# Is there a standard equivalent for effort between distance and elevation?

I'm starting to keep track of how far I ride on paper and while it's nice to have the data, I'd like to know how different rides stack up.

I have one path I like to take which is really flat and goes about 30 miles out (I usually only do 10 and then turn around), but I've found this ride is much easier for me than the other route I take which nets me about 12 miles and about 500 feet in elevation.

I saw on another SE that they calculate 500 feet elevation gain into one mile, but this doesn't seem that accurate for a bike.

So, what I want to know is if there is a standard calculation for "effort" or distance between distance and elevation gain, given perhaps average speed or any other relevant factors.

• No, given that you're using different muscle groups and what not. Commented Jul 2, 2015 at 20:12
• @Ross has mentioned speed, and that's my question. How fast are you doing these rides? Commented Jul 3, 2015 at 0:49
• @andy256 The flat sessions, 15 MPH on a good day and the hills about 13 MPH average Commented Jul 3, 2015 at 1:21
• @DanielRHicks If you use a power meter you can estimate "fatigue profiles" that show how power decreases with duration of effort. People who have done so have shown individual consistency under wide (but not extreme) ranges of pedal force and pedal speed conditions. That is, your fatigue profile may differ from mine, and each can change with fitness and training, but otherwise, yes, there is some science to calculating fatigue and recovery rates. Commented Jul 3, 2015 at 14:44
• I also apologize if imperial measurements are not the norm on this SE. Should I just switch to metric for the sake of consistency on the SE or doesnt it matter? Commented Jul 3, 2015 at 19:33

If you really want to measure how much effort you're putting in, you should look into getting a power meter. It measures the actual wattage you output, and can therefore be used to calculate total energy output. However, they are quite expensive.

The other option is to get a heart rate monitor along with cadence and speed meters which together can give you a reasonable estimate of how much energy you used. This ends up being much cheaper than a power meter, but is less accurate.

These are the preferred methods if you a really want to track how hard you are training. This is because many things can effect how much effort it takes to cycle such as hills (like you mentioned), but also wind, your body position, and the road surface itself.

• While a lot of people knock heart rate monitors as "less accurate", they trade that accuracy for ease of setup (no need for a new rear wheel) and expense. They are also useful for other endeavors besides cycling as well. They are an excellent choice for more recreational cyclists. Commented Jul 2, 2015 at 20:29
• Cycleops has a heart rate power meter that does a bunch of number crunching (I suspect some sort of statistical time series model) that gives decent average power over longer time periods (I.e., > 30 sec). It won't give an accurate peak power estimate and is affected by condition (e.g., fatigue, hydration levels affect HR), but as an inexpensive means of getting a ball park estimate of average power it is surprisingly good. Ran it beside a regular power meter and it was pleasantly surprised. Of course this needs an Ant+ bike computer (\$\$) to read it. Commented Jul 2, 2015 at 21:12
• A third option is to track "session RPE" using the 10-point (or 20-point) Borg scale, which doesn't require any hardware investment at all -- and, once you know how to use it and get some experience, is roughly about as accurate as looking at average heart rate. Commented Jul 2, 2015 at 22:46
• @Rider_X Some phones have Ant+ built in, so you may be able to get by with that. Also, there are some alternatives to "bike computers". I have an Garmin Oregon 450 (discontinued), which is very similar to the Garmin Dakota 20. They both support Ant+ and can read cadence and HRM (not speed, although they display speed from GPS). While similarly priced to the Edge 500, the price is a lot easier to justify because the device is a lot more versatile. The Edge series is very focused on cycling whereas the Dakota series is suited to a variety of activities. Commented Jul 3, 2015 at 12:50
• @Kibbee I would be careful just because something is Ant+ doesn't mean it will record the power readings. The Cycleops power HR band sends both an HR and a power reading. My old edge 305 is ant+ but doesn't read power, same with most of the of the Garmin watches. I assume a non-cycling GPS may not look either. I have no idea about Ant+ phones. Software limitations yes, but limitations nonetheless. Commented Jul 3, 2015 at 14:42

Apologies for re-opening a 9-year-old thread, but this question has bothered me for some time and I've now found the means to answer it, thanks to some data science skills I learnt while pursuing my PhD. Here goes.

## Method

3223 of my bike rides were uploaded to Strava between 2014 and 2024. 35 of the rides were races and were omitted. 1473 rides known to be commutes were also removed, as were 494 rides less than 10km long. Heart Rate Stress Score (HRSS) was calculated using the Elevate app assuming a LTHR of 168bpm (I'm 25) for activities where there was heart rate data, which was 1116 of the remaining 1715 rides. Almost all of the rides had the same start and end point, meaning that elevation gain was almost equal to elevation loss. My weight had fluctuated between 71 and 74kg during this time.

I fit a non-linear regression model using R's `nls` function to estimate the effect of average HR and elevation gain (in m) on HRSS. Distance (in km) is included as a constant. The coefficients (ßs) for avg. HR and elevation can be interpreted like linear regression coefficients. The model is:

``HRSS =  ß1Avg. HR * (distance + ß2Elev. gain)``

And the R specification of the equation is:

``````HRSS ~ hr.coef * Avg_HR_bpm * (Distance_km + elev.coef * Elev_Gain_m)
``````

## Results

The included bicycle rides had a distance from 10.1 to 200.2 km (median 59.4km) and elevation gains of 0 to 2825m (median 330m). HRSS ranged from 11 to 423 (median 118) while average heart rate ranged from 101 to 166bpm (median 147bpm). `hr.coef` was calculated to be 0.0111 (95% confidence interval: 0.0109–0.0114) and `elev.coef` was calculated to 0.0263 (95% CI: 0.0239–0.0289). That is, each bpm of average heart rate is associated with a 0.0111 higher average HRSS. Each meter of elevation gain is associated with an increase of 0.0263 HRSS.

## Discussion

Taking the reciprocal of `elev.coef` yields 37.97 (95% CI: 34.60–41.84). That is, 1 kilometre of cycling generates the same HRSS as climbing between 35 and 42m. A bicycle ride of 75.0km with 1000m of elevation gain would in theory yield the same HRSS as a 93.4km ride with 300m of elevation gain. The study gives an estimate of the correspondence between distance and elevation gain for a 70-75kg cyclist. Further studies could repeat the analysis considering average power and training stress core (power data is available for 597 rides).

• The nls package is non-linear, correct? Also, it's just occurred to me that TSS (and HRTSS) is one way to measure total effort, which is part of the original question. If you tell some athletes your weekly average TSS, there are a few people who will understand it. Commented Jul 9 at 2:00
• Hello! The `nls` package is written for non-linear model, but with the formula that I have specified, I believe that `Distance_km` and `Elev_Gain_m` should be linearly related to `HRSS`. Commented Jul 10 at 14:07
• As for the second question, the Elevate extension for Chrome tells me that over the last 52 weeks I have averaged 334 stress score points per week (most of the entries are Power Stress Scores based on my FTP). However, it ranges from 100 per week in December to 700 per week in July. Commented Jul 10 at 14:15
• I am interpreting the formula as something like: mean HRSS = f[avg_hr * (distance + elevation_gain)], and that each coef means a coefficient is estimated. So you estimate coefficients for avg heart rate and elevation gain, but no coefficient for distance (so it's treated as a constant, like the offset in a Poisson regression)? You might consider more explicitly stating what's being estimated; readers used to regression probably are used to OLS and derivatives. Commented Jul 10 at 15:20
• Look at the proposed edit. I think it clarifies things a bit. You are free to discard it entirely or edit it, especially if I miscommunicated your model/findings. Commented Jul 10 at 15:34

The short answer is no, there is no standard equivalent for effort between distance and elevation.

Of course, as others have mentioned, they are connected by the amount of effort you put in. But as you ride faster on the flat, the power required to drive you along rises as the square (some say the cube) of your speed. So riding 10% faster takes 20% to 30% more effort. This is mostly down to wind resistance, but rolling resistance plays a part too.

On a hill, your (well, at least my) speed is lower. Lets say you're climbing at 10kph (6mph). So the wind resistance is way lower (less than 1/6th) than the resistance you would have at 25kph (15.6mph). The main work (5/6 of the effort if you're producing the same power) you're doing is levering yourself up the hill. If you go 10% faster then the wind resistance still rises by 20% or so, but the effort of getting up the hill rises linearly.

So the point is, it's complicated. That's why there's no equivalence.

This is why, if you look at Strava or other tracking apps, they distinguish distance and elevation. They also distinguish time in the saddle, because measures something else: endurance.

For a real example, take a couple of my own rides.

• Today I rode an out and back course over 28 km (17.5 mi) with 650 m (2132 ft) of climbing. It's an undulating course averaging about 2.3% I averaged 25.3 kph (15.8 mph), similar to your pace. While not flat, it is compared to the next ride ...

• In the summer (it's winter here now) I climbed a hill. It's 1100 m (3600 ft) of solid climbing in 17 km (10.6 mi). It averages 6%, with pitches up to 24%. It took me 1 hour 45 minutes, for an average speed of only 9.7 kph (about 6 mph)!

There are two things about these rides. They show that I'm just a cyclist of modest abilities, so such comparisons should be relevant to you and many others. They also show the effect of continuous climbing, the effect of steeper hills, and the effect of longer duration, all tangled together.

It's complicated.

• +1 for modestly claiming a climb of 1,100m as a hill, when by all definitions that is mountain. The biggest mountain in the UK is just over 200 meters higher at 1,344. The tallest mountain in England is only 978m tall. feel ashamed now that the biggest hills in range of me are around 220m Commented Jul 3, 2015 at 13:32
• It's complicated but not terribly so if you know power and speed. That's the basis for the "virtual elevation" approach for estimating cycling drag -- it converts effort (in terms of power and speed) into elevation gain. Commented Jul 3, 2015 at 14:53
• @Cearon Lol :-) I just didn't want people imagining it was a big tough climb like we'll see in the TDF. Commented Jul 3, 2015 at 14:55
• @andy256 its only (only! lol) 700 meters shy of Alpe D'huez... The 6% is a little tamer but still Chappeau for riding it. Makes me being proud of getting up some of our ~200m 9% climbs around here seem pathetic. Commented Jul 3, 2015 at 14:58
• Maybe some day I'll climb a "hill" thats 3600 feet lol. Around here, the best I have is 13% grade 472 feet climb (144m). I guess I'll just ride that up and down a few times ;) its a fun one Commented Jul 3, 2015 at 19:50

It really depends on speed. If you go slow (and there is no headwind) then riding on the flat is almost effortless (rolling friction is a very small factor with properly-inflated road tires). What slows you on the flat is wind resistance, and the faster you go the more wind resistance you face.

On the other hand, climbing a hill of a given height consumes a fixed amount of energy regardless of how fast you do it (once you factor out wind resistance), and, in theory, regardless of how steep the climb is (though of course once the hill gets too steep to climb "theory" falls apart).

You can probably come up with a way to convert a given % climb rate into an equivalent headwind for you and your bike, but there's no way to convert % climb into level miles with any degree of meaningfulness. At best you can come up with a conversion factor that assumes a given speed or energy output. And this would obviously be different for every rider.

• Sure there's a way to convert % climb into level miles. All you need to know are a handful of things: your total all-up mass (including you, your bike, and all your equipment), your coefficient of rolling resistance, your coefficient of aerodynamic drag, the wind speed, and your ground speed. Commented Jul 3, 2015 at 14:49
• @R.Chung - My point is that you have to factor in speed, so it's not a simple equivalent between climb and miles. Commented Jul 3, 2015 at 18:06
• So I need a way to calculate my resistance (and by that you mean friction between the tire and ground and the wheel and whatever it's attached to?), my drag, and the wind at any given moment? Sounds.....too complex haha Commented Jul 6, 2015 at 17:29
• @TomSterkenburg - You don't need to know "rolling resistance" for standard (non-fat) tire bikes, since it's so small it can be ignored. Wind resistance is the biggie. And wind resistance is very greatly dependent on speed. Commented Jul 6, 2015 at 20:04
• Calculating that alone is another ordeal, since I need to know my speed and the wind speed and direction (average across time). Would be fun to figure out, but I'm not sure if worth the effort lol Commented Jul 6, 2015 at 20:39

If you want to use your own figures from strava or some other program you can easily get get a feeling for the relative effort between biking a certain distance on the flat and on a grade. Use this calculator....http://bikecalculator.com/

For example I have a 2 mile hill at 5% which my best average speed is 9.3 mph. Entering the data gives me a calorie total of 185 calories. If I bike that same distance at a 0% grade at 20mph it gives me 68 calories. All of which feels about right to me, although if you asked I would have guessed that biking the 5% is easily 3 times harder.

For fun to see how fast I would have to go on the flat to equal the 185 calories at 5% I got 36mph, something I cant do for 1 second much less 2 miles.

It depends on other factors as well. A big one is speed. When you climb a hill, there is a minimum energy expenditure to get up the hill. On the flat, there is almost no minimum, but riding 10 mph is much easier than 15 mph (or whatever range of speeds is suitable for you). If you climb hills fast, you will get tired quickly.

Elevation gain can be measured in various ways as well. If you use Strava or MapMyRide to tell you that you have gained 500 feet, that is much less than looking on a map and finding you got 500 feet above where you started. The apps add up all the little ups along the way, so may report 1500 feet of climb when your altitude is just up 500 feet. You have to get used to it.