An Evidence-Based Approach to Measuring Cumulative Climbing Performance

Many attempts have been made to produce a reliable means of ranking climbing performances, especially in the context of cumulative performances (e.g yearly rankings, gym rankings, single-day competitions). Traditional methods, such as simply counting the number of ascents or the highest grade achieved, often fail to capture the nuances and challenges faced by climbers. A new metric, which we call the “Cumulative Performance Grade” (CPG), offers a more evidence-based approach to measuring cumulative performance in climbing.

The CPG is based on empirical studies that have fitted logistic regressions to ascent data, revealing a consistent pattern: for sport climbing (especially from grade 23/5.11d/7a onwards), the number of failed attempts before success, on average, doubles for each increment of grade (Drummond and Popinga, 2021). This means that for each increment in grade, a climber typically requires twice as many failed attempts before successfully climbing the route. For bouldering in the V-grades, the multiplier for failed attempts needed before success is approximately $e$ (2.71828) for each grade increment.

By taking into account the exponential nature of the relationship between grades and the average number of attempts required for success, the CPG provides a more accurate representation of a climber’s overall performance. The formula for calculating the CPG depends on the grading system used:

For sport climbing (Ewbank, French sport, and Yosemite decimal system):

\[CPG = \log_2(\sum_i 2^{\text{grade}_i})\]

For bouldering V-grades, the natural log is arguably a better scale:

\[CPG = \ln(\sum_i e^{\text{grade}_i})\]

where the i’th “grade” is the numerical representation of the climbing grade (e.g., 23 for 5.11d, 24 for 5.12a, etc. for sport climbing, and 0 for V0, 1 for V1, etc. for bouldering).

Choice of Slopes

While the fundamental empirical feature is that the slope between grades and failed attempts is exponential, the precise multiplicative factors are subject to estimation error.

The actual slopes estimated in the Drummond and Popinga paper are 2.1 and 3.17 for sport and bouldering V-grades, respectively. Rather than using these exact numbers (which may be refined by further research), we chose to use 2 and $e$ mainly for simplicity.

Since the numbers we have chosen are slightly lower than the estimated slopes, this could potentially slightly downweight harder climbs compared to their actual expected difficulty.

However, the data used to estimate these numbers was mostly based on amateur climbers, and there is some evidence that top-level and elite climbers have shallower slopes, meaning they are better able to translate their skills to harder climbs. For example, ascentstats.com estimated that Adam Ondra’s V-grade slope was actually 2.0, so using $e$ might even be too generous for some top-level athletes.

It is widely recognized that the bouldering grade bands are broader than sport climbing grade bands, thus the choice of 2 and $e$ reflects this difference while maintaining a simple and intuitive formula.

The CPG effectively weights each climb based on its difficulty, giving credit to harder ascents proportional to the expected increase in difficulty. This approach acknowledges the empirically measured increments in effort needed to succeed at higher grades and provides a fairer comparison between climbers with different preferences for allocation of effort.

One interesting interpretation of the CPG is that it represents the grade that a climber could have potentially sent if they had focused all their performance effort on a single route. This perspective highlights the cumulative nature of the metric and emphasizes the idea that the CPG captures the overall performance potential of a climber based on a body of work.

Michaela Kiersch’s Performance at the 28th Annual Hueco Rock Rodeo

To illustrate the application of the CPG, let’s consider professional climber Michaela Kiersch’s winning ticklist from the 28th annual Hueco Rock Rodeo outdoor bouldering competition in Feb 2024:

  1. Mr. Serious (V8)
  2. Better Eat Your Wheaties (V9)
  3. Swiss Crisp Mix (V10)
  4. Left Chupacabra (V10)
  5. Landjager (V11)
  6. Sunshine (V11)
  7. Rumble in the Jungle (V12)
  8. Phantom Limb (V12)
  9. Thrilla in Manilla (V12/13)
  10. Crown of Aragorn (V13)

Using the formula for bouldering, we calculate Michaela’s CPG as follows:

\[CPG_{MK} = \ln(e^8 + e^9 + 2e^{10} + 2e^{11} + 2e^{12} + e^{12.5} + e^{13}) \approx V14.01\]

This CPG value suggests that Michaela’s performance over the course of the competition was equivalent to sending a V14 boulder problem (if we had chosen base 2, then it would have been equivalent to V14.83!). A remarkable achievement.

Vadim Timonov’s Rocklands Campaign

For an example of sustained bouldering performance at the highest level, consider Vadim Timonov’s sends across 4 trips and 35 climbing days in Rocklands (data from Tenaya Blog):

  • 8 × V15 (8C) sends, including the first ascent of G-Master
  • 5 × V14 (8B+) sends/flashes
  • 4 × V13 (8B) flashes
  • Plus 4 more flashes from V10-V12

His 48% flash rate across these grades is remarkable. Calculating his CPG:

$CPG_{VT} = \ln(8e^{15} + 5e^{14} + 4e^{13} + …) ≈ V17.35$

With 35 climbing days: $\text{CPG/day} = 17.35 - \ln(35) = 13.79$

This means Vadim averaged the equivalent of sending V13.79 every day for 35 days. Interestingly, while his total CPG of V17.35 far exceeds Michaela’s competition CPG of V14.01, her single-day performance actually represents a higher daily output. This highlights how the CPG metrics capture different aspects of climbing performance - Michaela’s explosive single-day power versus Vadim’s sustained campaign at the cutting edge of difficulty.

Adam Ondra’s Historic 2022 Season

For an example at the absolute pinnacle of sport climbing, consider Adam Ondra’s 2022 tick list (data from GearJunkie), which included 51 routes from 5.13b to 5.15b:

  • 2 routes at 5.15b/9b (including two first ascents: Bomba and Wonderland)
  • 3 routes at 5.15a/9a+
  • 5 routes at 5.14d/9a
  • 4 routes at 5.14c/8c+
  • 6 routes at 5.14b/8c
  • 12 routes at 5.14a/8b+
  • Plus 19 routes from 5.13b-5.13d (8a-8b)

Remarkably, 25 of these 51 sends were onsighted or flashed, including multiple 5.14a/8b+ onsights.

Using the French sport grade numerical system (where 8a = 29, 9b = 37), we calculate:

\[CPG_{AO} = \log_2(2 \times 2^{37} + 3 \times 2^{36} + 5 \times 2^{35} + ... + 7 \times 2^{29}) \approx 39.6\]

This CPG of 39.6 is approximately equivalent to grade 9c in the French system. To put this in perspective: Ondra’s collective performance in 2022 was equivalent to focusing all that effort on establishing another 9c route (like his famous Silence).

The breakdown is particularly revealing:

  • His two 9b sends contribute 32.5% of his total CPG
  • The six routes at 9a and above account for 91.4% of the score
  • All routes below 8c contribute less than 2.5%

This perfectly illustrates how CPG’s exponential weighting captures the true difficulty of elite performances.

My performance on a week-long trip to the Blue Mountains

During a week in the Blue Mountains over Easter 2024 I compiled the following list of sends:

  1. Snappy Dresser (24/7a+/5.12a), Bell Crag - flash
  2. Good Big Dog (24), The Freezer - red point
  3. Survival Day (24), Sublime Point - second go
  4. Cutopia (23/7a/5.11d), Porters Pass - onsight
  5. Aroma Gunsmoke (23), The Freezer - flash
  6. Searching the Light (23), Bell Crag - second go
  7. Hairy Horrace (23), Centennial Glen - flash
  8. So, Said the King (21), Porters Pass - flash
  9. Nice Vice Baby (20), Porters Pass - flash
  10. Old Blobby (20), The Freezer - flash
  11. Double Standards (19), The Freezer - flash

The CPG of this tick list is 26.4 (approximately 7b+/c or 5.12c/d), which is to say, that if I had picked an appropriate 26/7b+/5.12c and spent the week on it, I would have had a good chance of sending. And indeed, the four 26’s I have sent have taken me around 5 sessions each on average.

The CPG allows you to quantify a body of climbing work and compare compulsive onsight/flash climbers with those climbers that prefer longer projects.

Daily Performance: CPG per Day

Another useful way to interpret CPG is to calculate the average grade sent per day, which gives us a “daily sending grade equivalent.” This is calculated as:

$\text{CPG/day} = \log_2\left(\frac{2^{CPG}}{\text{days}}\right) = CPG - \log_2(\text{days})$

For my Blue Mountains trip:

  • CPG = 26.4
  • Climbing days = 7
  • CPG/day = 26.4 - log₂(7) = 26.4 - 2.81 = 23.6

This means my performance was equivalent to sending a grade 23 (7a/5.11d) or better each day of the trip. This metric provides an intuitive way to understand sustained performance over time and allows for meaningful comparisons between trips of different lengths.

For example:

  • A weekend warrior sending one 25 (7b) in 2 days: CPG/day = 25 - 1 = 24
  • My week-long trip: CPG/day = 23.6
  • A two-week expedition with CPG of 28: CPG/day = 28 - 3.81 = 24.2

This shows that while longer trips may accumulate higher total CPGs, the daily performance grade provides insight into the intensity and sustainability of the climbing effort.

Cumulative Effort Grade: Measuring Total Climbing Effort

While the CPG measures the difficulty of successfully completed climbs, it does not capture the full picture of a climber’s effort and learning process. The Cumulative Effort Grade (CEG) extends the CPG framework by accounting for all attempts, including failed ones, providing a more complete measure of the total effort invested by a climber.

The CEG is calculated similarly to the CPG but includes both successful sends and failed attempts:

For sport climbing: \(CEG = \log_2(\sum_j 2^{\text{grade}_j})\)

For bouldering V-grades: \(CEG = \ln(\sum_j e^{\text{grade}_j})\)

where the sum is over all attempts (both successful and unsuccessful) and grade_j represents the grade of the route attempted.

The CEG can be interpreted as the CPG you would have achieved if every attempt had been successful - in other words, if you had sent every problem you attempted on the first try.

The Efficiency Relationship Between CEG and CPG

The relationship between CEG and CPG provides valuable insight into climbing efficiency:

  • CEG represents the CPG you would have achieved if every attempt had been successful (i.e., if you had flashed everything you tried)
  • CPG represents what you actually achieved (the grade-equivalent of your successful sends)

Therefore, climbing efficiency can be measured as:

For sport climbing: \(\text{Efficiency} = 2^{CPG-CEG}\)

For bouldering: \(\text{Efficiency} = e^{CPG-CEG}\)

This efficiency metric ranges from 0 to 1:

  • Efficiency = 1 (CEG = CPG): Perfect efficiency - you flashed everything you attempted
  • Efficiency ≈ 0.5: Moderate efficiency - half your attempts resulted in sends
  • Efficiency < 0.1: Low efficiency - less than 10% of your weighted attempts resulted in sends

The CEG - CPG difference essentially represents the average gap between your flash grade and the grades you’re attempting. A difference of 0 means you’re flashing everything (CEG = CPG), while larger differences indicate you’re attempting problems increasingly beyond your flash ability.

Example: Analyzing a Bouldering Session

Consider a climber’s session at the gym:

  • 5 attempts on V5 (1 successful)
  • 8 attempts on V6 (1 successful)
  • 12 attempts on V7 (0 successful)
  • 3 attempts on V4 (1 successful)

CEG calculation (if all attempts were successful): \(CEG = \ln(5e^5 + 8e^6 + 12e^7 + 3e^4) ≈ \ln(17,293) ≈ 9.76\)

This means if the climber had flashed every problem they attempted, their CPG would have been V9.76.

CPG calculation (actual performance): \(CPG = \ln(e^5 + e^6 + e^4) ≈ \ln(606) ≈ 6.41\)

This means the climber’s actual sends were equivalent to sending a V6.41 boulder.

Efficiency calculation: \(\text{Efficiency} = e^{CPG-CEG} = e^{6.41-9.76} = e^{-3.35} ≈ 0.035 = 3.5\%\)

This low efficiency indicates that only 3.5% of the climber’s weighted effort resulted in sends. The CEG - CPG difference of 3.35 suggests the climber was attempting problems about 3.35 grades above their flash ability on average. The 12 failed attempts on V7 contributed enormously to the total effort but produced no sends, dramatically reducing efficiency.

Applications of CEG

The CEG metric has several practical applications:

  1. Training Assessment: By tracking both CEG and CPG over time, coaches and climbers can evaluate whether training efforts are translating into successful sends or if adjustments to approach are needed.

  2. Competition Formats: In competitions where attempts matter, CEG provides a natural scoring system that weights both difficulty attempted and efficiency.

  3. Personal Progress Tracking: CEG allows climbers to quantify their total effort investment, which can be motivating during periods where sends may plateau but effort and learning continue.

  4. Style Comparison: The efficiency metric (e^(CPG-CEG)) allows for objective comparison between different climbing styles - from conservative onsighters to bold projectors.

Considerations and Limitations

While CEG provides valuable information, several factors should be considered:

  1. Data Collection: Accurate CEG requires recording all attempts, which may be impractical in some settings.

  2. Attempt Definition: What constitutes an “attempt” may vary (e.g., does falling on the first move count the same as falling near the top?).

  3. Context Matters: The optimal efficiency varies based on goals - training for improvement often requires lower efficiency (attempting harder grades) than performance competitions.

  4. Interpretation Nuance: When attempts include problems you cannot currently send, the relationship between CEG and CPG can produce very low efficiency values that require careful interpretation.

In conclusion, the Cumulative Performance Grade and Cumulative Effort Grade together represent a significant step forward in the way we measure and compare climbing performances. By leveraging empirical evidence and mathematical modeling, these metrics provide a robust and unbiased framework that captures both achievement and effort in climbing. The CPG rewards successful performance while the CEG acknowledges total effort invested, and their relationship reveals important insights about climbing efficiency and style. As the climbing world continues to evolve, the adoption of these complementary metrics has the potential to provide a universal index for outdoor climbing competitions, rankings, and the way we celebrate the achievements of climbers worldwide.

References

Drummond, A., & Popinga, A. (2021). Bayesian inference of the climbing grade scale. arXiv preprint arXiv:2111.08140.