Why Doug Martin is still the Bucs' best running back

Recall those days in Algebra learning about mean, median, and mode? Remember how you had to create that silly box-and-whiskers diagram from the median? You were probably wondering, "The process is simple, but when exactly is this ever going to be used?"

There is a debate on who is a better running back for the Tampa Bay Buccaneers between Doug Martin and Bobby Rainey. Some choose to neglect Martin's rookie season where he was selected to go to the Pro Bowl and want to focus on his 2013 production by looking at averages. In 2013, the Bucs had gone through three running backs due to injuries, Martin, Mike James, and Rainey.

2013 Tampa Bay Bucs' Running Backs Productions
Player	Att	Yds	Avg	Long	TD
Martin	127	456	3.6	28	1
Rainey	137	532	3.9	80	5
James	60	295	4.9	24	0

From this stat, Rainey does look like he's a more productive back than Martin, but James is head and shoulders above if we look solely on average, or yards per carry (YPC). All stats were gathered from ESPN.com.

2013 Tampa Bay RB per game Production
Game	Team	Rush Defensive Rank (Yds/A)	RB	AVG	RB	AVG	RB	AVG
1	NYJ	1	Martin	2.7
2	NO	28	Martin	5
3	NE	24	Martin	4.4	James	5
4	Ari	2	Martin	1.7
5	Phi	4	Martin	4.2	James	3.5
6	Atl	31	Martin	4.3	James	3.2
7	Car	12			James	3.9
8	Sea	7			James	5.6
9	Mia	18			James	8.2	Rainey	5.6
10	Atl	31					Rainey	5.4
11	Det	20					Rainey	1.9
12	Car	12					Rainey	3.7
13	Buf	23					Rainey	5.8
14	SF	9					Rainey	2.5
15	StL	3					Rainey	1.9
16	NO	28					Rainey	3.2

What is particularly nice about this chart is the overlapping of running backs as well as the defensive rush rankings of the teams the Bucs encountered. From this breakdown, we see Martin's production dropped facing the first and second best run defenses. Martin ran against the 1^st, 2^nd, and 4^th best run defenses in the league in 2013! In the first game, he had no blocking fullback. While in the fourth game, it was rookie QB Mike Glennon's debut. Rainey, on the other hand, only faced one top four rush defense.

In the first Atlanta game, Doug had 11 attempts for 47 yards.  James had 14 carries for 45 yards in that same game. While in the Miami game, James had five carries for 41 yards and Rainey had eight carries for 45 yards. Using simple transitive logic, Doug is better than James and James is better than Rainey. Therefore, Doug is better than Rainey. But of course, that is not a good way of comparing the running backs.

Now let us return to mean, median, and mode. Mean is another word for average. Median is the middle most number of an ordered list. (If the ordered list has an odd entry, then it is easy to find the middle number of that list. If the ordered list has an even entry, then you add the two in the middle and divide it in half to find the true middle of that ordered list.) Mode is the number that occurs most often, otherwise known as frequency. And frequency is what is being addressed in the box-and-whiskers' diagram, which utilizes the Median.

The box is the range from the lower median, or Lower Quartile, and the upper median, or Upper Quartile.  We rename the lower and upper medians as quartiles to not confuse those medians with the Median of the total list of entries. The Lower Quartile is the lower median between the lowest entry to the Median. Similarly, the Upper Quartile is the upper median between the Median and the highest entry. Again, we are finding the frequency of production in finding all three medians: the Lower Quartile, the Median, and the Upper Quartile. What this does is remove some of the extremes on both ends of the entry list, or aberrations. For example, in 2012, Doug Martin had produced a one game YPC of 10.0. Obviously, that is an extreme and not often repeatable.  We are looking for repeatability, or rather consistent frequency of an event occurring.

2013 Doug Martin Rush Stats
Median, Mode, Variance, and Difference
Per Game YPC, in order	1.7, 2.7, 4.2, 4.3, 4.4, 5.0
Median	(4.2 + 4.3)/2 = 4.25	Mean	3.6
Lower Quartile	2.7
Upper Quartile	4.4
Variance of the Box (UQ - LQ)	1.7	Mean-Median difference	0.65

2013 Bobby Rainey Rush Stats
Median, Mode, Variance, and Difference
Per Game YPC, in order	1.9, 1.9, 2.5, 3.2, 3.7, 5.4, 5.6, 5.8
Median	(3.2 + 3.7)/2 = 3.45	Mean	3.9
Lower Quartile	(1.9 + 2.5)/2 = 2.2
Upper Quartile	(5.4 + 5.6)/2 = 5.5
Variance of the Box (UQ - LQ)	3.3	Median-Mean difference	-0.45

2013 Mike James Rush Stats
Median, Mode, Variance, and Difference
Per Game YPC, in order	3.2, 3.5, 3.9, 5.0, 5.6, 8.2
Median	(3.9 + 5.0)/2 = 4.45	Mean	4.9
Lower Quartile	3.5
Upper Quartile	5.6
Variance of the Box (UQ - LQ)	2.1	Median-Mean difference	-0.45

Using the Box-and-Whiskers' method via Medians (lower quartile, median, and upper quartile), reveals the frequency of production on a per game basis. From here, you can extrapolate a plethora of conclusions from the Box-and-Whiskers' diagrams.

Median Comparison:

Martin: 4.25 YPC, Rainey: 3.45 YPC, and James: 4.45 YPC. The Median finds the middle of the frequency of per game production of YPC, or the middle of YPC production that occurs most often. Here, we see that Rainey's Median is the lowest among the three running backs. That means most of his runs per game productions are clumped around 3.45 YPC.

Included in the stats is the Median-Mean difference. This stat tells how the event that occurs most often differs from that of the total season's average of YPC. Looking at Martin's Median-Mean difference exposes that Martin is more likely to produce a higher YPC per game than state that Martin's season average YPC. This also reveals that Martin is more consistent to produce a higher YPC than Rainey on a per game basis.

Box Comparison (UQ - LQ):

Martin: 1.7, Rainey: 3.3, and James: 2.1. The Box describes the range of frequencies of a certain event occurring.  The Box eliminates extremes in the entry list while clumping the events, or entries, that which occurs most often.  Martin possesses the smallest range, 2.7 to 4.4 YPC, 1.7 YPC range variance.  Rainey has the widest range, 2.2 to 5.5 YPC, 3.3 YPC range variance. This stat informs us whose production is more consistent as well as whose production is more inconsistent. Rainey's production Box dictates that the team does not know what to really expect from game to game with that wide range of 3.3 yards per carry. Martin's variance is 1.6 yards per carry smaller than Rainey's variance range. The team can rely on Martin's production consistently than Rainey's.

Take that range, or Box, and apply the Median to that box, then it compiles an interesting image for productions.  Rainey's range is the widest and his productions occurs more often around 3.45 YPC per game, his Median.  Martin's range is the smallest, but his Median is on the high side with a YPC per game stat of 4.25.  Even though Martin's range is between 2.7 to 4.4 YPC, a majority of his productions occur more often around 4.25 YPC per game. Which means the team can rely on Martin producing at or around 4.25 YPC per game more often.

Median versus Mean:

Mean represents the average of the total production. What it does not describe is the reproducibility of said productions. Can the outcome be reproduced, or repeated, often or consistently? This is where Median and the Box-and-Whiskers' diagram becomes a helpful tool. Frequency is the number of occurrences of a repeating event per unit of time. We can interchangeably use reproducibility or repeatability with frequency, if that helps to understand what the Median and the Box-and-Whiskers diagram is trying to display.

While Rainey's mean was 3.9 YPC for the whole season, his ability to repeat it is not so.  In fact, Rainey's Box exhibits the inability to repeat it consistently. Whereas Martin's mean, 3.6 YPC, is 0.3 yards per carry lower than Rainey's, Martin's Box informs us that he can repeat his production more than Rainey. Not only that, but Martin's Median within that Box dictates he has surpassed his mean YPC with a 4.25 yards per carry production. Martin was able to repeat around 4.25 YPC more often. Rainey's Median was 3.45 YPC. That is 0.8 YPC less than Martin's Median. This information does not take into account the strength of schedule in respect to rush defense.

While average, or mean, is a common statistic, it does not necessarily paint a picture full of depth. It does not show a pattern, consistency, inconsistency, or reproducibility. There have been plenty of comments that Doug Martin is not as good as Rainey, with 2013 season being the reason why while noting Rainey's YPC total average. Doug Martin was a Pro Bowl running back in 2012, but some fans dismiss his 2012 production due to his injury in 2013.  Thankfully, I found a way to express how Martin is still a better running back by finally understanding why we were introduced to mean, median, and mode as well as learn how to plot and draw a Box-and-Whiskers' diagram. It was all about frequency production! Never in my life have I ever thought of using Box-and-Whiskers outside of school, but surprisingly there is a direct application!

Although many fans do want to discount Martin's 2012 season, I went ahead and did a Box-and-Whiskers diagram for Martin's 2012 production. Maybe we can see if there is anything worthwhile from 2012 to 2013 from presenting the diagram.

2012 Doug Martin Rush Stats
Median, Mode, Variance, and Difference
Per Game YPC, in order	1.8, 2.4, 2.8, 3.1, 3.3, 3.4, 3.5, 4.0, 4.1, 4.6, 4.7, 5.1, 5.3, 5.8, 5.8, 10.0
Median	(4.0 + 4.1)/2 = 4.05	Mean	4.6
Lower Quartile	(3.1 + 3.3)/2 = 3.2
Upper Quartile	(5.1 + 5.3)/2 = 5.2
Variance of the Box (UQ - LQ)	2	Median-Mean difference	-0.55

The Box-and-Whiskers diagram for 2012 reveals that Martin not only shrunk his variance in his 2013 season, but also improved his Median by 0.20 YPC. Martin's Median of 4.05 YPC differs from his total 2012 mean YPC of 4.6, just in case one was curious of Martin's total average YPC production. Comparing Martin's 2012 and 2013 Box-and -Whiskers' productions, Martin was having an improved year in 2013 before he fell to injury in the first Atlanta game.  Martin was rushing 1.1 YPC more than James in the same Atlanta game, where both had similar running attempts (as denoted in the article). And despite Martin's low production in two games against the top two rush defenses as well as the running against the fourth best rushing defense, his Median in 2013 was still better than his 2012 production. The 2013 season exposes that Martin actually improved his ability to reproduce, or repeat, gaining around 4.25 YPC, a 0.20 yards per carry improvement between the years.

I do not know why Doug Martin is underappreciated amongst Bucs' fans, but I hope this look at a frequency production on a per game basis paints a better picture of Martin, the competition he faced in 2013, and his importance to the team.  Also, it seems as though Mike James is not receiving any love for his work in 2013. Yet, I am happy to have this problem of having three more than capable running backs at our disposal.