4 ms·

Well, he does say 'seemingly paradoxical', and look what happens when you try and express this (naively) in English: "The shorter stick is typically (about) 2/

Well, he does say 'seemingly paradoxical', and look what happens when you try and express this (naively) in English: "The shorter stick is typically (about) 2/5 the length of the longer stick, but the longer stick is typically much more than 3 times the length of the shorter stick." To be honest, I can't figure out how to summarize the two results in English correctly (short of saying 'the mean of x/y~=2/5, but the mean of y/x does not exist, where x is the shorter and y the longer length')

_delirium 16y ago
There've been some psychological studies on averageness, and I believe one common case (though I'd have to look it up) is that people think of "typical" as a representative example in the middle of a distribution, something like a peak of a distribution and/or a median, and then imagine the distribution as a cloud of "stuff on either side of that". So the intuitive answer to the question would require there to exist a "typical" stick. Once you pick one specific stick, of course things work intuitively: the stick's dimensions are x=A, y=B, its shorter/longer ratio is min(A,B)/max(A,B), and its longer/shorter ratio is max(A,B)/min(A,B). An interesting question might be whether there's a definition of "typical stick" that in any useful way provides the answer that's intuitively wanted here.
- wlievens 16y ago
  Our brains aren't exactly wired to handle uniform distributions too, I guess. Nature is full of normal distributions anyway.
joshu 16y ago
I think the notion of the average ratio makes little sense. Maybe a geometric mean? I dunno.
jacobolus 16y ago
Your problem here is that “typically” and “the mean” are two very different concepts. Taking the mean of a set of ratios is frankly not very meaningful, and so saying that the shorter stick is typically about 2/5 the length of the longer is highly misleading.

_delirium 16y ago

There've been some psychological studies on averageness, and I believe one common case (though I'd have to look it up) is that people think of "typical" as a representative example in the middle of a distribution, something like a peak of a distribution and/or a median, and then imagine the distribution as a cloud of "stuff on either side of that". So the intuitive answer to the question would require there to exist a "typical" stick. Once you pick one specific stick, of course things work intuitively: the stick's dimensions are x=A, y=B, its shorter/longer ratio is min(A,B)/max(A,B), and its longer/shorter ratio is max(A,B)/min(A,B). An interesting question might be whether there's a definition of "typical stick" that in any useful way provides the answer that's intuitively wanted here.

wlievens 16y ago
Our brains aren't exactly wired to handle uniform distributions too, I guess. Nature is full of normal distributions anyway.

joshu 16y ago

I think the notion of the average ratio makes little sense. Maybe a geometric mean? I dunno.

jacobolus 16y ago

Your problem here is that “typically” and “the mean” are two very different concepts. Taking the mean of a set of ratios is frankly not very meaningful, and so saying that the shorter stick is typically about 2/5 the length of the longer is highly misleading.