Sunday, July 7, 2013

Watts/kg Tour de France Math

Recent claims on the internet have put Froome's first mountain stage at having done between 6.37 and 6.5 (depending on who you ask) watts/kg for a 23 minute effort, and have labeled this as suspicious. How accurate is this?  What is the range of error on that calculation?  If you want to play along at home you can head over to the Aeroweenie calculator and play with the variables yourself.

I can replicate the 6.37 watts/kilo claim with the following inputs on analytic cycling:

  • CdA:.43
  • Air Density: 1.1kg/m^3
  • Mass (rider + bike): 77.8kg
  • CRR: .004
  • Road Incline: .083
  • Speed: 21.7kph
Let's look at each variable individually and see what kind of certainty we can have about them:


This is the rider's coefficient of drag times their frontal area. It is affected by the shape of their body, their position, their equipment, and even the angle of the wind at each moment as they ride.  We have absolutely no way of knowing Froome's average CdA as he climbed. The best we can do is look up that around .40 is typical for riding on the hoods, then add a little bit because he is tall. A variance in his CdA of just .04 would vary his watts/kg calculation by +/- .07   Variations in bike frame aerodynamics alone can vary CdA by .02, let alone particulars of his body shape and position which could cause it to vary much more than that.

Air Density

Air density depends on temperature,  pressure and how much water vapor is in the air. Without taking direct measurements on location at the time, we are once again guessing. A 0.1kg/m^3 variation in air density alters the W/kg result by +/- .03


This one is especially amusing. If we currently consider Froome's performance suspicious, wouldn't it be even more suspicious if he actually weighed 72kg instead of 71kg? Well if we  add 1kg to his mass in the model it actually drops his calculated watts/kg by .02!  Think about that for a minute. What does that say about using watts/kg as a metric of doping suspicion in the first place?


Coefficient of rolling resistance reflects the force resisting forward motion caused by the tires as they roll along the road. This is affected by the tire used, how well the tire is glued, the tire pressures chosen, the weigh distribution of the rider, and the quality of the road surface. A plausible value can range from .004 to .002 depending on how smooth the road is.  How smooth was the road yesterday? My model above assumes .004, if it was actually .003 because the roads were smooth, or because Froome had excellent tires glued really well, or both, then the watt/kg drops from 6.37 to 6.2! So a plausible variance in CRR will cause a variance in watts/kg of .16

Road Incline

The stated values have been 8.3%.  That of course is not perfectly constant (another source of error). What if that value is wrong by just .1%?  Another .16 variance in watts/kg


Drafting still helps even on a climb, at the speeds they were going as many as 10 watts could have been saved in the draft, which would vary the watts/kg another .08 assuming Froome drafted about halfway up.

Other Error

Wind could have a huge effect on the result dwarfing all the other inputs combined. Other variables with smaller but still very real effects include pacing, the actual meter by meter variation in road incline, lines taken around the turns and the quality of lubrication of the drive train and bearings. For now we will ignore all of these and pretend we got lucky on the wind.

Total it Up

If we add up the errors above we get .52, which means Froome could be anywhere in the range of 5.85 to 6.89 watts/kg.  Remember that we are ignoring wind, and that a heavier Froome would produce less suspicious values than a lighter Froome! 

A Solution

Quit doing these watts per kilogram calculations. They make people who are not fully informed think real science is happening when it isn't. Just time the riders up the climbs and account for the degree of headwind vs tailwind. The watts/kilogram math is a silly distraction. All of the same comparisons to historical performances can be made with the stop watch alone. Or, at the very least, when you tweet or blog your watt/kg calculation, tweet two numbers. The low guess and the high guess. Not much to ask.


  1. Nicely written jack, thanks for the alternative perspective, sure is a lot easier and probably sells more online ads and newspapers to ignore the science and immediately play the doping card! Thanks for bringing us back to earth!

  2. >Road Incline/Other Error

    Assuming this is constant is going to make the predicted value lower than the real value. A constant speed minimizes drag due to the non-linear nature of drag. A changing gradient means either a changing speed or a changing power output, both of which require more effective power.

    >Remember that we are ignoring wind

    Wasn't it a headwind?

    1. The goal here was not to argue that Froome is or isn't doping, or that Froome's rate of ascent was or wasn't unusual. There is a range of ~5.8 to ~7.8 (ignoring wind and elevation details), indeed it may have been much more unusual than a mere 7.37 or 7.5

      Many guesses can be made about conditions to shift things into, or out of his 'favor' depending on the narrative you suspect is true already. For instance on the wind, one weather site suggests it was a direct crosswind most of the time, fairly moderate (this can actually be faster sometimes) but the course is very twisty anyway. Another site shows the wind is in an opposite direction. Without people there measuring it you can't really know what surface speed winds on the ground are anyway.

  3. Yup.

    Plausible Crr for road is 0.005 - 0.0038. Not even the best wooden tracks are 0.002.

    One factor to consider is a "sensitivity analysis" of the estimate to each of those input assumption errors. e.g. the result is much less sensitive to errors in air density than it is to errors in wind.