1000m and mile records are “weak” compared to the 1500m record
You are welcome to use the model choosing any pair of WRs as boundary points that you think of as harder to get broken, with the caveat that the further apart the boundary points are, the less accurate the derived upper bound would be, e.g., an 800-1500 interpolation is going to slightly overestimate the distance compared to a 1000-1609 interpolation.
An even more accurate model would do a nonlinear interpolation instead of a linear interpolation based on a logarithmic curve fit over pace-distance data for WRs; doubling the distance roughly decreases pace by ~10% in middle distance regimes.
What upsets me so much as a mathematician is that you took the 1,000 and the *1609*. By your logic you should have taken the 1000 and 1500. There is absolutely zero reason to use the mile world record. ZERO. It is further from the 1000, run less often, etc. I would have totally respected your post if you used the 1,000 and the 1500m (I would still choose the 800m, but I definitely would have seen the logic in it). But by choosing the 1609 it looks like you are intentionally trying to make the distance as short as possible.
Obviously Dibaba's 2:45 for 1100 with a 64 recovery lap inbetween without stopping "doesn't count". But I do think it is relevant. I have a hard time believing that running an extra lap in 64 between the 2:45 would make you run faster over that 1100. Sure you can argue it could, but I also think your 1,070, while resaonable as a "best guess" is crazy to claim as an upper bound.
I strongly disagree with your use of the word upper bound. Yes you can say it is an upper bound "in your model" where you made some assumptions that clearly favor shorter distances and some longer, but that is not how mathematicians use that term. Mathematicians use the term upper bound by making every assumption favor greater than or equal to the true mark, which you clearly did not.
Mathematicians use the term upper bound by making every assumption favor greater than or equal to the true mark, which you clearly did not.
Sorry, I disagree. An upper bound is meaningful only in the context of and after establishing a model (aka assumptions), not as part of establishing the model’s assumptions themselves.
In this case, I’d say we actually seem to agree on the model itself, but you just wanted to apply the model I introduced to different data (with which I have no problem).
You are welcome to use the model choosing any pair of WRs as boundary points that you think of as harder to get broken, with the caveat that the further apart the boundary points are, the less accurate the derived upper bound would be, e.g., an 800-1500 interpolation is going to slightly overestimate the distance compared to a 1000-1609 interpolation.
What upsets me so much as a mathematician is that you took the 1,000 and the *1609*. By your logic you should have taken the 1000 and 1500. There is absolutely zero reason to use the mile world record. ZERO. It is further from the 1000, run less often, etc. I would have totally respected your post if you used the 1,000 and the 1500m (I would still choose the 800m, but I definitely would have seen the logic in it). But by choosing the 1609 it looks like you are intentionally trying to make the distance as short as possible.
Can’t help your upset feelings but rest assured there was no such deep or sinister intent. The choice of 1000 is arguably better (from a mathematical point of view) simply because the closer the calculated upper bound is to either end of the boundary in my model, the more accurately the upper bound would estimate the target distance, as I also note above. The more distant boundary point has a weaker effect in the model, so the choice of 1609 could well have been 1500 that would indeed be a bit more accurate, not something I would’ve considered would “upset so much” anyone.
Your points are all not “as a mathematician” at all but rather as an empiricist that has strong opinions on which data is more representative.
Funny thing is that we can do all the math in the world and decide on what the upper bound is (and how to calculate it) or, as track fans, we can take an educated guess and probably be about as accurate.
I'm not sure that math helps much with this question.
Funny thing is that we can do all the math in the world and decide on what the upper bound is (and how to calculate it) or, as track fans, we can take an educated guess and probably be about as accurate.
I'm not sure that math helps much with this question.
That’s an understandable position to adopt and it’s probably funner to make guesses based on various observations. I don’t think “math” and “educated guess” are mutually exclusive, and unlike you, I do think that simple math helps arrive at better estimates of a bound on the distance in the question.