Overview

Bio

My name is Sargent Burns and I am a senior at U-32 High School in East Montpelier, Vermont. Last year I was the 5th runner for the U-32 boys cross country team which became the 2nd Vermont boys team ever to win the New England Championship. I also play basketball in the winter, run track in the spring, and compete as a "mathlete" on the school's math team. I am a huge fan and connoisseur of both high school cross country running and mathematics, and with "Burns Scores" I seek to combine the two.

My Opinion of Tully Runners

Tully Runners is a New York/national high school XC database created by Bill Meylan which provides "Speed Ratings" to runners from across the country by adjusting race times based on the difficulty (hilliness, terrain, weather conditions, etc...) of well-known courses. These speed ratings, or "Tully Scores", attempt to set us high school runners equal so that we can convert (roughly) race times from Thetford, for example, to a fast course in California, as well as providing a basis for comparison/ranking of runners on the regional and national level. Scott Bliss wrote a fantastic article on his VTHSXC website with a much more detailed explanation of these Speed Ratings-please click here to read it. I encourage you to check out this VTHSXC site as well the Tully Runners website; both are great resources if you are interested in cross country.

Personally, I love Tully Runners and frequently explore the website because of my interest in data and statistics, especially in sports. I appreciate Meylan's dedicated work and I believe that his ratings are, for the most part, very accurate and fair. I understand that Meylan faces the near-impossible task of quantifying performances from all nooks and crannies of America and that he is scrutinized far too much for his amazing service to the running world. So I don't want to complain about slight differences in opinion between Bill and myself; he has far more experience and knowledge in this field and it doesn't do any good to argue about a 199 score which I think deserved a 200 or anything like that. However, there are a couple bones that I wish to pick with Tully Runners because I feel it less-than-optimally covers Vermont XC for two reasons.

1. It only speed-rates Woods Trail and States (and New England's every 5 years), meaning that several meaningful local races go unrated.

2. The mathematical formula for speed ratings simply isn't the best.

The first reason is by no means Bill Meylan's fault, in fact we all should be grateful that he pays ANY attention to our small state. He is primarily focused on New York races and larger national meets, so omitting Vermont results is totally justified. But as someone with a lot of knowledge about local race courses and historical results for Vermont XC, I feel capable of accurately rating smaller meets in the state. This way, instead of getting speed-rated only a couple times a year, Vermont high schoolers can measure themselves practically every week to observe progress. I will attempt to formulate "Burns Scores" for basically every Vermont meet throughout the season.

Mathematics of Speed Ratings

Now for reason #2, I don't want to dive too deep into the mathematics of speed ratings, but it is necessary in addressing the issues and possible fixes for Meylan's Tully Score equations. My main gripe with the Tully equation is that it treats 1 second as the same on every single course. If race time in seconds is the X-variable on a graph and Speed Rating is the Y, then each Tully Score equation is linear with the same slope of -1/3 for every course, regardless of difficulty. This -1/3 value means that for each second more it takes an athlete to run the 5k, the Tully Score will decrease by .333... points, and so every 3 seconds faster=1 unit of Speed Rating higher. This rate of 3 seconds per point makes things really convenient for comparison because the calculations are simple. For example, if a kid from another state runs a race that equates to a score 8 points higher than mine, I know that, according to Tully, he will beat me by 24 seconds, if I beat someone by 30 seconds my Tully Score will be 10 points higher, etc....

The problem with this universal, unadjusted slope is that it ignores the differences in difficulty of the courses. It implies that if I beat you by 40 seconds at Thetford then I will also beat you by 40 seconds at Woodbridge (super fast 3-mile race in California), when in fact this is not true. At New England's this year, Gavin Sherry got a 198 speed rating with a time of 15:54, and at the Woodbridge race Leo Young ran 13:38 for a Tully of 197.3. Therefore, a 13:36 at Woodbridge, according to Tully, is equivalent to a 15:54 at a muddy Thetford course. 816 seconds/954 seconds yields a .855 proportion between the two race courses so that, when applied to all results, a 40 second difference (lets say 18:40 to 18:00) at New England's would only be a 34 second difference (9:58-9:24 are the equivalent times using the .855 ratio) at Woodbridge.

The equations that Tully used for these races were: y(speed rating)=516-1/3x(total time in seconds) for NEs and: y=470-1/3x for Woodbridge. If you solve for x, you see that Tully thinks that a universal difference in race times of 0:46(3)=2:18 for runners of the same speed rating is accurate, but it is not. This is because when racing Gavin Sherry at New England's, you should be EXPECTED to lose by more time per unit of speed rating (in other words you are losing fewer points per second) than California kids up against Leo Young, due to the substantial difference in how long it takes to complete the course. Although less extreme, it is really no different than saying that if I beat you by 10 seconds in an 800m track race, I should also beat you by 10 seconds in a 1500m race, when in reality I should beat you by about 20 seconds in the 1500. I understand why Tully surmises that since the courses are all mostly the same length in XC (5k or 3 miles), the gap between Runner A and Runner B on each course should remain the same, but if a course is literally 15% faster to run like Woodbridge, it might as well be an entirely different distance of race. It is about TIME, not DISTANCE.

When graphing the Woodbridge and NE equations simultaneously, the flaws in using the same slope become evident: it displays a constant 46-unit gap in Speed Rating (the space between the parallel lines), for each and every race time, when really as you take more and more time to run the course, the greater the differences in rating should be for a similar race time. 198 guys may be 2:18 apart, but two 160 guys on the courses shouldn't also be 2:18 apart- it should look more like 2:38 apart because everything is harder at Thetford. If this explanation isn't sufficient for the flaws of Tully's universal-slope linear equations, I offer this analogy: In which scenario is a driver more likely to get a ticket: going 30mph in a 15 mph speed limit area or going 80mph in a 65mph? Obviously going 30 in a 15 is worse because of the larger ratio. But Tully's formula implies that since it is a 15 mph difference in both, the driving is equally dangerous.

New Rating System: "Burns Scores"

I applied this concept of proportional speed ratings>differential speed ratings to my Burns Scores. I believe that runners are proportional to each other, rather than a constant gap from each other. For example, Runner A will always be 5% faster than Runner B, instead of Runner A always being 30 seconds faster than Runner B. Yes, the proportional speed rating method isn't perfect, but I think it is much more accurate than linear speed rating methods. Another flaw with the linear Tully system is that at a certain race time in seconds, the equation crosses the x-axis meaning that the rating goes to 0 and then becomes negative, which is clearly unreasonable and actually discriminates against slower runners. For most courses, you have to break 25-26 minutes in a 5k to just have a positive value for your speed rating, and any slower would be a negative speed rating(?). This speaks to the inadequacy of Tully speed ratings for slower runners (especially in Vermont with some of the hilliest, slowest courses). Tully is suited for and most accurate for the top male and female runners in the country and as soon as the times drop off its flaws become apparent.

So when brainstorming possible types of equations for my new and improved scoring system, I thought about the theoretical extremes. As a runner's time approaches infinity seconds for a 5k, shouldn't their score approach zero? Yes. And as a runner's time approaches zero seconds (lightning fast), shouldn't their score approach infinity? I think yes. The parent function that achieves these two extremes is a reciprocal function, one which has a constant in the numerator and a variable, x, in the denominator. This allows for the score to be a decimal representing the ratio of a certain time and your time, x. I chose to make the "certain time" something that I've dubbed a "'Vermont Elite' Time". This "Vermont Elite" time represents a time that would indicate that a male or female runner is "likely" the best in the state or would typically be a top runner in the state. And this allows for the Burns Scores to be scaled to Vermont runners specifically.

I chose times that would roughly equate to a 135 Tully score for a girl and 185 Tully for a boy as the "Vermont Elite" times (I disagree with Tully in some areas but I trust the high speed ratings like 135 and 185 because Tully uses well-known elite runners as a basis for the equations). Evan Thornton-Sherman achieved a 188 in 2021, Brady Martisus a 184 in 2020, Henry Farrington a 187 in 2019, Tyler Marshall a 189 in 2016, I'm sure Mint Henk would have been in the mid 190s in the early 2000s, while some years no one in VT even broke 180, so I am comfortable with this mark of a 185 Tully score-equivalent as the numerator in my reciprocal function. On the girls side, Ava Thurston has achieved a 130 in 2021, Rena Schwartz ran a 138 in 2017, Autumn Eastman a 144 in 2013, and Erin Sullivan ran a 168(!) at Footlocker in 1997, while, again, some years no one broke 130. Currently in 2021, 214 girls have broken 135 in the Tully Runners Highest National Speed Ratings, and 213 boys have broken 185, so I feel that my chosen values are accurate relative to each other and to Vermont.

The next step in my journey for maximally accurate VT speed ratings was determining what exactly these "Vermont Elite" times would be. The equation was going to look like y=a/x, with 'a' being race time, in seconds, for a 185 guy/135 girl on a specific course. Thetford (and just recently Hardack as well) is the only course in Vermont to be Tully Scored so I have to use historical data, knowledge of performances, weather conditions, and my own judgement to find the correct numbers to plug in to the equation for the previously-unrated local races. With Thetford it was easy to locate these "Vermont Elite" numbers because it is a well-known, well-scored course that all Vermont runners will run at least 1 and probably 2 times a season. And with those other VT courses, I already knew so much about their speed and difficulty from running on them and from relentless athletic.net observation.

A final touch that I added to my scoring equation was multiplying the result by 100 in order to create a "percentile-based" scoring system which pretty much measures how "close" a runner is to being an elite Vermont runner. This percentile-based system is less arbitrary than Tully's system which have scores that go below zero on the low end and above 200 on the high end. Although some great performances will break that 100% barrier, the scoring system is essentially from 0-100 which I feel makes sense.

Just so you can visualize Burns Scores I will again use Woodbridge, California as an example. The Tully equation for this race is 470-1/3x, whereas the Burns Score equation (this time scaled to California's elite times) for girls is y=100(939/x) and for boys: y=100(816/x). This means that a girl who runs 15:39 would earn a Burns Score of 100 because they have reached that elite mark of 939 seconds and likewise a boy who ran a 13:36 would earn 100 as well. So the winner of the girls race, Dalia Frias with a 15:43.5, would receive a Burns Score of 99.523 because 939/943.5=.99523, and multiplying by 100 makes it 99.523. On the boys side, Leo Young's 13:38.1 would earn a Burns Score of 99.743 because 816/818.1=.99743, x100=99.743. A girl who ran 18:00 would get a score of 86.94 because 939/1080=.8694, and a boy who ran 20:00 would get a 68 because 816/1200=.68. These are very rough numbers because I am not well-versed in California XC, but I hope you get the point and understand how this novel rating-system works.

Key Points

1. Burns Scores=reciprocal equation, not linear.

2. I believe in proportional evaluation of runners as opposed to differential (constant gap).

3. I will rate most, if not all, VT races.

4. ~185 for boys and ~135 for girls=elite Vermont Tully Scores.

5. Separate equations for girls and boys derived from these elite Vermont marks.

6. Equation = (Vermont Elite Time in seconds)/(your time) and then multiplied by 100 for a percentile-based value.

7. Higher score=better, 0-100 scale, above 100 means that a performance was better than the typical "Vermont-best" performance.

Thanks for reading this and please contact me at burnssargent@gmail.com or @sargentburns1 on Instagram with questions, complaints, critiques, contributions to the site, or any other feedback.

Enjoy!

Sarge