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It was during the first European 100k Championships (Winschoten, 1992) that Gerard Stenger proposed in the "IAU European Newsletter" a formula to predict 100k results from known marathon results: [100k time = 2.85 * marathon time].
Roland Vuillemenot from France in the successive newsletter published a more accurate formula: [100k time = 2.79 * marathon time]. Both formulas were familiar, because Peter Stein from Netherlands, a Dutch coach (who coached for example Bruno Joppen, Dutch 100k record holder with 6:38:12) used as a rule of the thumb: [100k time = 3 * marathon time].
The discussion seemed to be about the accuracy of the constant k, that is: [100k time = k * marathon time]. The answer could not been given to the question "What is the most accurate estimate of k?". The value k = 3 by Peter Stein was used for recreative runners; those of Gerard Stenger (k = 2.85) and Roland Vuillemenot (k = 2.79) where obtained from elite runners. The value k was probably not a constant, but a value that depends upon the level of the runner.
From earlier research it was clear that not the running time, but the running velocity is probably the best quantity to scale running times from one distance to another. So a formula like: [100k velocity = marathon velocity - dv km/hour] would probably be more appropriate; dv being the difference in velocity.
Soon it turned out that: [100k velocity = marathon velocity - 2.925 km/hour], that is dv = 2.925 km/hour. In fact, the 2.925 km/hour was estabilished later during the study; the early values where lying between 2.8 and 3.0 km/hour. How the final velocity difference was obtained will be explained below.
Accepting the formula [100k velocity = marathon velocity - 2.925 km/hour] it is easy to show that the earlier formulas [100k time = k * marathon time] have a k value that depends on the level of performance. The result, based on the velocity difference of 2.925 km/hour, is shown in the table:
|t 100 km||v 100 km||v marathon||t marathon||k|
The late seventies at Philips Reseach Laboratories It was in the late seventies that Anton Smeets was involved in formulas for Scoring Tables of Athletics. One of his friend and colleague was interested in formulas for the prevailing Scoring Tables for Combined Events and Inter Club Competitions. Anton was familiar to the structure: (for running events) [Points = A/t - B] where t is the running time in a fixed distance event; (for throwing and jumping) [Points = A * sqrt(d) - B] with d being the distance thrown or jumped (in fact jumping is throwing your own body).
Both formulas are based on the physical principle that a performance is based on the escape velocity. For objects thrown, the object follows a parabolic curve, which explains the square root in the formulae for throwing. For running events the escape velocity is proportional to the running velocity, which explains the reciprocal of time for fixed distance running events.
In the late seventies the values of A and B were obtained for the athletics events from the "Leichtathletik-Mehrkampfwertung des Deutschen Leichtathletik Verbandes, Ausgabe 1969" (Scoring Tables of the German Athletics Federation, 1969). Later, the KNAU (Royal Dutch Athletics Federation) published values of A and B for several athletics events in their "Puntentelling voor Senioren an A/B Junioren, uitgave 1989" (Scoring Tables for open class and juniors, 1989). Here are some examples of those constants: (3000 meter men) A = 1077300, B = 1234.9; (5000 meter men) A = 1786833.9, B = 1145; (long jump men) A = 1094.4, B = 2075.3; (shot put men 7.26 kg) A = 462.5, B = 1001.8.
What can be obtained already is that A/d for running events is almost constant (at least for the long distance running events). So instead of using: [Points = A/t - B] where A and B depend upon the athletics event, [Points = a * v - B] can be used, where a is probably independent on the running distance, and obtained from [a = A/d].
Problem that remains is that B also depends upon the distance. The value B is decreasing by increasing distance. The decrease of B is linear if the distance is put on a logarithmic scale: [B = b - c 10log d]. From this, [Points = a * (d / t) - b + c * 10log d].
For men, for distances from 3000 meters up to and including the marathon, a = 359, b = 2483.39, c = 359. The value c = 359 implies that the mean running velocity v drops with 1 m/s if the running distance is 10 times longer, or 3.6 km/hour if the running distance is 10 times longer, or in another way, 1.084 km/hour if the distance in doubled. For women, for distances from 3000 meters up to and including the marathon, a = 399, b = 2425.39, c = 359.
Beyond the marathon, the running velocity drops more than in the sub-marathon region. This affects the value of c in the formulas. For men, for ultramarathon distances, a = 359, b = 4426, c = 779. For women, for ultramarathon distances, a = 399, b = 4368, c = 779. The formulas for women give higher points than those for men; it is a phenomenon that was known for a long time since the 1954 IAAF Scoring Tables. The constants a, b and c are an extrapolation of the 1954 IAAF Scoring Tables. In the Sixties only small adjustments are made to these tables, mainly due to the introduction of timing by photo-finish.
The structure of the tables were checked up in several ways. From numerous athletes personal bests are fitted by which we obtained the different velocity drops in sub-marathon and ultramarathon region. They asked themselves if the marathon was the point where the velocity drop changes, or if that was instead a time threshold somewhere between 2 and 3 hours.
The change of points when the level of performances drops, i.e. the choice of the values for a, were checked out by arrival statistics from marathon and ultramarathon events, by plotting the main running velocity of the participants versus the number of arrived participants on a logarithmic scale. For marathon events the velocity drop was about 0.85 km/hour each time the number is doubled. This was obtained by simply taking the running velocity of the first, second, fourth, eighth, sixteenth and so on athlete and obtaining the velocity slope. Marathons are needed with numerous participants, typically more than 1024 finishers. For ultramarathon no such races were available at that time, except the "100km Biel" results. However, soon the curve dropped more than 0.85 km/hour, probably due to the fact that we reached the region of walkers. From this, it was expected that the drop, expressed by the value a, was going to be similar in sub-marathon and ultramarathon regions. Furthermore was expected that the constants become doubtful as soon as we reach the region of walkers (i.e. when walking becomes more effective to cover the distance than running).
Finally it has to be recognised that the formula for points became a formula with both, running time and running distance in the right hand side of the formula: [Points = a * (d / t) - b + c * 10log d].
This formula can be used for both, fixed distance running and fixed time running (like 1 hour or 24 hour running events). The distance run in a fixed time running event determines which constants must be used; this holds in particular for 6 hour running for women where the lowest points are determined from the sub-marathon region constants while the others (above km 42,195 running distance) are determined from the ultramarathon region constants.
Finally, the feeling was that the constants for ultrarunning are at least reliable for distances up to 300 kilometers. Above 300 kilometres they probably do work also, but the amount of statistics available is too little to tell. Probably distance is no longer the determining factor, but speed is. Additionally the formulas become unreliable for running speeds less that 8 km/hour (i.e. 5 miles/hour).
The 1984 IAAF Scoring Tables for Combined Events It was clear that the 1984 Scoring Tables for Combined Events where not suitable for ultramarathon running. The fundamental structure of Scoring Table formulas, as introduced by the Austrian Dr. Karl Ulbrich, based on physical fundamentals, was replaced by a system that gives more point differences for elite performances, while it was intended to reward performances of combined events athletes equally. It is known however, that combined event athletes perform badly on the longer running events. The 1984 Scoring Tables where therefore not suitable to the task of rewarding road race performances in general and those of ultramarathon running in particular, which was the goal.
A lecture at the Dutch Masters Athletics Club Somewhere in the middle Nineties, Gerard Stenger inspired Anton Smeets again by facing him with the results of the WAVA Age Grading Factors at that time. It was obvious that the WAVA Age Grading Factors for 100 kilometres where not satisfying the objectives of fair scaling the masters results to an open class result. Later Smeets was invited to give a lecture at the annual meeting of the Atletiekvereniging Veteranen Nederland (the Dutch Masters Athletics Club).
Preparing his lecture, Smeets found out that all Master World Records could be scaled using the 1954 Scoring Tables, with several extrapolations. It turned out, that for all events the performances drop by 90 points for each age group of 5 years, that is in general the performance of a master drops with 18 point each year, regardless of which athletic event is considered. Some of the events gave some problems, due to the fact that different standards are used in the different age groups. Most of them could be solved by using the Junior tables of the earlier mentioned "Leichtathletik-Mehrkampfwertung des Deutschen Leichtathletik Verbandes, Ausgabe 1969" (Scoring Tables of the German Athletics Federation, 1969). Taking for example Shot Put, different weights are used in the different age groups. The German tables contain Open Class tables for 7.26 kilograms, and a 4.0 kilograms table for the young athletes (10 to 14 years). From that, the following constants for Shot Put for men were obtained:
|Shot Put 7.26 kg||462.5||1001.8|
|Shot Put 6.0 kg||51.7||1023.5|
|Shot Put 5.0 kg||443.1||1040.7|
|Shot Put 4.0 kg||434.5||1058.0|
It was unexpected that the "old" IAAF Scoring Tables worked so well also on age groups. It was even more surprisingly that the ultramarathon formulas fit in the whole system as well. It proved that the job of Smeets' Scoring Tables was done properly. However it proved something else: the system of Age Grading Factors does not work!
Age Grading Factors are based on the principle: [Open Class performance = k * Age Group Master performance]. From earlier work on Scoring Tables it was already clear that the factor k in such formulas depends upon the level of performances. If the WAVA Age Grading, factors are based on World Record Standards, thus they are not fair to combined event results. If World Record Standards develop, the Age Grading Factors have to develop with them. Life could be so easy, if the old IAAF Scoring Tables were used, and 90 points added for each age group of 5 years.
In case of Hurdles and the throwing events, the constants for different standards as showed above for the Shot Put had to be obtained. The formula became: [Points in Open Class = Points for Master performance + Basic master + 90 * age group number]. The "Basic master" turned out to be different for different athletic events; the age group number is 0 for M40, 1 for M45, 2 for M50 and so on.
The 2000 IAAF Scoring Tables In 2000 the IAAF introduced a new set of Scoring Tables, named the "2000 IAAF Scoring Tables for Athletics". They are intended to overcome the basic problem that combined event athletes are not distance runners. However, the structure is still not suitable to create a consistent set of formulas for fixed distance running and fixed time running. Nonetheless, Smeets tried to obtain the "best possible" formulas for road running. The structure of these formulas is: [Points = A * (B - t) ^ C] where the symbol ^ stands for raised to the power.
For distance running the power C is 2.0 for al long distance distance events. The 2000 IAAF Scoring Tables for Athletics only include 5000 and 10.000 metres on the track and the marathon as the only road running distance. The constants are: (Men 5000 meters) A = 0.002654; B = 1460; C = 2.0; (Women 5000 meters) A = 0.0008177; B = 2100; C = 2.0; (Men 10.000 meters) A = 0.0005567; B = 3120; C = 2.0; (Women 10.000 meters) A = 0.0001742; B = 4500; C = 2.0; (Men marathon) A = 0.000014682; B = 16800; C = 2.0; (Women marathon) A = 0.000007296; B = 21600; C = 2.0.
Smeets extended these formulas with the distances of the half marathon and the 100 kilometres: (Men half marathon) A = 0.000087555; B = 7380; C = 2.0; (Women half marathon) A = 0.00004167; B = 9600; C = 2.0; (Men 100 kilometers) A = 0.00000087484; B = 58800; C = 2.0; (Women 100 kilometers) A = 0.00000067079; B = 66600; C = 2.0.
For events where the result is expressed in a distance, like in throwing events or the fixed time running events, the formula becomes: [Points = A * (d - B) ^ C]. It turned out that a value of C = 1.2 is probably the best value for the fixed time running events, so: (Men 1 hour) A = 0.020444; B = 11350; C = 1.2; (Women 1 hour) A = 0.019018; B = 8320; C = 1.2; (Men 6 hours) A = 0.0028184; B = 50550; C = 1.2; (Women 6 hours) A = 0.0028549; B = 42195; C = 1.2; (Men 12 hours) A = 0.0012054; B = 76420; C = 1.2; (Women 12 hours) A = 0.0012516; B = 64000; C = 1.2; (Men 24 hours) A = 0.00055897; B = 118000; C = 1.2; (Women 24 hours) A = 0.00063207; B = 108000; C = 1.2.
These constants are obtained to meet the accepted mathematical structure of formulas of the IAAF. Smeets still believes the "old" formulas are better; moreover, they can be used to all road race events, while the proposed IAAF extensions are limited to those presented here.
Final remarks During the Nineties people where impressed by the running performances on 100 kilometres of Birgit Lennartz and later Ann Trason. Far before Tomoe Abe ran her fabulous 6:33 on 100 kilometres, the "old" Scoring Tables showed already that, based on the womens performance level on a marathon, a 100 kilometre time somewhere near 6:30 should be possible. After all it turned out to be correct, but at that time, nobody believed such times could be possible for women. During that time, the "old" Scoring Tables (with the constants a, b and c as given here) turned out to be very accurate for road running events. Not only recent development of World Record Standards showed the accuracy, also the observations from Age Groups of Masters give reason to believe that those Scoring Tables are very accurate for a fair comparison of performances in different athletics events.
Run The Planet thanks Anton H.M. Smeets of the IAU record committee for the permission to reprint the article "Scoring Tables for Road Running". Anton H.M. Smeets can be reached at the e-mail address Ton.Smeets(at)ou.nl with comments in English, German and Dutch.