The Runner’s WOD

The most important skill in the early days of human development is catching prey or getting away from a predator — running. This statistical investigation takes apart the anatomical components of running and looks at the variables which make or break a good runner. In my previous post Weightlifter’s Normative I came upon some athlete data… let’s start at the bottom.

Deadlift

8,635 men, 2,380 women. ±5% jitter. Original data clipped to 25kg – 300kg, 40s – 150s.

This amorphous cluster tells us that men, as a group, are faster than women; fastest women deadlift around 125kg, fastest men around 180kg; and that higher deadlift seems to correspond to faster running but, up to a point.  Further analysis is complicated by the fact that our data is sparse, kidney-shaped blob.  To extract [more] quantifiable insights, the analysis must internalize the functional meaning of these exercises. Namely, what matters for survival is that you outrun your fellow cave(wo)man to get the food, or outrun her while bolting from a sabertooth, thus not becoming the food.  Whether you are ahead by one second or one minute is irrelevant.  Consider the following experiment:

• Two individuals sprint 400m.
• If the winner’s deadlift is greater than the loser’s
  • record it as [win-win, |deadlift difference|];
• If the winner’s deadlift is less than the loser’s
  • record it as [win-lose, |deadlift difference|];
• Repeat for every pair of individuals.

Plot win-win percentage vs. |deadlift difference|. In other words: plot the victory probability of a stronger athlete vs. his/her [leg] strength advantage. This [computational] experiment produces an impressive amount of data because there are 8,635 men and 2,380 women available for this calculation. Any two individuals constitute a valid race: 37 million male pairs and 2.8 million female pairs. Recall that the number of pairs in n elements is given by n(n-1)/2.  The plot below encompasses 40 million races.

±5% jitter is added to counteract rounding and recording biases.  Mean filter with a radius of 3 points applied to the results. Transparent bands indicate uncertainty of a population measurement (a.k.a. standard error), mostly proportional to n^-½. Translucent, inverted semi-parabolas are logarithmic counts of the number of races for every deadlift difference, corresponding to the axis on the right.

The trends deteriorate and can no longer be trusted after the race count drops below 5k or so. Removing all data generated from less than 5k races yields:

8,635 men, 2,380 women. ±5% jitter. Original data clipped to 25kg – 300kg, 40s – 150s; results clipped to 5,000 races or more.  3-point radius mean filter applied to the results.

First feature to note is that both trends begin at (0, 50%). This desirable characteristic indicates that the samples are random and unbiased: if two athletes of the same gender have identical deadlifts, ignoring all else, neither has a statistical advantage in a 400m race.

As the deadlift difference between two athletes increases, so does the stronger athlete’s chance of winning the race. One may wonder why the curves aren’t steeper: why, for example, does a 100kg deadlift advantage only corresponds to a ~ 20% victory probability increase for men? Three answers: athlete height, body mass liability, and the fact that running isn’t entirely below the belt; in ascending order of importance.

Athlete Height

Athlete height is a correlate of leg length. Longer legs = longer stride = more distance covered per breath and heartbeat. Common wisdom suggests that taller athletes are better sprinters. Does this assertion stand up to mathematical scrutiny? Kind of…

8,760 men, 2,403 women. ±5% jitter. Original data clipped to 135cm – 250cm, 40s – 150s; results clipped to 5,000 races or more. 3-point radius mean filter applied to the results.

For women, height correlates to a modest increase in sprinting speed. Height appears to be irrelevant for men; which is peculiar and deserves an explanation.

If I am magically lengthened by 10%, the taller me will be able to keep up with the old me only if he is endowed with the amount of muscle commensurate with his new mass, which increases by ~ 33% because volume is proportional to length cubed. It is safe to assert that when it comes to running, men in this data set scale proportionally, while women scale super-proportionally. Meaning that as men get taller, on average, they are just strong enough to keep up with their shorter cohorts. Women’s strength increases at a rate greater than necessary to merely keep up, women get much stronger with increased height; again, on average.

Body Mass

For the most part, body mass is not a runner’s friend. However, it doesn’t appear to matter much up to a difference of ~ 30lb, acts as a handicap thereafter, especially for women.

8,821 men, 2,275 women. ±5% jitter. Original data clipped to 40kg – 150kg, 40s – 150s; results clipped to 5,000 races or more. 3-point radius mean filter applied to the results.

Upper Body Strength and Endurance

Put your hands in your pockets; now run. Primates don’t swing their arms while walking and running because it looks cool, we do it because bipedal locomotion is little more than a series of controlled falls. To control these falls, we must constantly control our center of mass. The faster we run, the more adjustments are needed per unit time. Every movement of the leg is countered by, or synchronized with, a movement of the arm and a slight twist of the torso.  If the arms and the torso can’t move as fast as the legs, upper body imposes a speed limit regardless of the leg strength.

A kipping pull-up encompasses the coordination of the upper and the lower body, core strength, arm strength, and cardiovascular endurance.

6,621 men, 1,548 women. ±5% jitter for the sprint time, none for the number of pull-ups. Original data clipped to 1 – 100 (athletes unable to do one kipping pull-up are excluded from this analysis), 40s – 150s; results clipped to 5,000 races or more. 3-point radius mean filter applied to the results.

Ignoring all other athlete parameters, unbroken kipping pull-ups appear to be a better predictor of sprinting ability than a deadlift maximum.

Correlation vs. Causation

One will be wise to question whether the correlations presented are meaningful. Would it not stand a reason for more experienced athletes to be able to lift more weight, do more kipping pull-ups unbroken, and run faster? If so, then we are merely looking at the athletes’ temporal progression. Much like plotting a person’s height vs. vocabulary: both grow naturally but, aren’t caused by one another. [Artificially] increasing one will have no effect on the other.

Chicken vs. Egg

If we assume that deadlift, kipping pull-ups, and running are anatomically related, in which direction is the causation arrow pointing? Are we looking at athletes who can do a lot of kipping pull-ups because they’re great runners, or athletes who are exceptional runners because they deadlift a lot of weight?

Part of the answer can be gleaned from the origin of the data: CrossFit. Sprinting, or running of any kind, is not emphasized in the CrossFit community to the same extent as, say, kipping pull-ups. CrossFit athletes are rarely instructed on proper speed or distance running techniques. Therefore, sprinting performance in the CrossFit community is at least in part an effect of the CrossFit training, and not a component thereof.

A more convincing piece of evidence are the contour plots of the 400m sprint victory probability as a function of the deadlift and the kipping pull-up differences:

Women’s 400m Sprint Victory Probability

Men’s 400m Sprint Victory Probability

6,490 men, 1,516 women. ±5% jitter for the sprint time and the deadlift, none for the number of pull-ups. Original data clipped to 1 – 100 (athletes unable to do one kipping pull-up are excluded from this analysis), 40s – 150s, 25kg – 300kg; results clipped to at least 5% of the maximum number of races per cell measuring 1s by 1kg.  10-point radius mean filter applied to the results. Thick contour is the 50% mark, the contour of no statistical advantage.

Non-trivially slanted contours are strong indication of the deadlift and the pull-up differences contributing to the sprinting speed, as opposed to solely correlating with it.

Consider the violet reference points. For women, being able to deadlift 70kg more than her opponent and do a few more kipping pull-ups provides the same statistical advantage, 70%, as being able to do 24 more kipping pull-ups than her equally-deadlifting opponent. For men, 70% statistical victory can be achieved by being able to do 25 more kipping pull-ups or just 12 more with a 100kg deadlift advantage.

The data clearly shows that when it comes to running, the upper body can compensate for the shortcomings of the legs and vice versa. This is so because the speed of bipedal (and quadrupedal) locomotion depends on two variables: stride length and frequency. If an athlete has strong legs, (s)he can make long strides not requiring as much coordination of the upper body. Athlete with lacking leg strength can take shorter strides at a higher frequency. Stride length is mostly limited by the muscles in the legs, stride frequency is mostly limited by the upper body strength and endurance.

Conclusion

Short of, or in complement to actual running, it would behoove anyone aspiring to lower sprint times to practice kipping pull-ups and deadlifts, in that order. Especially the former, because it much easier to get 10 more kipping pull-ups than deadlift 100 more pounds. As it pertains to sprinting, an increase of one kipping pull-up is equivalent to ~ 7kg deadlift increase for men; ~ 4kg for women:

∆

1 kipping pull-up

≈ ∆

10 15

lb deadlift

for women for men

A Word Of Caution

When one looks at, for example, the victory probability vs. height graph and notes a flat line for men, the takeaway is not that height has no effect on sprinting speed. The correct interpretation: it is not possible to use a man’s height alone to place a wager on his performance in a 400m race. Such interpretation is only valid for the data analyzed, and neither necessarily extends nor rules out its relevance to the entire population of Earth.

Weightlifter’s Normative

The Question

I once asked my weightlifting coach about a correlation among typical lifts. If I can Deadlift 300lbs, how much should I be able to Back Squat? I’m not talking about Olympic athletes whose records are public and who were as much born to be the best as they have trained to be so. I’m referring to an average human being who decided to play around with a barbell.  My coach was not able to answer my question definitively.  I found a website here and there with scant numbers and lacking citations but, nothing solid.

The Data

There’s a relatively large community of weightlifters who share their achievements publicly: CrossFit. Athletes create profiles such as this one and update them regularly.  I wrote some code to download the profile of every athlete on one of the CrossFit Open rosters, all 132,355 of them.

The Analysis

Back Squat vs. Deadlift

Consider the plot of the athletes’ Back Squat maximum vs. Deadlift maximum (hereafter the terms maximum and one rep max shall be omitted and it shall be understood that all quantities discussed are such).

The plots are made up of 13562, 30284, and 43846 samples respectively. Jitter is added to the data and opacity to the dots so that the plots are easier on the eyes. The jitter is necessary because the weights are multiples of five and if plotted as are create a caustic image; opacity helps to ignore the outliers.

Deadlift and Back Squat are two of the least technical lifts. Correlation between them is therefore likely entirely due to the athletes’ physical characteristics, as opposed to skill. The plot has a few stark features worth noting.

• The correlation is linear.
• The correlation does not depend on gender.
• Amateur athletes are typically able to Back Squat ~ 81% of the Deadlift, the solid line through the middle of the data.
• There appear to be sharp upper and lower bounds for the bulk of the data.
  • The upper bound is — suspiciously exactly — the line y = x.

Both lifts derive vast majority of the power from the legs. The same set of muscles is engaged in the same exact directions. There are, however, key differences.

• Back Squat requires one to lower and raise the bar in a controlled fashion. Athletes lower the bar sans control and sometimes even drop it after a Deadlift, thus dispensing with a significant fraction of the work.
• Athlete’s hip crease must drop below their knee during a Back Squat. It is more difficult to stand up from than position than from the bottom of the Deadlift, where the athlete’s legs aren’t nearly as bent (more on that later).

The points above hint as to the linearity of the correlation — identical muscles pulling in identical directions — and provide some insight into why an athlete can Deadlift more than (s)he can Back Squat — differing range of motion and the amount of work. Let’s now look at the apparent upper and lower bounds.

Deadlift ~ Back Squat

Given an uninjured athlete, there are three explanations for the case of one’s Deadlift and Back Squat approaching one another.

• An athlete is not squatting low enough during the Back Squat, not breaking parallel with the ground. His/her Back Squat maximum is exaggerated and must be lower if the exercise is performed properly.
• An athlete is placing an undue amount of the load on his/her back during the Deadlift, not using enough legs during the drive phase of the exercise. His/her Deadlift can increase with improved form.
• An athlete is taller than the average, requiring him/her to almost go into a full squat to grab the bar for the Deadlift. Shorter athletes are closer to the bar at the start of the exercise and need not drop into a full squat to pick it up. The lower is the squat, the harder it is to stand up from it. Little can be done to increase the Deadlift to Back Squat ratio for taller athletes.

Deadlift ≫ Back Squat

With the same assumptions as previously, there are three explanations for the Deadlift to significantly exceed the Back Squat.

• An athlete is not keeping his/her back erect enough while descending into the bottom of the squat, i.e. face too close to the knees, thus limiting the amount of weight (s)he can raise from that position. His/her Back Squat can increase with improved form.
• An athlete is descending too much during the Back Squat, thus limiting the amount of weight (s)he can lift from that position. His/her Back Squat can increase with improved form.
• An athlete is shorter than the average, allowing him/her to barely bend the knees at all to pick up the bar at the start of the Deadlift and lift significantly more than a comparably-developed, taller athlete.

At the time of writing of this post, Richard Froning‘s Back Squat is 83% of his Deadlift. Julie Foucher‘s ~ 82%.

Clean & Jerk vs. Back Squat

In the order of ascending difficulty, the next lift is the Clean (& Jerk). Consider the plot below (40,191 samples).

Clean & Jerk is a compound lift involving both raw strength, vast majority of which is provided by the same muscles as in the Back Squat and the Deadlift, and skill. Timing, shoulder strength and stability, coordination, which are all but absent from Back Squat and Deadlift, play an important role in Clean & Jerk. Two physically identical athletes max out at two significantly different weights depending upon their skill. The data places an average Clean & Jerk at ~ 69% of the Back Squat.

Whereas Richard Froning fell almost squarely into the average performance on his Deadlift vs. Back Squat, his Clean & Jerk is 78% of his Back Squat. This is so because the Deadlift and Back Squat are more about raw strength than skill. As an experienced weightlifter, Mr. Froning is expected to perform in the above-average skill range for the Clean & Jerk.  Julie Foucher’s Clean & Jerk is 76% of her Back Squat, also above average as expected.

Snatch vs. Clean & Jerk

The Snatch is the most technical lift. Consider its correlation to the Clean & Jerk (38,847 samples).

Requiring even more training and skill, the Snatch averages ~ 76% of the Clean & Jerk. Richard Froning’s ratio is ~ 82%, Julie Foucher’s is 85%; comparable to the modern Olympic weightlifters.

A Note About Olympic Lifts.

For the Clean & Jerk and the Snatch, athletes of above-average skill are expected to be above the average ratios, those of lesser skill are below. If an amateur athlete is significantly below the average ratio on one or both of the lifts, it could signal underdeveloped or insufficiently mobile shoulders and/or lack of coordination.

Conclusion

Table of average relationships between the lifts analyzed, column divided by row.

	Snatch	Clean & Jerk	Back Squat	Deadlift
Snatch	1	1.32	1.92	2.38
Clean & Jerk	0.76	1	1.45	1.77
Back Squat	0.52	0.69	1	1.23
Deadlift	0.42	0.56	0.81	1

Notes

An astute reader will notice that each plot shows less variance than the one before it. The thinning of the correlation is primarily caused by the fact that the athletes attempting more complicated lifts such as the Snatch are typically more skilled and therefore fall within a narrower distribution than novice athletes who primarily train with Deadlifts and Back Squats. This is a common phenomenon. If 100 individuals’ 1mi run time is sampled at random, the spread is likely to lie between 5min and 20min due to the variance in natural predilections and lifestyles. If said individuals receive one year of comparable training, the dispersion will shrink markedly. An extreme example of this effect are Olympic athletes, who often win by mere (milli)seconds and kilograms.

A trained eye may also notice that the intercepts of the plots are not zero. The analysis enforces zero intercepts to simplify the interpretation of the results, which are approximate a priori. Allowing the intercept to be derived from the data does not meaningfully alter the conclusions.

Future Work

These scatter plots represent a fraction of the data. More analysis is coming soon.