The lanes of your Pinewood Derby track may not be exactly equal. Which lanes are the most closely matched? Four-lane tracks, when used for head-to-head double elimination racing, present a special problem... which lanes should we use?
Our district's Tiger race took place on an older 4-lane track with electronic finish line sensors. It was a double elimination race in which cars raced two at a time.
Following that race, I hosted a demonstration race involving the scouts which placed second in the first through fifth grade district races. The chart below was used for that race. The actual scoring is shown in red.
Knowing the results of the demonstration race, which lanes would you have used for a double elimination race?
| Lane 1 | Lane 2 | Lane 3 | Lane 4 | Car's Scores | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Heat | Car | Pl. | Car | Pl. | Car | Pl. | Car | Pl. | 1 | 2 | 3 | 4 | 5 | ||
| 1 | 1 | 3 | 2 | 2 | 3 | 1 | 4 | 4 | 2 | 3 | 4 | 1 | = | ||
| 2 | 2 | 2 | 3 | 1 | 4 | 3 | 5 | 4 | = | 3 | 4 | 2 | 1 | ||
| 3 | 3 | 4 | 4 | 3 | 5 | 1 | 1 | 2 | 3 | = | 1 | 2 | 4 | ||
| 4 | 4 | 4 | 5 | 3 | 1 | 1 | 2 | 2 | 4 | 3 | = | 1 | 2 | ||
| 5 | 5 | 4 | 1 | 3 | 2 | 1 | 3 | 2 | 2 | 4 | 3 | = | 1 | ||
| 6 | 3 | 2 | 2 | 1 | 1 | 3 | 5 | 4 | 2 | 4 | 3 | 1 | = | ||
| 7 | 1 | 3 | 5 | 2 | 4 | 4 | 3 | 1 | 2 | = | 4 | 1 | 3 | ||
| 8 | 2 | 1 | 1 | 3 | 5 | 2 | 4 | 4 | 2 | 4 | = | 1 | 3 | ||
| 9 | 4 | 4 | 3 | 2 | 2 | 1 | 1 | 3 | 2 | 4 | 3 | 1 | = | ||
| 10 | 5 | 2 | 4 | 4 | 3 | 1 | 2 | 3 | = | 2 | 4 | 1 | 3 | ||
| Car's Scores: | 19 | 27 | 26 | 10 | 18 | ||||||||||
| Finish Place: | 3 | 1 | 2 | 5 | 4 | ||||||||||
Table 1: Frequency count of finish places on each lane. For instance, lane 3 recorded 6 first place finishes.
Since the chart above satisfies the Complementary-Perfect criteria, each car raced the same number of times in each lane, raced each of the other cars the same number of times, and raced each competitor in lane reversal pairs.
If the lanes were perfectly equitable, the fastest car would probably have won his heats in each lane. The two heats in which the fastest car was not racing would probably have been won by the second fastest car. The numbers across the first row of the following chart would have been 2's and 3's only.
| Finish Place | Lane 1 | Lane 2 | Lane 3 | Lane 4 |
|---|---|---|---|---|
| 1 | 1 | 2 | 6 | 1 |
| 2 | 3 | 3 | 1 | 3 |
| 3 | 2 | 4 | 2 | 2 |
| 4 | 4 | 1 | 1 | 4 |
| Mode | 4 | 3 | 1 | 4 |
| Median | 3 | 2-3 | 1 | 3 |
| "Mean" | 2.9 | 2.4 | 1.8 | 2.9 |
Three measures of "central tendency" are shown below the tabulation of finish places above. Since the numbers are "ranks", it is technically incorrect to use the "mean" or "average," so pretend that I didn't do that. The several measures are consistent in the indication that lane 3 is substantially better than the other lanes. Lane 2 is better than the two "outside" lanes. (This is pretty common among multi-lane tracks... the central lanes are "faster than the outside lanes.)
But when racing head-to-head, what we want are "equitable lanes," not necessarily "fast lanes!" By all measures, lanes 1 and 4 are most equally matched.
Table 2: Finish places by lane and car. For instance, in lane 1, car 2 had a first and second place finish.
| Finish Place | Car # | Lane 1 | Lane 2 | Lane 3 | Lane 4 |
|---|---|---|---|---|---|
| 1 | 2 | 1,2 | 1,2 | 1,1 | 2,3 |
| 2 | 3 | 2,4 | 1,2 | 1,1 | 1,2 |
| 3 | 1 | 3,3 | 3,3 | 1,3 | 2,3 |
| 4 | 5 | 2,4 | 2,3 | 1,2 | 4,4 |
| 5 | 4 | 4,4 | 3,4 | 3,4 | 4,4 |
In summary, the track's lanes are rather poorly matched. Racers 2 and 3 are very evenly matched. Had they drawn different racing numbers, the results might have been reversed. Similarly, racers 1 and 5 appear to be very evenly matched.
None-the-less, the results obtained in the above competition are probably pretty accurate.
Lacking finish line electronics, similar results could be acquired by a series of two-lane round-robin races. Of course, if a reliable electronic timer is available, the whole issue is much easier to settle!
Latest update: 2/17/98
Copyright 1998 © by Stan Pope. All rights reserved.