Speed= Stride length X Stride frequency

This simple equation is a staple in sprinting and biomechanics material. It simply means that to see a change in speed you’ve either got to increase the ground you cover (stride length) or increase your turnover (stride frequency) or some combination of the two. The bottom line is that something has to change. But what is it that changes when we go faster?

If you are a Chi Running proponent, your answer will be stride length, as Danny Dreyer says to keep the frequency constant while only changing length. In the real world though, the answer is that it depends and that limiting yourself to only being able to change one or the other is a mistake. Let’s look at what the elites do.

One study looked at the stride length and frequency of the top 3 finishers in the 10k at the 2007 world championships. This included Bekele (1st) Sihine (2nd) and Mathathi (3rd). They calculated their individual speed, frequency, and length for every 400m lap of the 25 lap race.  The graph below depicts their speed, stride length, and stride frequency:

The first thing you notice is that these three different athletes ran the same speed until the last 2 laps or so. That’s to be expected, but what is interesting is that they had different strategies to do it. Bekele had a low stride frequency with a long stride length for the first 9,000m. On the other hand, Mathathi had a small stride length and a very high frequency to run the same speed. Sihine was somewhere in between these two. What is also interesting is that Mathathi who is 1.67m tall and Sihine who is 1.71m tall both had considerably smaller stride lengths than Bekele who is only 1.60m tall. So, all you shorter runners complaining about how your height keeps you from having a long stride, just look towards Bekele. The point is though that we have three runners all employing different strategies to run the same speed.

It gets even more interesting when we look at the last 1km when the pace changed dramatically. For reference they went from hitting around 2:42-2:45 for each km to final km of 2:30, 2:33, and 2:36 respectively, and had last laps of 55.51, 58.66, and 62.16 so the pace picked up considerably. The question is how did these athletes do it?

Bekele did it by changing his stride frequency from roughly 190 strides per minute to an astonishing ~216 strides per minute, all while keeping his stride length about the same. So he went from having the lowest stride rate to the highest by far, thus using an increase in stride rate to increase his speed, while maintaining stride length. On the other hand, Sihine showed an interesting pattern. On the 3rd to last lap, he picked up his pace slightly with an increase in stride frequency. But then on the last lap, he increased his speed dramatically with an increase in stride length. The exact opposite approach of Bekele. Lastly, Mathathi, who had the shortest stride length, increased his stride length on the 3rd to last lap to increase his speed. What is really interesting is that on the last two laps, Mathathi’s stride frequency, which was the highest during the race, decreased slightly, while he increased his stride length significantly on the last lap. What this tells me is that he was suffering from the most fatigue and to compensate for the drop in stride rate, he tried to increase his stride length. The net result on the last lap was a maintenance of speed, not the giant increase like in the other two.

What does this all mean? Three different runners all had different ways of running at there race speed and then chose different methods of increasing speed when it was time to do. Interestingly, they seemed to increase the one factor that was lowest during most of the race. In Bekele’s case he had a low stride rate, so he increased that dramatically. Mathathi who had a small stride length tried to increase that during the final kilometer. And Sihine who was in between both during the race in terms of rate and length, did a little bit of an increase in both. It’s almost as if the runners subconsciously chose to rely more on rate or length for most of the race, to “rest” the other factor, and then went to the other factor when it was time to increase the speed.

This opens up a lot of questions, but before delving into those let’s look at one more study.

Sprinting- Stride length and Frequency

The above examples showed that there are several different ways to maintain a steady pace and different ways to increase the pace. But what about during maximum speed? Obviously when we sprint all out we are trying to maximally optimize the rate/length combination. Rate and length have an interesting relationship in that at max speed increasing one will lead to a decrease of the other, so it’s balance between the two that matter.

Which brings us to a study Salo et al. (2010) entitled: “Elite Sprinting: Are Athletes Individually Step Frequency or Step Length Reliant?”

In this study they looked at world class 100m dash competitions and calculated the individuals stride length and rate to see what the best athletes did. What they found was very interesting.

Individual athletes seemed to favor either longer strides or higher cadence. Just as we saw in the earlier study where Bekele versus Sihine and Mathathi adopted a different combination of length and frequency, it seems to occur in sprinters too. In the study, they found that individual athletes differed in their reliance on stride length. Some were very heavily reliant on having a large stride length, while others trended more towards relying on both, and even one athlete being reliant on stride frequency.

What’s interesting about this is that the author’s speculate that perhaps differences in the nervous system or power generation explains why some athletes rely on a longer stride or higher frequencies. For sprinters who are powerful athletes and can generate a large amount of force through their stride, they are more likely to cover more ground with each stride, thus being stride length dependent. On the other hand, some sprinters seem to rely on the nervous system’s ability to rapidly turn the legs over and contract and relax the muscles extremely quickly, thus making them more reliant on stride frequency. Driving this point home, the author’s stated that “Thus, it is possible to reach the absolute top level of sprinting in the world (run under 10.00 s) with widely varying pattern of SF and SL reliance.”

In recent years there has been a trend towards focusing solely on maximum force production as a way to train sprinters. It was fostered by Weyend et al.’s early 2000’s study that showed sprint speed was partially determined by vertical Ground reaction forces. Some coaches took this to mean that the only way to increase speed was to increase force production. In the Salo et al. (2010) study they mention that increased ability to produce force has been shown to be a determinant of stride length in animal models. What the current study shows us though is that for some athletes the limiter is not force production, but how quickly an athlete can contract and relax the muscles and ultimately turn the legs over faster. The author’s conclude by saying:

“Overall it is reasonable to conclude that SL is related more to increased force production, and SF is associated with faster force production during the contact and quick leg turnover requiring neural adaptations. Higher SF requires cross-bridges within the muscles to be built at high rates, and thus these need a high rate of neural activation.
Consequently, it is proposed that the SF reliant athletes are required to concentrate on neural activation in their final preparations for the major races and have a nervous system ready such that they can produce the quick turnover of the legs. On the other hand, the SL reliant athletes need to keep their strength levels up throughout the season and have the required flexibility in the hip area to produce long steps. Naturally, athletes cannot totally forget the non-reliant variable, as any disproportionate reductions in one variable cannot be generally compensated for by the other variable.”

Another interesting fact, which we saw in the 10k study too, is that the height of the athlete did not impact whether or not they would take longer strides or have a higher cadence.

Putting it all together
The point of this is to show you that there are many roads to running the same speed, whether it is sprinting or running a 10k. What the above studies show is that perhaps it is highly individual. Runner’s like Bekele may have adaptations that lead to him being able to hold a massive stride length for a long time, and then switch and be able to hold a large stride rate during the latter stages of the race.

The question is should we individualize based on our preferred method of running? Stride length dependent athletes might need to do more strength/power work, while stride frequency athletes might need to do more turnover work/neuromuscular work. Or should we work on our weaker side of the equation? The ultimate answer for us distance runners may be that we should be like Bekele and be able to run submaximally with one strategy and then be able to switch strategies when fatigued. Maybe this is why he can kick so well? He’s able to run both ways.

The great Hungarian coach Mihali Igloi was a big fan of what Bekele seems to have done in the 10k. Igloi believed that runners had a natural stride they preferred, but then should use their “unnatural” stride during heavy fatigue. He called them short swing and long swing strides. Short Swing would be equivalent to a shorter stride length with a high frequency, while long swing would be a long stride with a reduced frequency. His contention was that you needed to work on going back and forth between these two stride types during practice. If you could become adept at using both strides, then you could run the majority of your race in your preferred stride style, but when fatigue started to happen, you’d switch to the other stride style. His contention was that by switching, you are changing the muscle fibers that are recruited slightly, and you are also changing how they work (powerful as in a long swing stride, or rapidly as in a short swing stride). In essence, this is what Bekele did. He went from a long swing to a short swing style in the final kilometer.

Hopefully this mini stride frequency versus length topic makes you think a little bit. In particular it should be obvious that there is no magic stride frequency that everyone should run. It depends on the athlete and their speed. Some rely more on stride length, while others will rely more on frequency like Mathathi. But you have to be able to change but frequency and length throughout the spectrum of running speeds. If you, like Chi running advocates, try to hold one constant, you’ll reach your limit in increasing the opposite one before you should, and thus will be slower. In an ideal world, you should train to be able to do both when needed. Perhaps that’s what makes Bekele the man to beat on the kick?

References:
Salo et al. (2010) Elite Sprinting: Are Athletes Individually Step Frequency or Step Length Reliant? Medicine and Science in Sport and Exercise

Enomoto et al. (2008). Biomechanical analysis of the medalists in the 10,000 metres at the 2007 World
Championships in Athletics, New Studies in Athletics

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1. Mark U. on November 3, 2010 at 11:42 pm

Excellent post, especially through reflecting upon the diverging running tactics of Bekele, Sihine and Mathathi. The conclusion, that we're all different and that we utilize different stride length / cadence strategies depending in part on our fatigue is consistent with Matt Fitzgerald's book "Run: The Mind-body Method of Running by Feel" which I recommend.

2. GZ on November 4, 2010 at 1:45 am

216! That is insane. Excellent post.

3. flowrunningproject.com on November 4, 2010 at 2:50 am

Another great and informative post Steve. Towards the end it did get me to thinking. Which method of increasing speed generates less impact force with the ground, and increase of stride rate or an increase of stride length?

Patton

4. Anonymous on November 4, 2010 at 7:00 am

Do they do this explicitly ie are they conciously aware and thinking "now I'll increase my stride length/ frequency" Or are they less conciously aware of the mechanics and just aiming to run faster and doing that in naturally different ways? "Tactics" was mentioned which hints that it's a concious decision. What do you think Steve?

5. RICK'S RUNNING on November 4, 2010 at 9:54 am

Back in my cycle racing days the book I read by Eddie B [ ex national US coach]recommended a cadence of 86-92 revs per min for time trials, this I followed until I started doing road racing as well.
Due to the changing pace and much faster speeds of road racing my cadence often reached 120 Rpm and more!
The side effect of this was, I found that in Time trials my average cadence increased so i felt more natural at holding higher revs around 95-100 rpm.
End result was faster 10 and 25 mile TT's.
We often here that a turnover of 90 is best, but is it?
Often new runners and cyclists will have a much slower turnover of 60-70 but with training can quickly improve into the 80-90 range, I guess my question is would training for even higher average cadence lead to greater performance.
Arthur Lydiard used downhill strides with his runners, should we be adding this type of session to our workouts?
Did arthurs runners have a higher leg turnover than the other runners of that time period?

6. Wes on November 4, 2010 at 2:44 pm

Agreed. Excellent post. I've increased my stride frequency from 160 to 176. Getting there slowly 🙂

7. Anonymous on November 4, 2010 at 3:18 pm

Steve,

I'm a Chi Running instructor and a competitive runner. I enjoy reading your blog to get a scientific perspective on training.
I'd like to clear up the misconception that Danny Dreyer suggests to ALWAYS keep the stride rate the same. Danny suggests an increase in cadence while sprinting, kicking and potentially other situations such as running down a really steep trail hill. So while Danny does suggest keeping the stride frequency constant in *many* situations, this is not *always* the case. Please bear in mind that Chi is mostly training runners who are not elite and have no intentions of being, or training like an elite. Many runners have benefited greatly from getting their cadence up and running at a steady frequency. You'll note that elite runners make up a very small percentage of the entire running population.
That being said, I'm not even sure why it is necessary to bring Chi Running into your blog posts. You have mentioned it several times now, and many times have included a snide remark ("pseudo guru style" "in the real world")
I'm not sure I understand what exactly your issue is with Chi Running/Danny Dreyer, and why the need to mention it in so many posts?

Finally, my main point in posting here is to clear up the misconception that Chi Running is promoting some dogmatic – one size fits all program that never changes and is professed to work for all people in all circumstances. This is entirely not the case. Danny has helped me and many others to break away from a lot of the dogma in the running community and explore other options outside of the conventional discourse. In all of his material, workshops, and interviews he practically pleads with people to listen to their own body and find what works for them. He often says "don't take my word for it, go out and try it for yourself." The last thing Danny wants is to become an expert on running that people listen to blindly without personal experimentation. This is part of the reason why he named his book "Chi Running" as opposed to "The Danny Dreyer Formula"

While there are several physical suggestions, guidelines etc…the main point of Chi is to find a comfortable, efficient running stride that will help each individual runner meet there goals and objectives. We are working to find principles that work for about 80% of people. We fully acknowledge there are things that will not work for some people. Based on the huge positive reception to Chi Running, massive amounts of emails/testimonials and passion behind it, I'd say we are off to a great start.

It's certainly beneficial to dissect and analyze different teaching methods. I welcome the discussion. I do think you could improve in your approach as you seem to be more emotional about Chi than scientific. (basing that on your sarcastic comments)

Hopefully, this clears up some of the ideas you have about what and how we teach. I think a further exploration into Chi Running will help you find that a lot of instructors are very similar to yourself. Runners looking to break from the dogmatic conventional paradigm and find evidence based practices that enhance the overall experience of running whether it be running for pleasure, speed, or both.

8. Anonymous on November 4, 2010 at 7:28 pm

Speed=stride lengthX frequency/running gauge measurementsXdrift rate.

Stride and frequency are determined by skeletal efficiency and how easily the entire body moves. If optimum running form isn't measured or described, then optimum speed will never be achieved.

One's running gauge measurement reduces the body's ability to even run straight. If it isn't measured, then the original formula can't be right.

If two runners run at the exact same speed, the runner that runs straigher will cross the finish line first.

If your stride and frequency push you further off a straight line course, then optimum speed measured in linear distance travelled will be negatively imacted.

Sport Jester

9. stevemagness on November 4, 2010 at 7:40 pm

Sport Jester-

The deviation from a straight line is minimal to non-existant in almost every good runner. It's so much so that no one really measures it except for in the start of a sprint race where they are coming out of the blocks.

There's a reason that it's not in the research literature at all, because it's not a big factor .
Mark- Thanks for the comment, I'm reading that book right now.

Rick- Thanks for the cycling info. There seems to be a lot of connections that you can make with cycling. I always found it interesting that for decades (and some still do) insisted that cycling at a low RPM was more efficient than a high RPM. Only recently have people conceded that the pro's might be right. I don't know about Lydiard's guys. That would be interesting to look at.

Flow- An increase in stride length seems to cause more impact

10. stevemagness on November 4, 2010 at 7:47 pm

Anonymous-

Thanks for the post and thanks a lot for claryifying the Chi running stuff. My only source is the Chi Running Book. In that book, Dreyer stated that cadence should be kept the same with stride length being the variable that you change. So that's what I was basing it off of. If he's changed some, then I'm glad.

As for my comments on Chi or Pose or whatever. First off, I'm a bit of a perfectionist. Second off, I feel like those groups all have their websites and such where you can go to learn about their methods. My goal is to provide that counterbalance that says "hey wait a minute, that might not be correct."

As far as my past "pseudo-science" comments, it's a pet peave of mine. If you want to train runners a certain way and you've had success with it, go for it. Just don't try and make up a scientific sounding explanation to justify it. With both Pose and Chi, the whole gravity thing is what I call pseudo-science. It's using science to justify your beliefs instead of looking at the science, research, and practical and forming beliefs off of that.

Lastly, I deal with the competitive side of the sport. It bugs me when pose/chi/whoever cross over into the performance side and say this will make you run faster. I'd have more respect for Chi running if it didn't use pseudo-scientific explanations to justify it, and didn't try and bridge a method that seems to be for recreational runners, into the competitive side of the sport.

11. Anonymous on November 4, 2010 at 7:56 pm

If drift is "minimal" then why are track events held in counter clockwise rotation and not the other way around?

Sport Jester

12. Anonymous on November 4, 2010 at 8:11 pm

"within .03seconds of qualifying for NCAA Cross Country nationals as individual"

Drift rate may be minimal, but it adds 20-40 seconds to your 5K times…

Your drift rate cost you a chance for National competition…. And you think it was minimal?

Sport Jester

13. stevemagness on November 4, 2010 at 8:12 pm

We run counter clockwise because that's what International governing body of track declared. There used to be some clockwise races in Europe way back in the day.

14. stevemagness on November 4, 2010 at 8:15 pm

Sport Jester- Sorry, no mean spirit intended, but you sound like those SOMAX guys who think bounce, crossover are what is keeping americans from running sub 2hr marathon.

Drift doesn't matter. It does not take 20-40sec off your 5k. That's ludicrous.

15. Anonymous on November 4, 2010 at 8:34 pm

I'm definitely not a "SOMAX" guy…

Clinical studies show a 3 meter drift rate per 20 seconds of running if uncontrolled (gladly link the studies). The reason little literature exists is because its a mental skill, not physical one so coaches exclude it from any training routines.

Right handed runners (and the majority) as athletes drift to the left because of side dominance in strength, so that drift is taken advantage of to maximize times.

Left handed athletes have to be trained completely different in technique to overcome the right handed drift that's natural to them.

How do you train for the side dominance advantage to track running as pertaining to speed? It has nothing to do with stride length or frequency.

It also applies to street running when corners are in opposite to a runner's natural drift tendencies as technique has to change since increasing speed during the curves of any race is impossible to do.

And for elite marathon runners, are measured at 29-31 miles and amateurs at 36 miles, in physical distance travelled calculating in drift rates.

Cutting 20 seconds of a 5K is easy when the biomechanic factors which induce drift are eliminated in track competition.

For cross country runners, without the physical visual guidelines of a track lane to give drift reference, a 40 second cut in time is easy…

Sport Jester

• David Peters on November 25, 2015 at 3:59 pm

Do you actually believe what you are saying? If this was the case, then running a sub 2 hour marathon wouldn't be that hard. Just work on drift and they can run 1:50 or under.

16. Anonymous on November 4, 2010 at 9:49 pm

Steve,

What's your opinion on the recently Salazar/Ritz article discussing his form changes? There is an accompanying video with the article that shows a comparison btw Bekele and Ritz. Have you seen the video? Is the hip angle/separation legit, I see you mentioned SOMAX in your above post and they describe a similiar hip angle comparison btw Sammy W. and Hall & Ritz, SOMAX also has another article describing hip angle with various UK runners showing that Coe and Cram had the 100+ degree as oppose to their recent runners that were in the 85-90 range. Does this concept of hip angle/separation really mean proper hip extension?

Thanks,
Ken

17. Anonymous on November 5, 2010 at 1:29 am

Great, honest post and very interesting. KEep it up. You are helping lots of runners. Thanks Steve.

18. stevemagness on November 5, 2010 at 7:00 pm

Ken-

The Salazar/Ritz piece was a good article. It's an article though written by someone who has no running background most likely, so it's hard to tell what the exact changes are and such.

As for hip angle/seperation= It's essentially a way to sort of quantify hip extension/knee lift. Is it useful? It can be, but it has to measured correctly, which is where the SOMAX people go wrong.

In order to take any 2d angle measurement worth anything, you have to make sure the camera is exactly parallel to the runner and that the runner is as close to directly in front of the camera when measurements are taken. If you don't do this, you can't measure angles. Secondly, you have to make sure the angles are measured at the exact same point in your two comparison pictures. Lastly, you have to have a marker to insure that you measure the angles from the same spot in the two comparisons. Meaning that the 3 points better be somewhere consistant.

Somax just takes random youtube clips, regardless of the camera angle and measures stuff. THe angles mean nothing.

To give you guys a visual, I took two screen shots of video I had of myself running. This is during the exact same 150m sprint. It's exagerated to make a point, but if you go look at the SOMAX website they take random angles no matter the camera angle.

What you get is a "stride angle" of 71deg and 111deg on the same run, going the same speed, depending on the camera angle…So you see how it's easy for them to manipulate angles to prove their point? They use similar tricks in all their other "analysis". The pictures are linked below:

19. Fitz on November 6, 2010 at 9:16 pm

From my own personal experience, increasing stride rate at the end of a race is easier for me than increasing my stride length. It seems like I rest the neural component for most of the race because in most races I can turn it over quickly in the last 600-800m.

I wonder if there is a way to "feel" whether your body prefers a longer SL or a faster SF? I know that when I'm finishing a race and trying to sprint, I tend to focus on the fastest turnover I can – rather that a more forceful muscle contraction which will give you a longer SL.

Great post Steve and I love the pictures you linked to. Cheers!

20. Anonymous on November 8, 2010 at 5:38 am

Hi Steve,
This post is one of my favorite topics about running and IMO one of the most misunderstood. Speed = Stride Length x Stride Rate is an incorrect and outdated equation. Stride Length and Stride Rate are the results of increased speed not the causes of speed. Your mention of the 2000 Weyand et al study of faster running speeds is only partially correct, "sprint speed was partially determined by vertical Ground reaction forces." What you neglect to mention is the fact that the study also concluded that rate of force delivery and ground contact time available for force delivery are also determinants in faster running speeds. To me, the rate of force delivery and the onset of fatigue are the critical issues affecting speed. I'm not certain how trainable rate of force delivery is, but improving muscular force and ground contact time have training applications.

Coach Doug Robinson

21. Anonymous on November 11, 2010 at 8:53 pm

Hey Steve,
I was wondering about anaerobic work and its effects upon the aerobic system. Say I wanted to do start introducing sessions such as 20x200m or 10x300m in say January once a week at about 60s 400m pace to prepare for indoors (for a sub 14min 5k guy). Do you think this hurt the aerobic development that would need to be had for later in the year? I understand that too much anaerobic work can alter the blood ph; esp if one was to do that kind of work for a long amount of time. But I'm wondering how much is appropriate to be implemented. thanks.

22. Anonymous on November 19, 2010 at 12:15 am

When I run workouts barefoot on the grass, it seems that my stride frequency naturally increases. Does anybody else believe the same happens to them? If so, why do you think that might be?

23. Anonymous on July 27, 2011 at 12:46 pm

Frequency and stride length seem to be inversely proportional. whenfrequency of stride increases the length gets shortened. when the length of stride increases that puts a premium on stride frequency. It all depends on how good the athlete is in combining both to produce optimum performance. because of the room for improvement within limits this writing has good implication on training. As a coach i experimented a longer stride length by putting 2+ meters line marks on the track so that athletes keep this interval of stride for running. I mimicked frequency with a computer generated beeps with which athletes adjust their stride frequency by hearing. There is improvement but there is also a risk of body injury (such as the groin)with higher stride length.

24. Anonymous on January 5, 2012 at 9:16 pm

Love the post! I think Usain Bolt is a great example of how speed is derived from a combination of stride length and stride frequency. His stride frequency is about equal to his competitors, but his stride is so much longer thanks to those long legs of his. When you watch him run you can tell that he covers so much more ground with each stride, but he still manages to keep his stride rate high.

25. Anonymous on April 5, 2012 at 5:04 am

I searched dozens of posts to find any with actual numbers for the stride length. I have heard for years I should increase my stride rate, but what if I was already fast? This is the only post where you can see what other good runners actually get for stride rate and length – thanks. Now I know where I stand in relation to other runners.

26. Anonymous on June 7, 2012 at 7:53 am

Steve, this is very useful and practical material. I've been searching for my ideal walking and running pace since I run my first marathon in 2011. Your article has set me on a more sensible track; to vary my strategy for different stages of the run and to get the right balance between length and frequency for each stage. The example of what the top runners do was a great help. Thanks.
Nana

27. Gershon ben Franja on March 8, 2014 at 4:42 pm

Thanks for a great article. I've been trying to increase my walking speed and I'm experimenting to see if the stride rate affects my speed. I'm using a click generator (metronome) on my mp3 player to keep a constant rate and Fitbit to check it. So far, I don't have any conclusive results.

What I can say with certainty is without the click generator, I tended to slow my rate going up mild hills and increase it going down the same hills. When I started using the click generator, my speed increased.

28. Dutch Skater on May 7, 2014 at 1:10 am

Great food for thought, thanks for sharing.
Preference may be more a product of what one is accustomed to doing rather than best built for. There may be other ways of running we cannot execute right now, but could be conditioned to master, and do better with.
Also, Bekele might be more efficient in long stride when he's mainly aerobically. His high-frequency finish may not just be what he uses when the "preferred" technique fatigues away, he may simply not be able to reach such speeds at a relatively low frequency.
His relatively low cruising frequency might not point to raw strength after all. It may well be much more complicated than that, or result of a lazy technique when just running along. More study necessary 🙂

If an athlete's height is no factor to stride frequency, perhaps calf length or girth is?
I am 6ft4 and have switched from 145spm to much higher, while getting rid of heel strike which ruined my body and had me given up by a sports doc. No treatment for my louzy feet. But when I changed my running, I quickly bested PB's in a big way.
Anyway, my frequency is often higher than even lightweight girls and women around me. Watching myself jog in the shop windows' reflection is looks silly and weak. My strides are tiny for my height. Thus, my range of motion is tiny. If I'd open up the range of motion, I'd look like doing high knee drills when just running along. Not exactly efficient.
If cadence is not height related, then range of motion is inversely related to height. That does make us wonder why the given stride frequency for a given speed (or should we think intensity?) is so relatively constant.