Rowing Splits vs. Workout Duration

Rowing is a fun sport that can can improve both aerobic and anaerobic fitness. It is possible to row fast for a short duration or row slower for a long duration. There are various attempts on the internet to predict the rowing power (in watts) or split (in minutes per 500 meters) for one workout duration based on the results for another duration. Sander Roosendaal has done a nice analysis of statistics from the Concept 2 online rankings, improving on the approximations from Paul’s Law, or the Kurt Jensen model or the Critical Power Model of Scherrer and Monod (1960).

Somewhat frustrating to me, Roosendaal does all analysis in power (watts), and doesn’t publish the parameters of his best-fit model. Instead, he has set up an interesting website Rowing Analytics to make use this model, along with other aspects of his modeling of the physics of rowing, to help people train. That’s all very well and good, but I’d like to have a predictive model I can play with.

So I scraped all of the 2020 rankings from the Concept 2 rankings site, and analyzed the performance for different workouts of anyone who had posted a time for a 5000 meter workout during the year. That’s 52,684 workouts from 12,044 different individuals.

I’m going to reference all of the splits to a split of 2:00 minutes for a distance of 5000 meters. That’s because this is one of the more popular distances, and sort of in the mid-range of durations.

I’ll compare the Concept 2 statistics to two different predictions:

First, a simple power law, with the split falling off with the duration of the workout:

Second, a formula similar in spirit to Roosendaal’s equation, but with a different parameterization. It’s got two components: a “burst” power on short timescales and an “aerobic” power on long timescales. The aerobic power takes a little while to build up and drops off with a powerlaw behavior on long timescales. There is a sigmoid function that connects the two sources of power such that one picks up while the other drops off. There is a visual representation of this in the figure below. (This is actually the best fit model for the Concept 2 data, referenced to a 2k pace of 2:00/500).

The math is below:

So let’s take a look at the concept 2 data. The figure below shows all of the splits plotted as a function of the duration of the workout. The splits are re-normalized so that everyone achieved a split of 2:00/500m for a 5k workout. The small dots are the individual measurements and the vertical bars are the inter-quartile regions for groupings of 500 data points. The two models described above are overlayed as the curves. The two-component model does much better than the power law model, predicting the split to ±5.8 seconds vs. the powerlaw model’s ±12 seconds (quoting the semi-interquartile range).

I was curious whether there is any trend with age, so I split the sample into “young” (ages 20–35) and “old” (age > 60) and fit the two-component model to the separate data sets. There does seem to be a different trend, with the older rowers seeing less drop in performance of long distances, but more of a drop at the middle distances. The points shown here are the medians over 50 points. The model predicts the splits to ±6.7 seconds for the younger rowers and ±5.5 for the older rowers.

For completeness, and in case anyone one wants to use these models, the best-fit parameters are below:

Powerlaw: b = 0.066

Two-component, all rowers:

a, b, t_c, β, t_long = 4.15, 0.94,-0.17, -0.20, 3.48

Two-component, younger rowers:

a, b, t_c, β, t_long = 4.26, 1.00,-0.21, -0.22, 2.59

Two-component, older rowers:

a, b, t_c, β, t_long = 5.21, 0.81,-0.65, -0.15, 1.32

Setting t_long = 1 and refitting with just 4 parameters results in fits that are just as good.

A different way of normalizing to the same split

In the analysis above, I calculated the multiplicative factor needed to bring all the workouts to a 2:00 split for 5000 meters. That seems like the sensible thing to do, but I wondered how different things might look if instead I used an additive adjustment. That is, I calculate the difference in seconds in the split between the reference 2:00 for 5k and that individual’s 5k workout, and then the same number of seconds to adjust the splits of that individual for all the other workouts. I think doing it multiplicatively makes more sense, but, somewhat interestingly the scatter looks just about the same.

The model shown above is the version with t_long = 1. The residuals are only a little bit worse than for the multiplicatively-normalized version, predicting the split to ±5.9 seconds.

The handy thing about this way of doing things, is that the adjustment to your split for workouts of different durations is independent of your split at 5000 meters. This gives you a decent idea of a target split for different workouts. If we reference a “5 minute assessment,” then the recommended adjustments to the target split for different durations are given below.

Recommended target split adjustment relative
to the split for a 5-minute workout

Duration       Adjustment
 (min)          (sec)
 1               -16
 5                 0
10                 4
15                 7
20                 9
30                12
45                15
60                17
90                20

This works pretty well for me. My best splits these days are around 2:00 for 5 minutes, 2:09 for 20 and 2:15 for 45.

(The best-fit parameters of the model to the data with the additive adjustments are a, b, t_c, β = 4.69, 0.87, -0.63, -0.18, with t_long having been fixed at 1.0.)