### The 1999 Mercier Scoring Tables: A How-To Guide

J. R. Mureika
Department of Physics, University of Toronto, Toronto, ON Canada M4L 3N8

Derek Covington
Athletics Canada, Suite 606-1185 Eglinton Avenue East, Toronto, ON, Canada M3C 3C6

Daniel Mercier
Département de kinésiologie, Universite de Montreal, Montreal, PQ, Canada H3C 3J7

Published: Athletics: Canada's National Track and Field / Running Magazine (April/May 2000)

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:

Weighted avg = ( 4 x P1998 + 3 x P1997 + 2 x P1996 + 1 x P1995 ) / 10 ,

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     10.270
```
The 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	  0
```
The 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:

Swomen = ( Smen + 370.23683 ) / 1.10218405
Here, Swomen is the adjusted women's score, and Smen the original men's. Additionally, for certain events which differ by techical reasons between genders (100mH, 110mH, 400mH, Javlin, Shot Put, Discus, Decathlon, Heptathlon, etc...) a similar method to that previously described is applied, with the exception that only five sets of points are used for the linear regression. For the women's performances in these particular events, the pointage for the ranks adjusted as follows:

```Rank		Score (women-only)
===================================
5th			971.1
10th			951.6
20th			933.7
50th			902.0
100th			873.8
```
Since 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)

• Mercier Scoring Development Center
• Athletics Research