How Steep Can I Go?
Franz| March 14, 2010 6:56 pm
What is Percent Grade
The term “grade” comes from civil engineering and is the most common method of specifying the sloop of a hill. By definition, grade is defined as: 
It is not the angle of the hill, which is measured in degrees. A very steep section could be a 20% grade, which is about a 10 degree angle. A 45 degree angle would be 100% grade.
How Do We Use It?
There are two factors associated with a hill climb, the average grade and the grade at any given point. The average grade is easy to calculate, assuming you know the distance of the climb and the total elevation gain. Let’s take a hill that is 1 miles long with an elevation change of 1,000 feet (0.189 miles). You can calculate the grade as follows:
On our bikes we don’t actually measure the “run” but measure the road along the slope of the hill. Using a little trigonometry, we can determine that for this particular set of numbers the “run” is 0.98 miles and the grade is 19.2%. We could therefore use the measured distance on the bike and the error here would only be 2%, even less so for lower grades. We all know from climbing a very steep hill that a grade of 19% is very difficult, but none of the climbs we track have an average grade of 19%. The steepest of climbs usually average no more than 10% grade, or about 500 vertical feet per mile. In California, Bolhman On Orbit averages only about 10.5%. So although average grade is certainly a factor, there are several factors that need to be considered.
- Maximum Grade
- Total Distance
- Total Elevation Gain
How each of these impact you as a climb is very much a personal thing. Some can power over a very steep, short section, and yet fade with a long climb, while others have a very difficult time with a short, steep grade (or maybe they are not using a low enough gearing) but can climb strong for 3,000 vertical feet.
Of all the parameters we could use to describe a hill climb, the hardest to determine is maximum grade. Unless you are a surveyor, you are usually limited to measuring elevation gain using an instrument that is using barometric pressure (or even less accurately, GPS only). How accurate is such a measurement? Pretty good over a significant elevation change, but not so good over a short distance. Couple that intrinsic error, with the aspect that a very short, but very steep pitch, is not nearly the same factor as a steep climb for 1/4 mile. So what are the parameters that should be used calculating maximum grade?
Last year I was biking in the beautiful island of Hilo Hawaii and ventured down about the steepest road I have ever attempted on a road bike. This is a view from the top that shows the vertical descent down to the ocean.
There was a sign at the top of the road that said 25% grade. This picture gives you the idea.
Going down was tough enough, going up was impossible and only one person in our group made it all the way up without stopping. To excuse myself for walking a section, I stopped and used an inclinometer application in my iPhone to measure the grade, resting in on the top tube. I measured 35% grade.
Does that mean the maximum grade was 35%? Even if the measurement device was accurate, it was still over a distance spanned by my two wheels so a bump in the road could have a big impact. To calculate the maximum grade, we need to decide over what distance. It is a decision that the designers of all cyclometers that read out percent grade, need to wrestle with. Make the distance too long and people don’t get the instant feedback they expect. Make it too short and you get some numbers that don’t reflect what you feel and that fluctuate too rapidly. So leave your comments here on:
- What is the minimum distance we should use to calculator maximum grade.
- How best to measure it.
We will use your feedback on developing some factor for maximum grade on the hills we track.
Website Changes
We are updating our hill climbs on the Ultra Cycling website (http://www.ultracycle.net). We are working on maps and hill profiles for the various climbs. Look for those changes to be coming soon.
4 Responses to “How Steep Can I Go?”
Asking how to calculate average grade open so many pandora’s boxes in statistical experimental design.
One change form what’s commonly done, and as used by John Summerson in his toughest hillclimbs book, could be to calculate the median of all valid data points for average grade rather than the mean. That would be a stronger correlation to actual climbing difficulty than the mean because a median is more resistant to outlying data points. But what makes a data point valid?
In college I had about 40 credits of mathematical modeling and statistics, and from that, sometimes different experimental designs give different results that each can be argued are equally correct. For statistics fans, the type 3 error is fascinating.
Wrt calculating road grades, if we start with Franz’ basis that the bump over one bike length is too short to consider a grade measurement at that single point to be accurate, then where to draw the long end is a paradox. The road cannot be assumed to be planar, but what is the maximum surface irregularity and change in actual grade and over what length of roadway compared to the rise of the measured run at that point that would not skew the calculation within the desired accuracy? How much variation from the previous and succeeding data points should be considered an erroneous measurement? Should acceptable variation within the previous and succeeding points affect the variation we allow a point to have and still consider it valid? The opening wall on Welch Creek is a good test for proposed criteria, going from zero to >15% in a few pedal strokes. Quinnhill also crests from 21% to zero in about the same distance.
We also need to decide what to do with an erroneous value: discard it, take an average of the previous and following points, or try to fit a curve using the mean or median of the N points before and after.
For now, I’m going to use raw data from a Garmin 500, with obviously bad points such as 40% grades removed.
4 bike lengths seems about right for me. That’s where I generally sight when I’m grinding up a hill.
I’m not technical enough to give you any suggestion on how to calculate grade, over and above what I’ve already learned from your blog and basic geometry.
Thanks for sharing.
I think you have to consider the practicality of measurement. For a manual measurement of maximum instantaneous gradient I think Lucas Pereira’s gradiometer is a neat solution:
http://graphics.stanford.edu/~lucasp/bike-grade.html#Build
It measures the gradient across a bike length using a gradiometer attached to the bike frame, and calibrated to zero when the bike is level. One would of course apply common sense in its application to eliminate any bumps that would significantly affect the measurement. Another common case is a switchback where the line through the apex is often significantly steeper than the line that one would ride. There the measurement should IMHO be done on the obvious ride line, though one can debate where this actually is.
For a more automated approach I think one could take a large number of measurements using a Garmin Edge with altimeter. I suspect more than 10 samples is needed, and they could easily be collected from many different riders on many different bikes in many different conditions. One would then need to align each point along across the samples, discard outliers, and average the elevation at that point. Fortunately, GPS has good accuracy in location though of course riders will take different lines. Some work is probably required to good satisfactory point alignment. I would definitely exclude devices that calculate elevation by GPS. I would also normalize each sample to be elevation height at sample minus the elevation point at the beginning of the climb to eliminate “common mode” errors across a sample which can be very significant with an altimeter.
With the average elevation computed along the route one can then calculate the gradient for all desired lengths (up to the length of the route obviously). Calculating for a large number of lengths allows one to plot a gradient versus length profile for that climb. This is rather like a power versus time profile. I think these profiles would be a very good way to characterize a climb and to compare it with others. One can readily distinguish those climbs that have a very steep but short wall versus those that have sustained steepness.
Having computed these graphs one can look at the behavior as the length of measurement approaches zero. Typically I would expect to see a linear behavior that will intercept the gradient axis to yield a max gradient. But it might be that the line starts to diverge upwards toward some larger max gradient, and this would be a case worth investigating to see if that is a problem with the data or a real steepness in the road. One could extract the location from the GPS data and then go and look at it and measure using the manual approach. In practice, I think that the GPS data and averaging over multiple samples is going to flatten out the max gradient. One could argue that this is a good thing.
I would just like to point out that the iBike power meter uses an inclinometer to measure grade and so it is probably the most accurate of the cycling computers for that puropose. It can measure grade statically to a bike-length resolution if you want to get the max slope on a particular switchback and don’t mind stopping. While riding, the accuracy is insured since it monitors speed so any sudden acceleration won’t fool it into thinking the road just got steeper and it measures altitude both with the inclinometer and with a barometric atltimeter and then applies a correction factor to the inclinometer data to compensate for accumulated error. The resolution is time based at either 1 or 5 seconds depending on the user’s preference and filters can be applied in post processing to get more of an average grade. Although I have issues with the iBike in other areas, as an inclinometer it can’t be beat.
Care to comment?