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I don't know that there ever was an exact statistical argument that led to the requirement in the USA for repeat rides to be within 0.08%.

However, this value of 0.08% appears to me to be sensibly related to the aim that courses should not be short and the 0.1% SCPF included to prevent that.

Under good conditions with little calibration constant change due to temperature, good riders can get repeatability at the +/-0.02% level. However, with a difference of 0.08% or more one could use the shortest result and lengthen the course to ensure it was not short, but then there is a risk that the course could be excessively long.

I view the 0.08% rule or (as we apply it in the UK -- guideline) as a signal that a third ride may be needed to work out what is going on.
What is the USATF rule if the second ride is not within 0.08%? Obviously, a third ride is performed. Then it's clear what to do next if the outlier turns out to be the longest measurement. You just discard the outlier. But what if the outlier is the shortest measurement? That is, you have two measurements very close to each other, and then a 3rd one that is 0.12% shorter. Could we discard the shortest measurement in that case?

I realize this is an extremely unlikely situation, and the answer may just be let's cross that bridge if we ever come to it.
Mark;
What you describe happened to me a few years ago on a half marathon. It took 3 rides to get 2 within .08%. I took the longest of the first two rides as the "first counter" and then rode it again. Thankfully, this extra ride fell within the .08% of the first counter (whew) so I did what you suggested, discarded the shortest ride.
When measurements are not agreeing within 0.08% the problem might just be strict adherence to the "use the longer constant" rule, where you have perhaps calibrated in the early morning with cool conditions, and worked through the day as things warm up. I really think for experienced measurers that using the average constant will not only give you a closer approximation to the best possible measurement of the course, but will give you results that stand up well to a repeat measurement on a different day.
Bob,
I was talking about the scenario where you do a pre-cal, ride the course twice, and then do a post-cal. If the two rides are not within 0.08% when dividing by the larger constant, it's not very likely they will be when dividing by the average constant. We are not permitted to divide by two different numbers for the two rides in that case, right?
When I had a low-pressure mountain bike tire on my front wheel I used to do a quick cal ride between my course rides to make sure the constant wasn't changing too much. If I saw a big change I'd go ahead and do three more cal rides to help avoid a 0.08% problem due to cal-constant changes. After I switched to a high-pressure tire (which is much less sensitive to temperature changes) I stopped bothering with that.
Mark,
You are right about a measurement all on the same day. I have found that with measurements you do on different days there is more apparent consistency when you figure the distance based on average constants.
If I'm working over a long period of time I always try to do intermediate calibrations. It happens fairly often that the effective constant varies more than 0.08%, and there's always the chance that this will catch you with an excessive difference between measurements.
Interesting about the different tires. Another difference between those tires is the relative thickness of the layer of rubber. I'm thinking that is important as well-- and may relate to the problems folks have had in using small-diameter wheels as in fold-up bikes.
While I have rarely had this happen, it has happened to me as well. However, there is usually a reasonable explanation which mostly comes down to having one ride where I was either forced of line by a car or something or forgot where the next turn is and didn't have a good approach. In this case, if I felt really good about the shorter ride, I accept the distance as accurately measured. If I don't have a reasonable explanation, I ride it again.

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