# Dictionary of Mathematical Eponymy: Mercier tables

I’m writing this during the delayed 2020 Olympics, and the natural question of trying to compare athletic performances across disciplines came up. Of course it would. It’s a natural question. And it reminded me that I had a bookmark about that.

You can compare athletic performances across different disciplines using *Mercier tables*.

### So what is a Mercier table?

A *Mercier table* assigns points to athletic performances, based on the performances of elite athletes over the previous few years. Here’s how it’s worked out:

- Pick an event
- Take the weighted average of the 5th-best performance among men over the last few years (more recent years are weighted more heavily). (If it’s a running event, it’s the speed in metres per second that’s used as the benchmark.)
- Do the same for the 10th-best, 20th-best, 50th-best and 100th-best performances
- Do the same for the women’s competition
- Assign points to each performance following a scheme listed here
- Add in a “zero points” performance ((How this is computed, I don’t know.))
- Calculate the line of best fit through your eleven points
- Use this to calculate the number of points for any given performance.

There are a few adjustments for throwing events and women-only events, but this is the basic idea.

For example, in 2019, the best UK athletics performance (according to the tables) was KJT’s 6981 in the heptathlon (1006 points), ahead of Dina Asher-Smith’s 21.88 in the 200m (992) and Laura Muir’s 3:55.76 in the 1500m (986).

### Why is it interesting?

For me, it’s interesting as much because there’s a little bit of mystery behind it as for its usefulness. (I’m a bit curious about how it compares to the points-scoring in heptathlon and decathlon, but not curious enough to do any research.)

There are two mysteries for me: how are the points values assigned? and, where does the zero-performance come from?

I can’t speak to the second of those. I presume it’s based on the statistics somehow, but again, I’ve done no research here. If you know any better, let me know.

I have a hunch about how the points are assigned, though. It looks to me like the $n$th-best men’s performance earns $1000 - A\ln(n)$ points, with $A \approx 23$. It’s not a perfect fit, but it look pretty close. The women’s performances are something like $750 - B\ln(n)$, with $B\approx 30$.

There’s some logic to this: if you take $N$ samples from a normal distribution, where $N$ is large, the top few of these typically follow an exponential curve quite closely. Now, elite athletics performances are anything but normal ((in any sense)), but it seems a reasonable enough model out here in the tail – and the numbers are picked to reflect that.

As I say, I’m not privy to the precise details of how the system was implemented – my instinct is that it’s more of a model picked to fit observations rather than one derived theoretically (this is, of course, a spectrum). Again, I’d be curious to know more!

### Who is Daniel Mercier?

According to the Mercier-Rioux scoring site, Daniel Mercier is a former 1500m runner and coach who runs a sports health start-up in France. He was born in 1956.

* Thanks to @nicolapipe for asking about these in the first place!