Is Bobby Rainey really the Bucs' most productive back?

Tampa Buccaneer running back Bobby Rainey averages 4.3 yards per run attempt.  He is the most productive running back on the team because he averages about 1.4 yards more than his fellow running backs Doug Martin and Charles Sims.  Is he truly that much more productive?

Statistics are a form of measurement based upon production, namely dealing with numbers of a common measurement.  We can compare apples to apples indiscriminately.  Instead of saying, for example in sports, a player looks faster than that other guy from across the country.  Looking is a biased form of measurement.  It is not common.  It is not the same.

Even though statistics are based upon common measurement, sometimes different types of statistics present, or rather represent, a different outcome.  Math is a language.  English is a language.  Languages describe persons, places, things, and/or events.  The more you learn of a language, the more you are able to describe better.

Let us take the description of colors as an example.  We are taught, by proxy of children's toys, of the order and color of the spectrum.  Recall roygbiv, phonetically "roy-gee-bihv".  You can see this pattern on children's xylophones and ring toss toys.  Roygbiv is the order and basic colors of the spectrum: red, orange, yellow, green, blue, indigo, and violet.  Further along life, you are introduced the more colors than your basic roygbiv such as magenta, cobalt blue, tangerine orange, and pewter.  Do you find yourself often correcting non-Buc fans that our color is not gray, but pewter?

In English, instead of saying a person ran, one can describe a person as jogging, racing, prancing, or dancing as opposed to simply saying a person ran or a football player ran for a touchdown.  It is bland and does not give enough description of how a player or person ran.  Math is similar in that respect such that there are more and more formulae to describe that same incident.  Each statistic is a thread of fabric that when weaved together helps to describe a picture.

Bobby Rainey's average is 4.3 yards per attempt after 11 games.  What this implies is that Rainey will produce 4.3 yard on every attempt, also known as an event or occurrence.  Similarly, Rainey's average per game is also 4.3 yards per game, yet another occurrence.  Let us delve into Bobby's body of work for the 2014 season thus far. While we are at it, I will also include Martin's and Sims' per game occurrences as well.

2014 Tampa Bay RB per game Production
Game	Team	Rush Defensive Rank (Yds/A)	RB	AVG	RB	AVG	RB	AVG
1	Car	24	Rainey	3	Martin	1
2	StL	21	Rainey	6.5
3	Atl	15	Rainey	3.7
4	Pit	26	Rainey	0.5	Martin	2.9
5	NO	30	Rainey	3.5	Martin	3.2
6	Bal	5	Rainey	6	Martin	4.1
7	Min	25	Rainey	3.1	Martin	2.7
8	Cle	23	Rainey	4.6
9	Atl	15	Rainey	2.3			Sims	2.9
10	Was	11	Rainey	0.8			Sims	2.8
11	Chi	16	Rainey	3	Martin	2.5	Sims	3.7
12	Cin	19	Rainey	11	Martin	3.2	Sims	1.2
13	Det
14	Car
15	GB
16	NO

From the chart above, you can see how erratic the run production is for Bobby Rainey.  It varies as low as 0.5 yards average to 11.0 yards average.  Then there is everything in between.  Using average as a good barometer of production for Rainey seems as if it were fools gold as one would expect 4.3 yards average to occur more often.  Averages, in this instance, include extremes and do not paint an appropriate picture of production on a consistent basis.  This where medians are introduced along with a Box-and-Whisker plot to help with painting the imagery for production based upon game production occurrences.

Box-and-Whisker plots utilize medians to help find the middle of occurrences.  Several medians are utilized to create a Box-and-Whisker plot.  The Box-and-Whisker plot removes extremes as well as reveal where occurrences happen more often.  Median is the middle of an order set of entries.  Then one can find a lower median for the lower set of numbers and an upper median for the upper set of numbers.  This breaks the set of entries into fourths.  The lower median will be renamed lower quartile to help reduce the confusion of using the word median.  Similarly, the upper median of the set of entries will be renamed upper quartile.

2014 Bobby Rainey Rush Stats (After 12 games)
Median, Mode, Variance, and Difference
Per Game YPC, in order	0.5, 0.8, 2.3, 3.0, 3.0, 3.1, 3.5, 3.7, 4.6, 6.0, 6.5, 11.0
Median	(3.1 + 3.5)/2 = 3.3	Mean (Avg)	4.3
Lower Quartile	(2.3 + 3.0)/2 = 2.65
Upper Quartile	(4.6 + 6.0)/2 = 5.3
Variance of the Box (UQ - LQ)	2.65	Median-Mean difference	-1.0

Although Bobby's average is 4.3 yards per attempt as well as 4.3 yards per game, utilizing medians and expressing it in a Box-and-Whisker plot reveals the majority of occurrences revolve around 3.3 yards per game.  In short, Rainey's run production most happens much lower than his average reflects.  Bobby's average does not reflect reproducibility as his Box-and-Whisker plot designates most of his runs one yard less per outing.  Also, look at how wide a range that Box reflects, a variance of 2.65 yard.  Rainey could produce 2.65 yards per game or 5.3 yards per game, but the true reflection of Rainey's per game production falls upon 3.3 yards per game.

Last season, Bobby had an average of 3.9 yards per attempt and his median was 3.45 yards per game.  With more carries this year, Bobby's median has taken a slight hit, but his average has taken an increase of nearly half a yard.  That reveals the illusion that Rainey is a productive back when a majority of his run production is around 3.3 yards and not the 4.3 yards per attempt average that everyone seemingly is touting.

Now, let us look at both Charles Sims and Doug Martin utilizing medians and expressing them in a Box-and-Whisker plot.

2014 Charles Sims Rush Stats (After 12 games)
Median, Mode, Variance, and Difference
Per Game YPC, in order	1.2, 2.8, 2.9, 3.7
Median	(2.8+ 2.9)/2 = 2.85	Mean	2.7
Lower Quartile	(1.2+ 2.8)/2 = 2.0
Upper Quartile	(2.9+ 3.7)/2 = 3.3
Variance of the Box (UQ - LQ)	1.3	Median-Mean difference	+ 0.15

Although Sims does have a small sample, his median is better than his average.  That means is his production per game is on the high side, especially when looked upon his Box range as compared to Rainey.

2014 Doug Martin Rush Stats (after 11 games)
Median, Mode, Variance, and Difference
Per Game YPC, in order	1.0, 2.5, 2.7, 2.9, 3.2, 3.2, 4.1
Median	2.9	Mean	2.9
Lower Quartile	2.5
Upper Quartile	3.2
Variance of the Box (UQ - LQ)	0.7	Mean-Median difference	0.00

Finally, we come to Doug Martin.  If you look at his chart, then you will notice his median and mean are the same.  Now, look at his Box range.  Doug has the smallest range of all three of the running backs presented here.  Martin has two more entries (or games) than Sims, but notice his production box range is still smaller than Sims' Box range.  Hence, Martin becomes a very predictable running back with his production per game.  This information predicts that his production is most reproducible than both Rainey and Sims.

Compared to last season, Doug's median production has faltered like one would if they were not able to recognize the "sar-chasm".  In 2013, Martin had an average of 3.6 yards per game and a median of 4.25 yards production.  Many pundits and fans think Martin only produced well in his rookie year, but that is not true.  Martin's rookie season he had an average of 4.6 yards, but his median production was 4.05 yards per attempt per game.  On a per game basis production of reproducibility, Martin was better in 2013 than his rookie year.

2014 TB Bucs Running Backs
Average v Median Productions
Player	Avg per attempt	Median per game
Rainey	4.3	3.3
Sims	2.7	2.85
Martin	2.9	2.9

The utilization of average per attempt as a reference of production is misleading as one would believe that Rainey will always produce 4.3 yards per game.  Rainey does not.  This was proven so using medians, measuring the occurrences of said events re-occurring.  We are talking about reproducibility of said production.

Should we refer to the median per game statistic, then we notice the reproducibility of all three running backs do not vary greatly as it is quite significant in the average statistics.  Now factor in turnovers for the running backs.  As runners only, Martin has zero fumbles lost, Sims has one fumble lost, and Rainey has two fumbles lost.

There is so much information that one can extrapolate from the Box-and-Whisker plot in respect to reproducibility, consistency, inconsistency, re-occurrences as well as emphasize games as unique events that matter pertaining to reproducibility.  The Box-and-Whisker plot includes the extremes, or outliers, in the making of the plot, but it excludes them when it comes to producing the Box and finding of the median.  Averages are a good statistic, but it does not frame the image of production very well when adding a source such as median to mix to help aid paint that picture.  Also, averages can be skewed immensely by one or two extraordinary games.

Based upon the calculations, Rainey's average stat is inflated by a whole yard when compared to his median production.  Martin's average and median productions are the same.  With that said, Rainey's median production is still best among all three running backs within this article, but by only 0.4 yards.  Add into the conversation turnovers and relying on Rainey maybe a scary proposition.

I do not know how the Bucs' organization make their evaluations and why Rainey has continued to be move down the depth chart, but I am giving plausible reasons why Rainey is moving down the depth chart and that the theme that Rainey is the most productive running back is not as impactful as the consensus believes.