J. R. Mureika
Derek Covington
Daniel Mercier
Department of Physics, University of Toronto, Toronto, ON Canada M4L 3N8
Athletics Canada, Suite 606-1185 Eglinton Avenue East, Toronto, ON, Canada
M3C 3C6
Département de kinésiologie, Universite de Montreal, Montreal, PQ, Canada
H3C 3J7
Originally devised by
Daniel Mercier,
the Scoring Tables are the result of a
statistical comparison of all performances in Athletics. This is a joint
project with Athletics Canada, to be used in part for the purposes of
National Team selection and carding.
Briefly,
they are the end-result of a linear fit to the weighted average of
the 5th, 10th, 20th, 50th, and 100th World-ranked performances in each event
over the past 4 years (for these tables, 1995-1998). The performances from
more recent years are given a higher weighting (which can tend to skew the
comparisons if one of the events had a weak year). Explicitly, this
average is calculated as:
with the Pyear indicating the performance for each particular year.
The overall procedure is the same for every event, although there is one main difference for field events (also for the multievents).
For track events, the prescriptions is as follows. As an example, take the Men's 100m. The appropriate rankings for the 100m used in the tables are (from the Athletics Handbook for each year):
Rank 1995 1996 1997 1998 Weighted Avg ---------------------------------------------------- 5th 10.03 9.95 9.92 9.92 9.937 10th 10.07 10.01 9.98 10.00 10.003 20th 10.13 10.04 10.06 10.04 10.055 50th 10.23 10.17 10.19 10.18 10.186 100th 10.31 10.27 10.27 10.26 10.270The same procedure is applied to the Women's events, and so continuing the previous example, we have
5th 11.02 10.96 10.88 10.89 10.914 10th 11.09 11.03 11.05 10.99 11.026 20th 11.19 11.14 11.14 11.11 11.133 50th 11.33 11.31 11.26 11.26 11.277 100th 11.47 11.44 11.41 11.40 11.418
The weighted average performance (we'll call it Twt) for each event must always be converted to seconds before we proceed. In the case of the sprints and hurdle events it doesn't matter, but it will for 800m and up.
Once we have Vwt, we must calculated the weighted average speed for the event. As its name suggests, this is simply Vwt = D / Twt , where D is the race distance in metres.
The next step is to assign to each weighted speed (and hence each ranking) an associated score. For the 10 performance (5 men, 5 women), these are
Rank Men Women
5th 965.8 694.2
10th 947.3 673.9
20th 931.8 657.7
50th 910.0 625.3
100th 889.7 597.2
Thus, the average performances for each ranking are assigned the corresponding
score (up to the accuracy of the fitting, which is discussed later). A
zero-performance is added to each event (i.e. the performance
which would earn 0 points), bringing the total number of data pairs to
11. In the case of the 100m, this performance is 14.880s.Thus, the completed 100m data table looks like:
1995 1996 1997 1998 Twt Vwt Points 10.03 9.95 9.92 9.92 9.937 10.063 965.8 10.07 10.01 9.98 10.00 10.003 9.9970 947.3 10.13 10.04 10.06 10.04 10.055 9.9453 931.8 10.23 10.17 10.19 10.18 10.186 9.8174 910 10.31 10.27 10.27 10.26 10.270 9.7371 889.7 11.02 10.96 10.88 10.89 10.914 9.1625 694.2 11.09 11.03 11.05 10.99 11.026 9.0695 673.8 11.19 11.14 11.14 11.11 11.133 8.9823 657.7 11.33 11.31 11.26 11.26 11.277 8.8676 625.3 11.47 11.44 11.41 11.40 11.418 8.7581 597.2 14.880 6.7204 0The 11 (Vwt, Point) pairs are the subjected to a linear fit-- that is, we find the best straight-line equations
Points = A x Vwt + B Vwt = C x Points + D
which describes the data. Once the coefficients A and B (C and D, too) are found, then the point-value of any performance for the event in question may be determined.
Scores for field events are obtained in a similar fashion, but instead of using a weighted-average speed, the square root of the performance is used for the linear fit. For the Heptathlon and Decathlon, no further adjustment is made to the statistics, and the weighted average performance (score) for each year is used.
A ``Women's Only'' scoring table is obtained by a linear rescaling of the base tables: the associated scores/performances from the men's tables are adjusted by the following equation:
Rank Score (women-only) =================================== 5th 971.1 10th 951.6 20th 933.7 50th 902.0 100th 873.8Since the relays are a cooperative effort, the associated point scheme is shifted from the norm. Whereas aforementioned points are awarded for 5th, 10th, 20th, 50th, and 100th places, the same scores for the relays are given to 1st, 2nd, 4th, 10th, and 20th positions (in part due to the relatively smaller number of world-class rankings per country for these events). The lower score for a high performance can be attributed to the relay being a cooperative (not individual) event.
The majority of the linear regressions are good ones, giving linear correlation coefficients generally above r2 = 0.99 (for those who aren't sure what this means: a relationship is perfectly linear if r2 = 1, so that ain't too shabby!).
Work is currently underway at generating scoring tables for Junior performances, and should be available by summer 2000.
References:
Athletics 1999: The International Track and Field Annual, Association of Track and Field Statisticians (Peter Matthews, Ed.), Sports Books Ltd. (1999)
Athletics 1998: The International Track and Field Annual, Association of Track and Field Statisticians (Peter Matthews, Ed.), Sports Books Ltd. (1998)
Athletics 1997: The International Track and Field Annual, Association of Track and Field Statisticians (Peter Matthews, Ed.), Sports Books Ltd. (1997)
Athletics 1996: The International Track and Field Annual, Association of Track and Field Statisticians (Peter Matthews, Ed.), Sports Books Ltd. (1996)