On Sat, 21 Feb 2004, Robert Frick wrote: > Which reminds me of a story. I gave a student two numbers (say, 376 > and 378) and asked him the average. He answered 377, quick as a flash. > Then I asked him how he calculated the average, and he said, quick as > a flash, add the numbers and divide by two.
Of course he did. He knows you're a psychologist, and he knows that you know that the way to get an average is to add up the numbers and divide by their cardinality (not that he would put it that way!), so that's what he told you. And possibly he even thought (until you asked the next question) that was how he did it himself. > Then I asked him what was 376 + 378. He couldn't do that without > pencil and paper. (1) You don't know that he _couldn't_ do it without pencil and paper, only that when asked he used pencil and paper. (Absence of evidence is not evidence of absence; this situation is a corollary.) (2) The original question did not ask what the total was, so there was no reason to concentrate on remembering the total, even if one had more or less instantaneously calculated it. But your inference may be correct, even though the reasoning be fallacious: he may well have called on a different algorithm for finding the average of two numbers (as I recall, a couple of possible algorithms were suggested by others in this thread). And if the internal routine for finding an average has become sufficiently automatic, he may indeed not be able to trace the memory of his own thoughts in the calculation: as you say of yourself. Now the examples you've mentioned all involve reporting the average of two integers that differ by precisely two. This is a rather special case, for which one possible algorithm is both ridiculously easy and eminently forgettable (if one isn't paying VERY close conscious attention to one's thought processes -- close enough, possibly, to interfere with the processes): add 1 to the smaller number. Works every time, for two numbers that differ by 2. Do you get the same quickness (and correctness) if you ask about the numbers 376.3 and 378.3 ? Or 37.6 and 37.8 ? Or 376 and 398? Or 376 and 388 (which I would expect to be harder than 376 and 398)? How about numbers that differ by 3: 342 and 345, e.g.? As someone has pointed out, in this situation the mid-range = the mean of the two numbers. So try presenting situations in which this is not true. What's the average of 36, 37, and 39? Of 25, 27, and 29? Do you get different results (in either accuracy or latency) if you ask for "the mean" rather than "the average"? > So he wasn't adding the numbers. Non sequitur, as I have tried to point out. Possibly true, but non sequitur. > I thought for a while how he actually did figure out the average. Did you ever ask him how he did it, in view of the evidence that he did not _seem_ to be adding the original numbers? > I thought about the possibility that he stripped off the front > numbers, added 6 + 8, divided by two, and then restored the front > numbers. I don't think he knew math well enough to do that, Depends on what you mean by "math". Blanking out the common digits, and then restoring them, is not necessarily what some would mean by "math" -- it's more like Hofstadter's "typographical number theory", and hardly constitutes "calculations" in the usual sense. And if it's a dodge one had learned very young, without paying much attention to it or to its implications, one might very well DO it, just not find it so easy to TALK about it. It may be worth recalling that halving and doubling used to be viewed as arithmetic operations in their own right. Perhaps there was a reason for that. Halving a difference and adding the half to the smaller of two numbers may be in a sense "natural". (Of course, as remarked above, one cannot tell whether this would have been in the service of finding the mean, or the mid-range.) > or that he could do those calculations as quickly as he could know the > average. Also, I don't think he would have had trouble with the > average of 999 and 1001 (assuming he knew they were two apart). > > My memory is that this worked with several students, and anyway it > works on me -- I can tell you the average of 376 and 378 without doing > any addition, and the answer comes so fast it is not easy for me to > know what I am doing. > > I think this generalizes to more numbers. If I give you a set like > <62, 69, 72, 58, 64, 69, 61, 57> you can give me a rough estimate of > the average without doing any addition. You have a lot of statistical > expertise, but I think anyone could do it. Or at least anyone could do > it if I asked about the center. If I graphed the numbers out, it would > probably be even easier. Do you have some evidence about this? It would be interesting to know (a) how rapidly a result is reported and (b) how accurate the result is. (When you say "a rough estimate" I'm impelled to wonder "with what precision?".) > So I guess I am talking about some very primitive ability to find the > center, Sounds entirely possible.. > somehow without adding. This I think is still conjectural. Possible, but not proven. > We call it average, fine, but that might disconnect average from the > student's idea of center, Not sure I see your point. Wouldn't this be more like connecting "average" with "center", rather than disconnecting? > and then when average is defined by a formula and the formula involves > adding the numbers, they might think that the more numbers there are, > the higher the average. There may be some nomenclature to clarify. When I first learned statistics (and when I was teaching it), it was made very clear that "average" was a generic term, that might be operationalized (though this term didn't appear in stats textbooks!) by "mean" or "median" or "mid-range" or even by "mode". And sometimes "mean" was further classified into "arithmetic", "geometric" and "harmonic" varieities. You seem to be using "average" synonymously with "arithmetic mean" -- although it's hard to tell, since in your examples the arithmetic mean and the mid-range are equal, and are both equal to the usual definition of a median (as the mid-range of adjacent values: the "mid-range" part is not inherent in the definition of "median", but merely a ... ummm ... convenient side condition imposed in order to arrive at a unique value instead of an infinite number of values; without a side condition, the median of 376 and 378 is any number larger than 376 and smaller than 378: 376.1, 376 + 1/pi, 376.849, ...). Cheers! -- Don. ------------------------------------------------------------ Donald F. 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