uh-oh. Another math post! Feel free to ignore it.

Gary Robbins of Canada set a new course record at HURT (20:12) this week. But just how good is that? Comparing trail races is the holy grail of running math geeks.

I think I've found the best way to do it. One can linearize race results by plotting the finish time against the cube root of the place - don't ask why it works; I've spent 20 years justifying it. It works better as a first approximation than anything else does. For the H.U.R.T. 100 Mile, places 6-14 and 27-47 fall on a straight line (there's a weird divergence in between, due to the constraints of the model). The equation is 314.463x(cube root of place)+1027.59=minutes. The intercept, 1027.59, is the theoretical limit of fastest finish time, in this case 17:07.

Twice this time, or 2055, is the expected finish time of the average runner and that runner's expected place is the cube of (intercept/slope)= 34.89. A check of the results shows that this is close.

The effective field size is twice this place, or 70 runners. The expected finish time of runner #69.78 is 38:42, which is an expected cut-off time for the race. HURT is notorious for its short cut-off of 36 hours and its corresponding low finish rate.

The slope plus the intercept equals the expected winning time (22:32), which is almost always higher than the actual winning time. [It's worth noting that Kyle Skaggs' record at Hardrock is faster than the intercept!]

For those looking for a quick and dirty method, the intercept is approximately three times the eighth finisher's time minus twice the 27th runner's time (1052). For larger races, another approximation is 4 x the 27th runner's time minus 3 x 64th.

The intercept time should be the same for any race, year after year, with a variation due to weather. A runner can compare his time at one race to another by comparing fractions of intercept times.

For example: the 2007 Superior Sawtooth 100. The equation is 447.2364n(1/3)+1013.32=T.

T(1/2)=33:46.6

n(1/2)=11.63

T1=24:21 (23:17 actual)

T(last)=38:10 (cut-off=38 hours, 23 official finishers)

From this, one can see that HURT is about 1.5% harder than Superior (i.e. 1027/1013). The two are almost equivalent, though HURT has a reputation as a hard race and Superior is relatively unknown outside the midwest (where it's known to be hard).

No one did both races, so it's impossible to check how well they compare that way.

How would Wynn's winning time that year compare? 23:17x1027.6/1013.3= just under 24 hours, or third place. Does that mean Wynn would've finished third? No. One can't justify that; the method compares, it doesn't predict.

I'm going to try to record some top performances this way and see if my results match those of Ultrarunner magazine.

Hello world!

5 days ago

## 8 comments:

Wow. You are quite the mathematician. If you gave me 10 years and 100 monkeys with typewriters I would never be able to figure that one out.

love these posts - i have spent many many hours calculating and recalculating likely finish times using past data. and of course for the 100 mile distance it's extra difficult with the variables. but not an issue for me these days. the 50M is much easier on all fronts!

PS I am not just commenting so that I can get into my own blog :)

I really enjoy your math posts as well.

Were you out at Phalen on Monday? I either saw you or your doppleganger.

Hey it's me with the big boobs. I wish. You are seriously too smart for your own good.

I could have been in worse shape, but with more 100mile experience the year I did Superior. You don't have to be fast to win 100s, any extra bit of speed helps though. That said, it was my first time past the 50mile mark, so I pissed away a ton of time looking back. I did better with that at Heartland, but still not great. I'm in far better shape now, and would probably run faster at Superior if I were to do it again, but trails and 100mile are just so arbitrary. Anytime you have to voluntarily force yourself to walk just frustrates me to no end. Just does not seem like it should be "race" protocol, kind of like no crying in baseball. One should also be very weary at comparing 100mile finishing times to projected marathon time. That would be like taking a fast high school kid's 5km time and say he could run a marathon time parallel to that. Extremely doubtful! I'll leave the 100milers to you guys. I'm focused on a fast marathon.

Very interesting computations Steve. Superior 100 is a wonderful and at times hopeless journey. I remember totally perplexed and sitting alongside the raging caribou wayside in the pitch dark filling my bottles, contemplating why Larry had not put an aid station here this year and what I would have given even for a nasty gel at that given moment. Forging on I would find Sugarloaf to be an absolute oasis and my revival. What a long strange trip.

I see a list of what you won't be doing, does that change your goal races for the year? Wasn't one Superior 100 or marathon?

Yes Chris, I was at Phalen - I've done 20000+ loops of that lake in my life! Unlike the chain of lakes, the path there is usually ice-free.

Actually love reading all this stuff-Have to admit most of the time it's over my head, all those #'s ,figures,math, etc. Keep bringing it on Steve.

Never know what to expect next-

question: does all these blogs come together at night in a dream while your asleep,? If it does, I wish I had your dreams!!

Having returned from HURT where I DNFed due to injury, I'm just getting in the mood to read everyone's posts.

Having experienced both the HURT and Superior trails, my completely unscientific comparison is the Superior is difficult, but HURT is definitely more difficulty. In fact, ridculously hard.

Pure mathimatical equations miss on element; the conditions on race day.

HURT this year, for example, was the driest those trails have every been. Even from the years of living in Hawaii, I don't remember seeing the rainforest trails in that great condition.

This doesn't take away from Gary Robbins amazing performance. It truly was. I don't know if you did, but probably the more years' data you include, the more accurate you get.

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