A Better RB Speed Score: Z-Speed Score

Before I get to the Foot Speed Score post, I'm going to wedge in this one.

As I stated in the last post, Punch Score, every year there are armchair analysts who declare which running backs are bargains and which will be busts, based solely on one number: Speed Score.

While Speed Score enjoys moderate success - it has successfully identified bargain backs like Brandon Jacobs, Arian Foster, and Rudi Johnson, and issued warnings upon NFL duds Mark Ingram and Trung Candidate, among others - it both misses on several running back archetypes and has a questionable mathematical foundation.

Punch Score is a great start to resolving those misses. Foot Speed will help, too. However, before we get to Foot Speed and putting it all together, though, we need to give Speed Score an actual mathematical foundation.

Wait, what is this Speed Score?

I'm sure almost all of you are familiar with Speed Scores. For those of you who are not, it compares a player's speed - based on his NFL Combine 40 yard dash time - to his weight. The premise is:

When a 225-pound player runs a 4.48 40-yard dash, it's a lot more impressive than that same 40 time from player who weighs 185 pounds.

I brushed upon its success above, but it's worth re-emphasizing its usefulness: the original Speed Score allegedly accounted for 45% of future performance, defined by yards/carry, DYAR, and a few other statistics.

While I applaud the thought process, the mathematics behind the way the score is calculated leaves room for improvement.

Giving Speed Score a Mathematical Foundation

The current Speed Score is calculated in a way that your math teacher may have called "a hand waving argument":

Adjust for those factors and you get the formula for Speed Score: (weight * 200)/40 time^4. Multiplying the player's weight by 200 conveniently scales the metric so that an average Speed Score is right around 100.

In other words, multiply by an arbitrary number because it looks good and then divide by something that is also arbitrarily raised to the fourth power. Because everyone measures speed in units of pounds per seconds to the fourth, right?

Look, I get what the creator of the statistic is trying to do: scale the number so there's an intuitive mid-way point such that anyone above the round score - in this case, "right around 100" - is above-average and anyone below the round score is below-average. But there is a much prettier way to do that that makes mathematical sense as well as intuitive sense: z-scoring.

Z-scoring is a mathematical method that scales a value in such a way that zero becomes average - the mean. Numbers above zero are above-average and numbers below zero are below-average. (You can read more about the procedure here, but it's a fairly commonplace and easy-to-produce figure.) The value (either positive or negative) is based upon how rare that individual value is, based upon population statistics and assuming a normal distribution.

In that sense, z-scoring weights and speeds into our (aptly-named) Z-Speed Score accomplishes exactly what the current Speed Score tries to: it gives you an easy and intuitive way of determining if a player is better or worse than expected. But it's also done in a scientific fashion.

How the new Z-Speed Score is calculated

In order to calculate a player's Z-Speed Score, we need to know three things: the population mean, the population standard deviation, and the player's individual value. For our z-scoring, we are using the mean and standard deviation of combine scores since 1999, provided the player participated in both the weigh-in and the 40 yard dash. For running backs, this is a robust population of 338 players.

Note: Only players who run the 40 yard dash at the combine are included in the weight and speed measurements. Players who opt not to run do not count towards the rarity calculations. This ensures that our calculated Z-Speed Score averages exactly zero.

Mathematically, the old method determined that going from a 200 lb, 4.500 40 was equivalent to a 210 lb, 4.556 40. This wasn't based on science. Now, however, we can measure exactly how rare a specific weight and a specific 40 are, and compare those two rarities. This is scientific.

In other words, we'll subtract the player's z-scored 40 time from their z-scored weight to produce our Z-Speed Score. A number above zero means that the player's 40 time is more rare - with faster oriented as positive - than the player's weight - where lighter is denoted as positive.

We could, in theory, z-score the result in other to further normalize our figures, but that isn't entirely necessary. Remember, z-scores are used to identify potential busts and bargains - we don't really need to precisely rank each individual player.

As an example: Darren McFadden's 4.33 40 time at the combine in 2008 registers as a 1.91 z-score. His weight, 211 pounds, registers as a 0.15 z-score. So while he's just slightly lighter than the combine average (at RB), he's immensely faster. This gives him a Z-Speed Score of 1.76 - a very good score.

Some example Z-Speed Scores

Here are some benchmark Z-Speed Scores for the reader to chew on. I have also posted that player's colloquial Speed Score, so you can compare the two methods.

Player	Year	40 Time	Weight	Z-Speed Score	Speed Score
Brandon Jacobs	2005	4.56	267	3.87	123.5
Andre Brown	2009	4.37	224	2.36	122.9
Michael Turner	2004	4.49	237	2.29	116.6
Ryan Mathews	2010	4.37	218	1.93	119.5
Chris Johnson	2008	4.24	197	1.50	121.9
Johnathan Franklin	2013	4.49	205	-0.03	100.9
Mark Ingram	2011	4.62	215	-0.40	94.4
Brian Westbrook	2002	4.57	200	-1.07	91.7
Robbie Rouse	2013	4.80	190	-3.73	71.58

It may appear that the heavier guys receive the bulk of the advantage from the new Z-Speed Score, but that was always the point - guys who run faster than their size are potential bargains, while players who are fast in large part because of their light weight are potential busts.

For fun: a guy Chris Johnson's size, 1.17 on the z-scored weight scale, would require a z-scored 40 time of 5.04 to equal Brandon Jacobs' combined rarity of speed and weight z-score. That corresponds with a 3.96 40 yard dash time. Which is basically what scientists estimate Usain Bolt would run the 40 yard time at. Usain Bolt, however, would actually have a Z-Speed Score of 4.51 since his weight, 207 pounds, is only a little bit on the light side for a prospective NFL RB. In other words: Usain Bolt has a more rare combination of weight and speed than anyone who has run in the NFL Combine since 1999. I think we all expect that, but it's nice to see it proven scientifically.

Conclusions

I have given the theory behind why the new Z-Speed Score is "better" than the old one. However, proof they say, is in the pudding. So how well does the new Z-Speed Score correlate with future success?

Gives the Speed Score a robust mathematical foundation while preserving its predictiveness.

Unfortunately, I don't possess Football Outsiders' DYAR numbers in a ready-to-crunch format. So, instead, I analyzed how my Z-Speed Score correlated with the kitchen sink of what could be considered "future success": CarAV, peak value, yards / carry, etc. and even went through all the more trivial variables, like Pro Bowl appearances or All Pro nominations.

The results were roughly the same in how it directly correlated with future success, with moderate improvement, within the margin of error, on some of the metrics.

Replicating the promising results of a key analytic tool, as the Speed Score has become for armchair pundits, while giving it a solid mathematical foundation is a nice result - it's even nicer once you consider it's more orthogonal to other key metrics, like the aforementioned Punch Score and the soon-to-be-released Foot Speed Score metric.

So what's next?

When the combine finishes on February 26, come back to Bolts From The Blue to see our post on the Speed Scores and the Z-Speed Scores.

And, as mentioned above, stay tuned for the next post which will focus on the other new metric I have come up with in addition to Punch Score: Foot Speed Score. It won't disappoint.