Pareto is a good mate of mine, and we both enjoy running. When I’m not injured, I love the strava stats, my own spreadsheets, planning a marathon block, and tracking my progress. It would seem that running is simply an excuse for some more recreational maths in my life. Who is Pareto, you may ask? He’s my mate who I told about the 80/20 rule, and it seemed to have done well for him in his career.
The Pareto Frontier Threshold is apt for my own running tracking. Define a curve and determine how well I should expect to do in any given race. Amazing. But what form should the curve take? For this, I’ve done some more weekend maths.
I’ve had a couple of big races, which I have had some big training blocks captured in recent data. These are: 2022 Melbourne Half Marathon, 2025 Sunset Series (three 8 km races), 2025 Ballarat Half Marathon, 2025 Run Melbourne (for the Cure Brain Cancer Foundation). I also have a track record of treating some parkruns like races and have set my best 5km times at these events.
Interval running is of course, a problem. These are fast and hard runs, but have a very slow total elapsed time (and pace), so most of these are out. I have however taken some of my fastest individual 1 km intervals into their own activities as I have no data below about 4 km.
Running As Fast As I Can (to calculate my performance curve)
I’ve not raced every distance and I’ve never raced on a track. So putting all of my data into Power Query Editor, I’ve managed to sort it all out. Most of my runs naturally aren’t races, and most of the best races are the main Pareto Frontier points with a few extras. I thought it would have been a cleaner fit, but as I’ve not raced all distances with equal effort at a consistent level of fitness, I guess this is my staggered performance curve.
Quite a few non-continuous (interval, fartlek, etc.) runs are beyond the frontier, but they’re marked as crosses not considered in the Pareto curves seen below. I’ve calculates this as “if the elapsed time is more than 10% longer than the moving time, mark as interval”.
Having thought that I’d have smooth lines, I found all of the world records for all distances from Wikipedia and plotted them out below. This has not disappointed my smooth brained smooth line enjoyment. The data perfectly fits a nice looking curve.
To fit this data, we can use a double exponential for the short and middle distance, and then the long distance speed.
$$ \text{Speed} = A e^{-b \cdot D} + C e^{-d \cdot D} + F $$
\( D = \) Race Distance in kilometres (x-axis)
\( A = \) Sprint modifier (men = 12, women = 15)
\( b = \) Sprint decay rate (men = 0.8, women = 1.2)
\( C = \) Endurance modifier (men = 8, women = 8)
\( d = \) Endurance decay rate (men = 0.0085, women = 0.0075)
\( F = \) Floor (men = 15.5, women = 13.5)
Notably, you can see above that the endurance decay rate for women is less than men out to the marathon distance. It’s a well established fact that women are better super-endurance athletes and this slice can also show that women’s long distance endurance drops off slower.
Walking and Cycling
Just because I’ve already extracted all the data, I can also do walking and cycling. I’ve done a lot of hikes and walks, but it’s nice to show that there are some long flat walks which might show an aerobic threshold for walking speed/pace. Luckily as there’s no need for knowing when all my races have been, because there have been none, and we can see that the resultant trends are much smoother.
But again… cycling has let me down. With many commutes at 16km and the occasional 50km+ weekend warrior rides with a long coffee and croissant half way, the non-continuous category is absolutely upsetting the data here. All those green crosses on the right are lost to the data filtering. I’m going to absolutely push some longer continuous rides, simply for the data!
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