2011 Tour de France Summary

The table below summarizes the quality of my predictions in this year's Tour de France.

Stage	Actual	Predicted	Difference	% Diff.
1	4h 41' 31"	4h 40' 01"	-01' 30"	-0.53
2	0h 25' 16"	0h 26' 35"	01' 19"	5.21
3	4h 40' 21"	4h 44' 55"	04' 34"	1.63
4	4h 11' 39"	4h 09' 29"	-02' 10"	-0.86
5	3h 38' 32"	3h 52' 05"	13' 33"	6.20
6	5h 13' 37"	5h 20' 13"	06' 36"	2.10
7	5h 38' 53"	5h 11' 43"	-27' 10"	-8.02
8	4h 36' 46"	4h 49' 26"	12' 40"	4.58
9	5h 27' 09"	5h 19' 43"	-07' 26"	-2.27
10	3h 31' 21"	3h 41' 08"	09' 47"	4.63
11	3h 46' 07"	3h 46' 05"	-00' 02"	-0.01
12	6h 01' 15"	5h 59' 26"	-01' 49"	-0.50
13	3h 47' 36"	3h 46' 52"	-00' 44"	-0.32
14	5h 13' 25"	5h 02' 45"	-10' 40"	-3.40
15	4h 20' 24"	4h 36' 38"	16' 14"	6.23
16	3h 31' 38"	4h 05' 59"	34' 21"	16.23
17	4h 18' 50"	4h 32' 07"	13' 17"	5.13
18	6h 07' 56"	5h 51' 23"	-16' 33"	-4.50
19	3h 13' 25"	2h 57' 54"	-15' 31"	-8.02
20	0h 55' 33"	0h 51' 06"	-04' 27"	-8.01
21	2h 27' 02"	2h 27' 40"	00' 38"	0.43
TOTAL	85h 48' 16"	86h 13' 13"	24' 57"	0.48

I predicted six stages (I mistakenly noted five yesterday) to better than 1%.  Only the enigmatic Stage 16 came in over 8% (rounded) off.  I am quite pleased by my predictions!

Note the last row in the above column gives the sum of the stage-winning times.  My model cyclist completed the Tour de France in a time 0.48% slower than the sum of all the stage-winning times.  Note that that time is not the total time posted by this year's winner, Cadel Evans.  His winning time was 86h 12' 22".  Though that time is just a mere 51 seconds off from the sum of my stage-winning times, my goal at the outset was not to predict the overall time for any one cyclist, but the sum of the stage-winning times.

The overall error of 0.48% is a bit misleading because it comes from a lot of cancellation.  For example, I was 27' 10" fast on Stage 5 and 34' 21" slow on Stage 16.  Those "fast" and "slow" times tend to cancel at the end.  If I add error in quadrature, I wind up with a 1.16% error, which is still not bad!

Modeling the Tour de France is a lot of fun for me, and it enhances my pleasure in following the race.  My stats page tells me that people from 21 different countries checked out this blog during the course of the race.  I am humbled and flattered by the level of interest in my blog.  Please contact me with any questions you may have.  If you want more details about Tour de France modeling, check out Chapter 4 in my book, Gold Medal Physics:  The Science of Sports.

I plan to add more sports science posts in the future.  Feel free to send me suggestions of sporting events and/or athletic feats that might benefit from the eye of a physicist.

Nearly perfect on Stage 21!

Here is the result from the final stage of the 2011 Tour de France:

• Stage 21:  2h 27' 02" (actual), 2h 27' 40" (prediction), 38" slow (0.43% error)

Given how the last stage is usually run, I am quite happy with my prediction!  I asked yesterday if Mark Cavendish could win his third straight final stage -- and he did!  Here is his average speed:

• Stage 21:  10.77 m/s (24.1 mph)

Congratulations to Cadel Evans for becoming Australia's first Tour de France winner!

Tomorrow I will post a summary of my predictions for this year's Tour de France.  Five stage predictions came in under 1% off.  Just one stage came in over 8%, and that was my dreadful Stage 16 (still an enigma for me!).  I am happy with how my model did!

A bit fast on Stage 20 ...

Here is the result from today's individual time trial:

• Stage 20:  55' 33" (actual), 51' 06" (prediction), -04' 27" fast (-8.01% error)

Just like with Stage 19, I was 8% too fast on the penultimate stage.  I guess Stage 16 convinced me that I needed a little more power -- a little too much it turned out.  Here is what Tony Martin was able to average in his win today:

• Stage 20:  12.75 m/s (28.5 mph)

To answer my question from yesterday, "Yes, the Schleck brothers can be stopped!"  Let's hear it for Australia!!!  Cadel Evans came from down under to sit atop the cycling world.  Australia will surely celebrate its first Tour de France winner after tomorrow's ride into Paris.

The last stage is always the hardest to model.  I can take the terrain data and predict what a top cyclist would do.  The last stage, however, is usually not as hotly contested as previous stages.  This year's final stage is rather short at just 95 km (59 miles).  Here is my prediction:

• Stage 21:  2h 27' 40" (prediction)

I have dialed the power input back quite a bit.  Will Mark Cavendish win his third straight Tour de France final stage?  Were the final stage fought as hard as possible, I'd probably lop 15 to 20 minutes off the above time.  But, I am trying to predict what will happen, so I am going with the slower time.

A bit fast on Stage 19 ...

I thought someone could finish Stage 19 in just under three hours.  I was wrong about that!  Here is the result from Stage 19:

• Stage 19:  3h 13' 25" (actual), 2h 57' 54" (prediction), -15' 31" fast (-8.02% error)

Pierre Rolland had a great climb at the end to overtake Alberto Contador.  Here is Rolland's average speed:

• Stage 19:  9.44 m/s (21.1 mph)

I'm always glad to be under 10% on my predictions.  Even though I was 8% off today, I thought the stage could have been done a little faster.  Here is my prediction for tomorrow's individual time trial:

• Stage 20:  51' 06" (prediction)

Can the Schleck brothers be stopped?

Back under 5% for Stage 18!

Here is the Stage 18 result:

• Stage 18:  6h 07' 56" (actual), 5h 51' 23" (prediction), -16' 33" slow (-4.50% error)

I am happy to be under 5% again!  Did the poor weather slow riders down a little?  If so, I feel even better about my prediction because my model did not include adverse weather for Stage 18.  Then again, it rained on Stage 16, and that one remains a mystery to me.

The Schleck brothers got it done today.  Here is Andy Schleck's average speed:

• Stage 18:  9.08 m/s (20.3 mph)

Thomas Voeckler was able to hold on to the yellow jersey.  Tomorrow's stage, which is the final mountain stage, might just decide this year's winner.  Here is my prediction:

• Stage 19:  2h 57' 54" (prediction)

Stage 19 should be incredible!  Riders climb to an elevation of 2.556 km (1.588 miles) as they reach Col du Galibier in the French Alps at the 48.5-km (30.1-mile) mark.  The next 46 km (28.6 miles) will then make for a great downhill.  The final 15 km (9.32 miles) will have riders climbing a 7.9% grade to get to the famous ski village at L'Alpe d'Huez.  In 2004, Lance Armstrong dominated that climb in that year's Stage 16, which was an individual time trial.  Armstrong beat the second-place finisher, Jan Ullrich, by just over a minute.  That stage sticks out in my mind because I cover it in detail in Chapter 4 of my book.

If weather does not slow riders down, I hope to see the winner's time sneak under three hours.

Much better on Stage 17!

Here is the Stage 17 result:

• Stage 17:  4h 18' 50" (actual), 4h 32' 07" (prediction), 13' 17" slow (5.13% error)

As with Stage 16, I was slow on this stage.  I am obviously much happier with a 5% error than yesterday's 16% error!  Stage 16 is still a mystery to me.  That stage was mostly uphill in the rain, and the winner's average speed beat every other stage winner's average speed (except that in the team time trial of Stage 2).

The Norwegian Edvald Boasson Hagen had the following average speed today:

• Stage 17:  11.53 m/s (25.6 mph)

Despite the near 50-km downhill near the end of today's stage, Hagen's average speed was nearly 10% less than Hushovd's Stage 16 average speed.  Again, what happened on Stage 16?!?  I welcome your thoughts in the comments.

Stage 18 has three monster climbs and two great downhill segments.  By the time riders reach the end at Galibier / Serre-Chevalier, they will have gained 2.29 km (1.42 miles) of elevation from their starting point.  Here is my Stage 18 prediction:

• Stage 18:  5h 51' 23" (prediction)

Can Thomas Voeckler hold the yellow jersey after tomorrow?  It should be a wonderful climb to the finish line!

Worst prediction ...

This is, by far, my worst prediction of the 2011 Tour de France.  Here is the result from Stage 16:

• Stage 16:  3h 31' 38" (actual), 4h 05' 59" (prediction), 34' 21" slow (16.23% error)

I am stunned by how fast this stage turned out to be.  Here is Thor Hushovd's average speed:

• Stage 16:  12.80 m/s (28.6 mph)

Except for the team time trial in Stage 2, the above average speed is the largest so far.  I never thought that a stage that is almost entirely uphill could have such a large average speed.  The God of Thunder certainly shocked me today!  Was there a massive tailwind today???

Here is my prediction for Stage 17:

• Stage 17:  4h 32' 07" (prediction)

I am shocked that today's stage was well under four hours.  Stage 14 was won in over five hours.  Tomorrow's Stage 17 does not have as many brutal climbs as were found in Stage 14.  Stage 17 should end with a great downhill sprint into the Italian city of Pinerolo.

Stage 16 prediction ...

Here is my prediction for Stage 16:

• Stage 16:  4h 05' 59" (prediction)

Can someone finish Stage 16 in under four hours?  We shall see!  It will take a rider with a lot of power input because the majority of the stage is uphill.  The stage should end with a great downhill sprint into Gap.

Congratulations to Japan

It was agonizing seeing the US women lose today, but I am happy for the Japanese team.  They played with a great deal of heart.  To keep coming back the way they did, Japan will take home a well-earned trophy.


For me, I loved being able to share the women's World Cup with my two young daughters.  They now know names like Alex Morgan, Abby Wambach, and Megan Rapinoe.  Each time Rapinoe had the ball, my girls exclaimed, "Rapinoe has it!  Rapinoe has it!"  After the game was over, my two girls went outside and played soccer.  I thank my national team for inspiring my girls to put boot to ball and experience a little of the beautiful game.

A bit slow on Stage 15 ...

Here is the Stage 15 result:

• Stage 15:  4h 20' 24" (actual), 4h 36' 38" (prediction), 16' 14" slow (6.23% error)

This is the one stage I'm kicking myself over!  After three grueling mountain stages, I knocked my code's biker power input down just a little for Stage 15.  Had I not done that, my error would have been cut in half.  The fact that I was wrong to do that actually inspires me because that means the athletes at the Tour de France are even better than I first thought.  Here is what Mark Cavendish was able to average in today's impressive win:

• Stage 15:  12.32 m/s (27.6 mph)

This is almost exactly the same speed Cavendish averaged during his win in Stage 11.  Certainly impressive!

Monday is a rest day.  I plan to publish my Stage 16 prediction on Monday.  I am hoping to get one or two more profile points.  Right now, I need to get a few other things done so that I can watch the women's World Cup final.  Go US!

A tad more realistic on Stage 14 ...

When I started modeling the Tour de France in 2003, I thought predicting a stage win to better than 10% would be pretty good.  Anytime I got a stage better than 5%, I thought I had done a great job.  The last three stages of this year's race were a little surreal.  I just can't predict every stage to better than 1%!  Stage 14 brought me back to reality a little, but I am still pleased with my prediction.  Here is the result:

• Stage 14:  5h 13' 25" (actual), 5h 02' 45" (prediction), -10' 40" fast (-3.40% error)

Though thrilled with just a 3.4% error, I think my model was a tad fast because I did not add a "Gee, I'm tired on this third mountain stage in a row!" line to my code.

Here is what Jelle Vanendert was able to average for Stage 14:

• Stage 14:  8.96 m/s (20.0 mph)

Stage 15 will give the riders a little relief because it is mostly flat.  Monday is a rest day.  Here is my Stage 15 prediction:

• Stage 15:  4h 36' 38" (prediction)

Will riders be tired on Sunday?  Will the ride into Montpellier be a relief for rider's looking forward to Monday's rest?  I can't wait to see what happens!

After the Tour de France, I will be watching our US women's soccer team take on Japan in the World Cup final.

Just 44 seconds off Stage 13!

Here is the result from Stage 13:

• Stage 13:  3h 47' 36" (actual), 3h 46' 52" (prediction), -44" fast (-0.32% error)

I'm once again shocked how well I predicted a mountain stage.  Thor Hushovd had a great ride.  Here is what The God of Thunder was able to average:

• Stage 13:  11.17 m/s (25.0 mph)

Stage 14 looks to be a brutal ride.  There will be, however, some amazing vistas along the way.  Here is what I predict for Stage 14:

• Stage 14:  5h 02' 45" (prediction)

After such a grueling couple of stages, can someone finish Stage 14 in under five hours?  I'm sure the riders will be thrilled to reach the Pyrenees ski resort of Plateau de Beille at the end of the stage.  They will then be at an elevation of 1.78 km (1.1 miles).

Nearly nailed Stage 12!

Here is the Stage 12 result:

• Stage 12:  6h 01' 15" (actual), 5h 59' 26" (predicted), -01' 49" fast (-0.50% error)

Wow, I'm really happy to hit the first major mountain stage to within half a percent error!  I thought the winner might be just a tad under six hours, but it turned out to be just a tad over.  Here is Samuel Sánchez's average speed:

• Stage 12:  9.73 m/s (21.8 mph)

Stage 13 has a great climb up to Col d'Aubisque.  Here is my prediction:

• Stage 13:  3h 46' 52" (prediction)

After a monster climb to the 110-km mark, it'll be a great downhill to the end at Lourdes!

On to the final!

Our US women played the beautiful game beautifully!  Tension rose after the French equalizer early in the second half.  Abby Wambach's header after Lauren Cheney's perfect corner kick was magical, as was the patience Alex Morgan displayed on her goal.


Kudos to the French women's team; they played very well.  They also displayed a lot more class than what their male counterparts displayed at last year's World Cup.


The best moment for me?  Once the game ended, my two young daughters were screaming, "We won!  We won!"

Missed Stage 11 by TWO SECONDS!!!

This is by far my best stage prediction of this year's Tour de France!  Here are today's results:

• Stage 11:  3h 46' 07" (actual), 3h 46' 05" (prediction), -02" fast (-0.01% error)

Here is Mark Cavendish's average speed for today's win:

• Stage 11:  12.35 m/s (27.6 mph)

Stage 12 is a great mountain stage, which finishes in Luz Ardiden.  There are three big climbs in the stage.  Ben Hannas sent me a couple of extra data points so that I could adequately model the motion through the valleys.  The net elevation increase from start to finish will be 1.55 km (0.96 mile).  Here is my prediction:

• Stage 12:  5h 59' 26" (prediction)

I am hoping that the winner can complete the stage in just under six hours.

Okay, it's time to get home to watch the US women take on France in the World Cup semifinals.

Women's soccer and soccer physics ...

We in the US are rooting for our women's soccer team to beat France in tomorrow's semifinal World Cup match.  People are still talking about the amazing comeback win over Brazil.  For those of you interested in soccer (or football, depending on where you are from), I wrote an invited article for the July 2010 issue of Physics Today that coincided with the men's World Cup.  The article is short and intended for a general science audience.  Click here for a PDF version of the article.


If you are interested in a longer description of soccer physics, but still at a general science level, Chapter 7 of my book is devoted to soccer kicks.  I model free kicks and corner kicks.


For those interested in my technical research papers on soccer physics, click here to get to my webpage.  You will find paper links at the bottom of the page.

Stage 11 prediction ...

Here is my prediction for Stage 11:

• Stage 11:  3h 46' 05" (prediction)

Ben Hannas once again provided me with some extra data points.

Still better than 5%!

After a great sprint to end today's stage, here are the Stage 10 results:

• Stage 10:  3h 31' 21" (actual), 3h 41' 08" (prediction), 9' 47" slow (4.63% error)

It turned out the extra seven data points took me a little further from the result.  The model with fewer data points may have averaged a little better, but I stand by the more detailed stage model.  Riders were a little faster today than I thought they would be!  Yesterday's rest must have helped.  I'm still thrilled to have predicted the time to better than 5%.

Here is the average speed for André Greipel's ride today:

• Stage 10:  12.46 m/s (27.9 mph).

I'm not surprised being a little slow on what turned out to be the race's second-fastest stage (excluding Stage 2's team time trial).

My prediction for Stage 11 is coming soon.

Day of rest ...

On 8 July, I gave the average speeds of the first seven stage winners.  Here are the average speeds of the last two stage winners:

• Stage 8:  11.38 m/s (25.5 mph)
• Stage 9:  10.60 m/s (23.7 mph)

There is no surprise that Stage 9 is the slowest so far, notwithstanding the crashes.  The climbs in the middle of Stage 9 hit some high elevations.  Riders better get plenty of rest today.  After a couple of modest stages, the great Pyrenees climbs await!

I'm anxious to see how my Stage 10 prediction does tomorrow.  I'm also anxious to know who is reading this blog.  Feel free to add a comment and let me know what you think of the Tour de France, scientific modeling, and anything else on your mind.  Let me know your country of origin, too.

Extra valley points did the trick on Stage 9!

Here are the results for Stage 9:

• Stage 9:  5h 27' 09" (actual), 5h 19' 43" (prediction), -7' 26" fast (-2.27% error)

The four extra valley data points did the job on Stage 9!  I'm happy with a 2.27% error.

For Stage 10, Ben Hannas sent me seven extra data points.  If you examine the stage profile online, you will see a few missing peaks and valleys.  With the added data, here is my prediction for Stage 10:

• Stage 10:  3h 41' 08" (prediction)

Without those seven extra points, my prediction would have been 3h 35' 57".  By adding in more detail, specifically more peaks and valleys, the predicted time goes up a little.

New Stage 9 prediction ...

Ben Hannas sent me data for the four missing valleys in the middle of Stage 9.  Those four additional data points modify my Stage 9 prediction:

• Stage 9:  5h 19' 43" (prediction -- four extra data points)

The middle of Stage 9 looks to be an amazing ride!

Error cut in half for Stage 8!

Here are the results for Stage 8:

• Stage 8:  4h 36' 46" (actual), 4h 47' 37" (prediction), 10' 51" slow (3.92% error)

After yesterday's crashes helped bump my error up to about 8%, today's stage-winning time came in at under 4% of my prediction.  There were enough points in the final 6.5 km to give a decent description of the end of the stage.

Stage 9 is really the first stage were the Tour de France website severely fails to describe the terrain.  Four major valleys are left out of the data.  Here is what I predict based solely on the Tour de France website:

• Stage 9:  5h 5' 57" (prediction -- no extra points)

I hope to obtain reasonable estimates for the elevations of the four valleys.  Look for a better (I hope!) Stage 9 prediction later today.

Average Speeds

Here are the average speeds for the winners of the first seven stages of the Tour de France:

• Stage 1:  11.34 m/s (25.4 mph)
• Stage 2:  15.17 m/s (33.9 mph)
• Stage 3:  11.77 m/s (26.3 mph)
• Stage 4:  11.42 m/s (25.5 mph)
• Stage 5:  12.58 m/s (28.1 mph)
• Stage 6:  12.04 m/s (26.9 mph)
• Stage 7:  10.72 m/s (24.0 mph)

Stage 2's team time trial clearly has the largest average speed.  Today's Stage 7 has the smallest average speed.  As I wrote in my last post, I'm surprised that my prediction for such a flat stage was 8% too fast.  My colleague, Ben Hannas, suggests that riders might have been saving a little energy for the big climb awaiting them in Stage 8.  That's a great suggestion!  That's also not something I programmed into my computer code.  It's tough to program human psychology!

A bit fast on Stage 7 ...

Here are the results for Stage 7:

• Stage 7:  5h 38' 53" (actual), 5h 11' 43" (prediction), -27' 10" fast (-8.02% error)

Given the complexity of modeling the Tour de France, I'm usually happy to predict a stage-winning time to better than 10%.  Stage 7 was flat enough that I thought I could do a little better than 8% error.  I'll have to analyze the race footage to see what might have slowed the riders down today.  I saw there was a crash, but I'm not sure if that contributed to a time longer than what I predicted.

Stage 8 has a great climb near the end of the race.  Here is my prediction for that stage:

• Stage 8:  4h 47' 37" (prediction)

My colleague, Ben Hannas, has offered some valuable suggestions for increasing the number of data points.  We will include some additional points for Stage 9, which the Tour de France website's profile desperately needs (no data for the four valleys in the middle of the stage!).  Ben graduated Lynchburg College in 2003 and now has his own company that specializes in analyzing energy usage in buildings.  Click here for his website.

Better on Stage 6!

Here are the results for Stage 6:

• Stage 6:  5h 13' 37" (actual), 5h 20' 13" (prediction), 6' 36" slow (2.10% error)

I'm happy to have predicted the winning time to within 2.10% for a stage that is 226.5 km (140.7 miles) long.

Stage 7 looks to be pretty flat.  Here is my prediction for that stage:

• Stage 7:  5h 11' 43" (prediction)

High elevations are coming in Stage 8!

A tad slow on Stage 5 ...

Here is today's result:

• Stage 5:  3h 38' 32" (actual), 3h 52' 05" (prediction), 13' 33" slow (6.20% error)

As I thought, my prediction would be a tad off today.  There are just not enough data points given on the Tour de France website.  Getting a 6.20% error is still not too shabby!  I may be able to estimate more points for future stages.  Until then, here is what my model predicts for Stage 6 (using only the Tour de France website data):

• Stage 6:  5h 20' 13" (prediction)

There are a lot of missing peaks and valleys on the website.  As with Stage 5, I'll see how the averaging goes.

Almost nailed Stage 4!

Stage 4 just ended.  Here are the results:

• Stage 4:  4h 11' 39" (actual), 4h 9' 29" (prediction), -2' 10" fast (-0.86% error)

Hey, not bad!  That's my second stage under 1% error.

For Stage 5, here is what I predict:

• Stage 5:  3h 52' 05" (prediction)

This stage worries me a little because the profile (click here and then click on the "Profile" tab at the lower left part of the page) given on the Tour de France website has very few data points.  If you look at the stage online, you'll see lots of peaks and valleys that are not labeled.  Oh well, let's see how well my model averages over the missing data!

2011 Tour de France

After a six-year hiatus, I decided to model this year's Tour de France.  For those interested in the physics details, click here for a paper I published with Ben Hannas back in 2005.  The profile data available on the Tour de France website is, unfortunately, not as good as the data I've used in the past.  There are only about half the points available this year.  That means that the mesh I use is more coarse this time around.


Due to the fact that I've been in the process of moving, I've not been able to post predictions before the first three stages.  The list below shows my predictions, the actual winning stage time, and my error.

• Stage 1:  4h 41' 31" (actual), 4h 40' 01" (prediction), -1' 30" fast (-0.53% error)
• Stage 2:  25' 16" (actual), 26' 35" (prediction), 1' 19" slow (5.2% error)
• Stage 3:  4h 40' 21" (actual), 4h 44' 55" (prediction), 4' 34" slow (1.63% error)

I've a guess why my model was a tad slow on the team trial (Stage 2).  That stage was quite flat. My model distinguishes between positive and negative road angles.  I probably have too little power on some of the tiny, but negative, road angles.  I am quite pleased with how Stages 1 and 3 came out!

Okay, so the above three stages show my predictions after the stages were completed.  I'll give it a shot and predict the rest of the stages ahead of time.  Here is what I've got for Stage 4:

• Stage 4:  4h 9' 29" (prediction)

I'll keep my fingers crossed!