helpful? It sounded like the OP was looking for an explanation of the reasoning involved, so I tried to explain this. My explanation does not apply only to mathematics. In mathematics, the criteria would be more precise and slightly different. I simply explained in more detail how the reasoning was working, and thought such an explanation called for a slightly mathematical treatment. Math is the perfect tool when you want to understand a phenomenon generally and precisely. – Rachel Aug 29 '12 at 5:48
@Rachel Explaining the mathematics behind the mathematical meaning is moot. OP wants to know if the brother is right that extrapolation is improperly used because it means a mathematical procedure. The fact is that the brother is wrong about the meaning of the word. The word has a more general primary meaning which is the sense of the word used in the joke. – MετάEd Aug 29 '12 at 6:02
I didn't talk about the mathematics of extrapolation. I only borrowed and greatly simplified the mathematical idea of a function because this is the whole core of an extrapolation. If you want to understand what is going on, you need to have this idea. Otherwise, you can only be told dogmatically what others think the right answer is. The definitions provided don't distinguish a prediction based on logic and one based on coin-flipping. You need to consider how the input and output are related, so you need to think about functions. – Rachel Aug 29 '12 at 6:10
@ΜετάEd: There are certainly two senses of extrapolate, the mathematical one having the more rigid definition. I don't think that you have given enough evidence to support your claim that the "primary meaning is more general". The sense intended will be largely governed by the register. Without contextual clues, we're left with some confusion - an example may both involve and not involve extrapolation depending on the polyseme intended. This is usually the case where polysemy and hypernymy co-occur (ie more and less restricted senses of a word). – Edwin Ashworth Aug 29 '12 at 8:11
@Edwin Ashworth ~ the primary meaning is more general pretty much by definition, but even without that, most people are not mathematicians but speak about extrapolation anyway. Can you give an example of a word where the folk meaning is not more general than the expert one? – Roaring Fish Aug 29 '12 at 14:21

I'd be hesitant to claim myself to be a notable expert but, yes, that does infer extrapolation.

A more common example of extrapolation would be if a company's sales figures were:

• Year 1 - £10,000
• Year 2 - £15,000
• Year 3 - £20,000

Extrapolating that set of data, one would suggest that Year 4 would yield £25,000 of sales and Year 5 would see £30,000.

In your example, there are two items in the data set; two types of people.

Having been given the first type, those who can extrapolate (which is the observed interval), one will assume that the second type (the unobserved interval) is those who can't extrapolate.

Here, we know that there are two items of data and logic infers that the second will be the antagonist of the first.

It's essentially extrapolation on the smallest of intervals. It works because the range is defined as 2. If the range was defined as larger, as in "there are twenty types of people", one couldn't extrapolate the data.

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That was my initial thought, but based on the comment to my question, there's room for disagreement here. – DCookie Aug 29 '12 at 1:48
Did you just say infer when you meant imply? – tchrist Aug 29 '12 at 13:14

Speaking as someone who has used statistical analysis a bit, I agree with Ste's example of extrapolation, but I think that the reasoning involved in the joke is not quite as described. I also think that the definition is incomplete in that it misses1 a crucial component of what makes something an extrapolation: the function used.

If you're not quite familiar with the function concept, the pictures in the linked article are I think a good start. You can think of a function as a way of associating some input to some output. For example, if you turn on the burner on your stove, the amount of heat that it gives off is a function of the amount that you turn the control handle. If you drop a rock in a glass of water, the water level will rise. The distance that it rises is a function of the size (volume) of the rock. When you fill out a form, the box that you check for sex is a function of you. Different people will check different boxes, but a person has (normally) only one sex. These examples all use what we'd recognize as a rule, but you can also have arbitrary functions, where the value that comes out is not related in an obvious way to the value that went in; you might say the assignment is random (but I won't pursue the precise meaning of this).

In Ste's example, sales are a function of time (in yearly intervals).

Here's how I think extrapolation works at its simplest. You have two sets, an input set and an output set. In the terminology of the OP's definition, the observed interval is the input set, the variable ranges over the input set, and the value of the variable is the output set. The thing that relates a variable to its value is a function. An extrapolation assumes a definition for the function and then just plugs values into it. In Ste's example, the input set is the years {1,2,3}, and the output set is the sales {10000,15000,20000}. The function is:

5000*y + 5000 = 5000(y+1)

where y is the year. Thus, when you extrapolate and plug in a value to the function that is not in the input set, e.g., 4, you get:

5000*4 + 5000 = 5000(4+1) = 25000

Assuming that the input set and output set are defined by some particular function is the core power and risk of extrapolating. Ste and I both appear to have thought of the same function, but we might have picked different ones. Two functions might agree on the outputs for some inputs but disagree on the outputs for other inputs. This is why choosing the right function is tricky and important.

To see the importance of the function itself, consider whether making a prediction counts as extrapolating if you merely flip a coin to get the answer. I would say no, that it rather counts as something like guessing or leaving things to chance. Interestingly, mathematics also recognizes this possibly vague notion of what counts as a legitimate prediction function. According to the modern definition of a function, even something that arbitrarily assigns output to input counts. But this was not historically the case, and the difference between the old and new concepts is not trivial. Much mathematical research centers around the difference between arbitrary functions and those that are definable or concrete in some technical sense. This distinction probably captures what matters in the everyday case: rule-based vs. arbitrary.

So what is the function for the joke? How do you know how to complete the sentence? The joke depends on your being familiar with the cliche or snowclone:

There are two types of people in the world: those who __ and those who don't/can't.

There are some other variants too, which change the number of people or such. I like this one:

There are 10 types of people in the world: those who understand binary and those who don't.

Another variant would produce something like this:

There are two types of people in the world: those who don't understand quantum mechanics and those who only think they understand quantum mechanics.

(Um, this is funny (to me; I just made it up) because it implies that no one in the world does understand quantum mechanics, which was a favorite claim of one of its greatest teachers and practitioners, Richard Feynman.)

The OP's joke works by your knowing the pattern that these sayings follow; the pattern is the function. When you apply the function to the beginning of this sentence (the input):

There are two types of people in the world: those who can extrapolate from incomplete data

you have to extrapolate, using the function, to get the output:

and those who can't.

Or possibly, you could consider the whole sentence as the output. This is immaterial.

This is a little silly as an extrapolation since the output is only marginally a function of the input. You just have to know whether you should use don't or can't or whatever will make the sentence work grammatically. It is almost a constant function, which a rather trivial function. But I suppose it counts as extrapolation. The suggestion that it's induction is also interesting. I'm not sure what the difference between induction (as opposed to deduction) and extrapolation is off the top of my head. Maybe this would be a good other question.

Also, note that incomplete is really a pun here because the input datum is an incomplete sentence. :^)

Incidentally, there is another function related to the joke. It takes a person and assigns to them the value can extrapolate or can't extrapolate. The joke actually serves as an implementation of this function, which is kind of cute, methinks.

1. On second thought, perhaps they are trying to capture the role of the function with infer. If so, I think this is asking a lot of that word (and their readers' attention) even if you could technically argue that it implies some kind of rule-based process or such.
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+1 (actually +1 to all) - I have a background in math so I love this answer. I do like your quantum mechanics joke too :-) As for extrapolation vs. induction, I believe the answer lies in the confidence of your prediction, which you talk to in your discussion of the reliability of the function. Induction is less certain about the unmeasured value(s). – DCookie Aug 29 '12 at 13:20
@Rachel: A former maths teacher, I'd like to qualify the expression the function used in your second sentence above. Very often, as you say, one can match different variables; even then, there should be some reason for suspecting a relationship. Plotting a graph of number of dogs in a country against number of letters in its English name would give a relation (if you could count the dogs accurately) at a given time, but be meaningless. Where two variables would seem to be connected, ordered pairs are determined, then a tractable equation fitted to the data: a model / approximating function. – Edwin Ashworth Aug 29 '12 at 16:34

Extrapolation, given the MW definition, is used correctly (though the sentence itself is deliberately incorrect to make the joke, which may be why your brother doesn't think it counts).

Interestingly enough, professor cat is testing the reader's ability to extrapolate by using a rhetorical device known as omission, possible categorized as enthymeme or syllogismus. The power of omission in a logical argument is that the speaker intends for the audience to fill in the missing part, which persuades the audience for one or both of the two reasons:

1. The audience thinks they thought of the remaining bit, thus they take some "ownership" of the idea, having thought of it rather than heard it.

2. The audience thinks that the missing piece is so true it need not be spoken, hence the omission.

In the case of humor, it's not so much that we are persuaded or "sold" on the idea like in the second example, so much as we think we're the smart ones that got the joke since we came up with why it was funny in our heads.

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Actually, my brother thought it was something other than true extrapolation, e.g., induction - which is a very good word for it indeed. – DCookie Aug 29 '12 at 4:29
Is the distinction between the two that extrapolation suggests an independent or original idea or inference based on given data, while induction is more like filling in the missing piece, but there is only the one right answer to be filled in? – Anthony Aug 29 '12 at 5:48
@DCookie: Your brother is welcome to reword the joke when he tells it: There are two kinds of people in the world: those who don't have powers of induction. – J.R. Aug 29 '12 at 9:22
@Anthony, I believe induction is a line of reasoning where the answer is less certain, just more probable. Maybe an even better word is deduction? – DCookie Aug 29 '12 at 13:12
And your re-wording of the joke is nice. – DCookie Aug 29 '12 at 13:32

I think the correct and immediately understood word could be deduce — therefore ruining a good(?) joke for the sake of explanatory/correct etymology.

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