# Fontainebleau grades are unevenly hard

I analyzed the data of bouldering grades available on Kaggle and discovered that not all grades contain the same difficulty range. While a runner knows that shaving off a minute from his 30 minutes run is about as hard as shaving a minute from a 29 minute run, it is not the case for climbers. When using the Fontainebleau grading system, it is on average 60% harder progressing from a grade without a “+” then with a “+”.

The code I used to perform this analysis is here as a Jupyter notebook, it includes additional technical details.

For most people, the climbing grades are just a useful approximate way to track progression and know which climb will match their skills. While it is the same for professional athletes, more effort is put into accurately pinpointing the difficulty of a climb at the highest level. Questions get asked such as why there are so many 8Cs compared to 8C+s? Is being stuck on a grade a sign that there is no improvement? Why are those 2 boulder problems the same grade if one is clearly harder? My analysis brings some light to those questions.

While the act of grading a single problem can differ according to opinion, we can use a large amount of data to better understand the Fontainebleau grading and how it’s used. For many activities, the performances of a population follow a Gaussian distribution. For example, this distribution of the runners pace follows roughly a Gaussian distribution. This might also the case for the distribution of the hardest bouldering grade climbed.

My starting idea is to see how well the distribution of climbers’ performance follows a Gaussian distribution. Any differences might give us some insight on the grading system.

# Data

I will use the data available on Kaggle. It contains information scrapped from the website 8a.nu in 2017. There are a total of 4 111 877 ascents recorded for 62 593 users, 19255 of which recorded boulder ascents. As a comparison, more then 5 million people climb in the USA, which makes this sample a small fraction of the global climbing population. Most of the users are from Europe, followed by America (mostly from the US) and the other continents contribute a small amount. It’s hard to say if that makes the data skewed, because I couldn’t find data on the global climbing population.

I decided to only work with data from the European population. Grades of routes on 8a.nu are entered in the Fontainebleau system. This leads to some peculiarities with data from the US such as a number of problems being rated as V3/V4, or some unusual conversion value from the Fontainebleau system. Since I want to analyze how grades are distributed, I want to analyze the grades directly given by the climbers without conversions. This means my analysis only concerns the Fontainebleau grading system.

The data can be obviously erroneous at times. Some people live in Antarctica, others in a country that doesn’t exist. Some people have climbed 9C boulders. In consequence, I removed every ascent above 8C+ and all ascents of problems that were not repeated to avoid most of the erroneous data. The details are in the Jupyter notebook.

I said earlier that “performance” often follows a Gaussian distribution. What is performance in bouldering? It isn’t clear, but I’ll use the max bouldering grade as an approximation. The maximum grades can be as low as 2, but few people on 8a.nu have a maximum climbing grade below 5A so I removed it from my analysis. Here’s the resulting distribution:

On the vertical axis is the percentage of climbers achieving a given maximum grade, named frequency. For example, 14% of the boulderers on 8a.nu have a max grade of 7A. The trend was calculated using Kernel Density Estimation (KDE), which calculates a local average. It is a continuous probability distribution of the frequency.

The max bouldering follows roughly a skewed Gaussian curve with a few differences. There’s a very high peak at 7A. Probably because people who just got to 6C+ make the extra effort to go to 7A. Going from 6C+ to 7A sounds a lot better then going from 7A to 7A+. I’ve seen this phenomenon in other contexts where people aim for a particular benchmark. Another really interesting aspect of this graph is that grades with a “+” (positive grade from now on) always has lower frequency then grades without a “+” (neutral grade from now on), particularly between 6A and 7C+.

To showcase this, compare the grade frequency to the trend. Positive grades are always lower than the trend. I calculated the difference between the trend and the data and showed it in the graph below for each grade:

In the range from 6A to 7C+, the natural grades are always above the trend while positives grades are always below. On average, neutral grades are 26% above while positive grades are 26% below the trend. When comparing to the trend, there’s about 2 times more people who a have a natural maximum grade than otherwise.

It’s a lot, and it’s a recurring pattern which makes it unlikely that it is caused by chance. I think that it’s because there’s a bigger difficulty range in natural grades.

Let’s think a bit theoretically here. When we measure the time for a 5K marathon we use seconds. And we know that if we use seconds to measure the performance, the resulting distribution of times will follow roughly a normal curve. The key factor here is that a second is worth the same, whether someone finished his race on the 900th second or the 1800th. While the 900th second is not as easy to shave off as the 1800th, it’s nearly as hard to shave off the 901th and the 900th second, because each second is worth the same. The change in difficulty from a second to the next is gradual. That is not the case for the Fontainebleau grading system. The neutral grades encompass a greater range of difficulty than positive grades, which means that going from low 7A+ to 7B is easier then going from low 7B to 7B+.

This is on average, because Fontainebleau grades each include a range of difficulties, and going from 6A to 6A+ is fast if you go from the hardest 6A to the easiest 6A+. Again, if we go back to the running analogy, running the 5K at a 8B+ level could be running at a pace of between 6 minutes per mile to 5:30 per mile. However, for running we use the more precise measure, an exact time instead of a time range. Taking all those ideas together, we could picture the Fontainebleau scale on a continuous difficulty scale like this:

From the previous calculations, 62% of the difficulty progression will be found on neutral grades (without a “+”) and 38% on positive grades (with a “+”). This means that on average 62% of your time is spent having a neutral max grade.

To confirm this theory, I looked at the number of boulder problems recorded. If natural grades have a higher difficulty range, it should be reflected on the number of boulder problems in each grade. There should be more problems with a neutral grade than a positive grade.

It’s pretty similar to the max bouldering grade distribution. And the average difference between the trend and data is the same as for max bouldering (less than 0.2% difference). This fits with the theory that the Fontainebleau scale is not divided in part of equal size.

Whether it’s by design or simply how the community made use of it, the results is that neutral grades take 62% of the difficulty.

# Impact

Now we have good answers to questions we asked at the start.

Why there are so many 8Cs compared to 8C+s? There are lot more 8Cs then 8C+s because of 2 factors. In general, there are more boulder problems for neutral (no +) grades then positive (+). It’s also because, for most performance metrics, the number of people performing at a very high performance drops sharply.

Is being stuck on a grade a sign that there is no improvement? No, the Fontainebleau grading is a staircase with high and uneven steps. Even if you are not reaching the step, you can get closer. Also, by its design, you can have 2 boulder problems with grades 6B and 6C (or any pair of adjacent neutral grades) that are closer in difficulty then two 6B boulder problems, as shown below. This makes the tracking of progression with few data points inaccurate.

Grades are the method we choose to measure a problem’s difficulty, as well as our progress. There are some intricacies that can cause confusion and a lack of precision. However, range grading is the only method we have to quantify climbing difficulty. I hope a better understanding of the Fontainbleau grading system will help people better understand what the grades truly mean and how their progress is measured.