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Math Class: Averages are not your Friend

It’s a dirty little secret, but one you may have guessed by the title of this article.

If you’re assessing players based solely on the average number of points they score per game, you could be doing your team a grave injustice. Sure, in a pinch, without much data to rely upon, averages will do when you’re forced to make a quick and dirty decision. Unfortunately, most decisions made based solely upon average points per game are not these “quick and dirty” calls, but are rather the extent of some owners so-called “research.” I beg you, please don’t be one of them.

With the information you already have on hand to determine a player’s average points per game, you can much better assess which players are better for your team and which players you can let some poor other sap have. The information in this article requires a small degree of mathematical analysis, but it’s nothing most of us didn’t learn in high school.  Yeah, I’m going to make you go back to those painful times, but just for a little bit.

Here’s a fun exercise. Using the 2012 DLF IDP Dynasty League Mock Draft several of us on the DLF staff took part in on MyFanastyLeague.com, where would you guess Indianapolis linebacker Gary Brackett ranked among all linebackers last year (bear in mind that Brackett was injured in the very first game of the regular season and didn’t return all year). Based solely on average points per game, Brackett ranks out as the #14 linebacker for the year for the 2011-2012 season!

Now, Brackett is a good player and if he played all year, he would likely have posted decent numbers, but he would have likely never approached being ranked #14 overall and able to maintain his high points per game average. The average Brackett posted was based upon a single game and that average beat out stud linebackers such as Paul Posluszny, Stephen Tulloch, standout rookie Sean Lee and fellow Colts teammate Pat Angerer.

Clearly, based upon this exercise, average just isn’t a valid way of evaluating a player. What should take its place though? Well, let’s talk about the averages less known, and much sexier cousin, standard deviation. While averages attempt give you an idea as to how many points a player typically scores per game, it fails in many ways.

First, averages gloss over how many games a player actually played in as well as how well a player performed in those games. Standard deviation, however, illuminates players that had one amazing game, but didn’t do much the rest of the season or those players who simply appeared in one or two games. An added bonus of using standard deviation is using it also helps identify those who are inconsistent in their play, and as we all know, consistency is a highly sought after commodity in dynasty leagues. If you can’t depend on your players week in and week out, you’re going to spend much of the season second guessing yourself and giving yourself an ulcer.

Have I sold you yet on incorporating standard deviation into your player evaluations? I hope so, otherwise the rest of this article is going to pass you by.

Now then, how do we determine this magical number?

Well, if you know how to use Microsoft Excel then you’re in luck – it’s built right into the program. Referring back to the 2012 DLF IDP Dynasty League Mock Draft, we’ll examine two players who ever very close in overall and average points. Look at #7 Wes Welker (335.9 points overall/20.9938 average per game) and #8 Matthew Stafford (335.6 points overall/20.975 average points per game). As you can see, those players are about as close as any two players could be, but who was more consistent? Grab the values for both players from Week 1 through Week 17 and paste them into A1 through Q2. In R1 type with the quotes “=STDEV.P(A1:Q1)” and in R2 type without the quotes “=STDEV.P(A2:Q2)”. If you did this right for Wes Welker, you should get a value of 11.82 and for Matthew Stafford you should get a value of 7.44.

Great, but what does this mean?

Well, it’s actually pretty simple – this tells you how tightly clustered all the values for any given player’s season are around their average. The lower the standard deviation, the more consistent a player is. Take Welker for instance, his best game was 51.6 points, while his worst was 4.2 points. Stafford on the other hand pulled down 38.0 points in his best game and 9.1 in his worst game. Sure, we’d all love to have Welker’s 51.6 game, but that was the anomaly, not the rule. The closest he ever got in any other game last season to 51.6 points was a 36.0 game. Stafford, on the other hand, scored 29.2 and 29.5 in two different games and was fairly consistent throughout the season. On a game-by-game basis, you’d want to draft Stafford over Welker because the average points per game you’re seeing from him are going to be a better indication of what to expect from him. As you can see, while our two players appeared very similar at first, the actual story is very different!

If you’re not too good with Microsoft Excel, never fear, we’ll walk through calculating it the good old fashioned way as well. This is clearly not the preferred method as it’s very time consuming, but it is possible. This is going to be where many of your eyes gloss over. If you hate math,  just skip the next paragraph and number description series.

Just as with Microsoft Excel, in order to calculate the standard deviation by hand, you’ll need all the point values that the player you’re evaluating scored during the season as well as the average points the player scored. In this fictional scenario let’s say we have a five game season in which the player scored 2, 3, 4, 5, and 6 points. First we need the average of all the games:

2 + 3 + 4 + 5 + 6 = 20/5 = 4 points

Now we need to subtract the average from every score:

2 – 4 = -2

3 – 4 = -1

4 – 4 = 0

5 – 4 = 1

6 – 4 = 2

After that, we have to square each of those numbers and add the squared numbers together:

(-2)2 + (-1)2 + (0)2 + (1)2 + (2)2

4 + 1 + 0 + 1 + 4 = 10

We now have all the information we need and need to plug in the numbers:

√10/5-1 = 1.5811

As I said earlier, it’s time consuming, but not impossible.

For those re-joining us, there is a third way – simply look for an online standard deviation calculator. An online calculator of standard deviation requires the least knowledge and is the easiest method by far, but it doesn’t provide a way for determining this important tool you can add to your evaluation toolset. It works in a pinch, but please don’t depend solely on an online calculator.

So, now that we have standard deviation, what other calculations can we do to determine the best player to take out of two closely scoring players? We’ve looked at the sexy cousin of averages, but did you know the family is bigger than just averages and standard deviation? Indeed it is, average (sometimes referred to as mean) was born as one member in a set of triplets, so to speak – the other two triplets are mode and median. These two could be called the ugly sisters because it seems as if everyone wants to take average to the ball and leave mode and median sitting at home. Maybe they aren’t as sexy, but it’s inner beauty that counts with these two!

First, let’s look at how to calculate mode and apply it to fantasy football.

Mode is the number that occurs most often in a series of numbers. In fantasy football, this number tends to point to a better expectation of what to expect out of a player because these are actual numbers the player put up as opposed to average, which is an approximation of what to expect. In a league that scores fractionally I suggest rounding in order to find the mode. Given our Welker/Stafford scenario, we find that Stafford posts a mode of 14 while Welker posts a mode of 11. Once again, we can see Stafford deserves the pick based solely on how often he scores a higher mode than Welker.

It’s not looking so good for a Welker pick right now, but maybe he can turn it around with our median calculation.

Median is probably the easiest calculation to make out of all the options we’ve discussed. All you have to do to determine median is find the number right in the middle when you order the numbers numerically. If the number of values is even, then add the two middle numbers and divide that value by two. Once again, looking at our Welker/Stafford example, we find that Welker has a median of 21.60 while Stafford pulls down a median of 19.90. Welker wins one! Median is helpful to determine what you can expect as a middle value, it ignores very good or very bad performances. This is why home income in the United States is based upon the median as opposed to the average – it ignores the extremely rich and the extremely poor, aiming instead for what the households directly in the middle make.

With all of these tools the obvious question is, “Which is the best to use when evaluating players?” The honest answer is, whatever you value most. I prefer standard deviation myself because it tells me how consistent a player is. Others prefer mode because it’s based upon actual numbers the player has scored and not derived calculations. Others still prefer median because it ignores values with are extremes in favor of a more central number. No one is right, the answer is based solely upon you value system. One might even want to perform all of these calculations and go with the player that performs best out of the three calculations, Stafford in our example.

These calculations are particularly helpful in a start-up draft, but can be used throughout the dynasty experience. Perhaps you can’t decide between two players on the waiver wire or are offered a trade that swaps similarly ranked players. These methods of evaluation are also very useful when proposing a trade or simply evaluating your own squad. What better way to give yourself the greatest advantage possible then to go out and get the players you need as opposed to just waiting for those players to be offered up to you and evaluating them then?

As with any dynasty league player age must be factored into any player evaluation. Unfortunately it’s tough gauge just how age affects a player, some players can’t even make it to 30 in the NFL, and others can push playing until they are almost 40. There is no absolute way to gauge just how productive a player can be as those years tick away. However, using these evaluation methods to calculate outcomes year after year you can develop trend lines which can show how a player is trending of time.

Maybe a player is still performing at the top of their position but their play is becoming more erratic, or perhaps their median score is slowly trending downwards. Both of these trend lines would be red flags to look out for and a time to consider selling high while you can get as much as possible for a player on the decline. While a player on the decline might not be immediately apparent to most people, trend lines can help you be ahead of the pack and look like a genius as the player continues to slide as time goes on. Perhaps you can pull of a trade for a young player on the ascent but pulling up marginally lower numbers with the knowledge that they will “grow into” their numbers over time, just as your declining player is getting passed by.

With all these new tools, get out these and evaluate to your heart’s content. At least now you have some actual numbers to rely upon when trying to choose between to similarly performing players. No longer do you have to rely upon the flipping of a coin.

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justin.moilanen
10 years ago

Nice article. I also like to use the Coefficient of Variation to assess the Standard Deviation as a percentage of the mean. Allows for an easy comparison of players that do not have similar means. For example, a QB with a mean score of 30 and a SDev of 10 is still preferred to a QB with a lower StDev of 5 but a mean of 15. CoVar lumps everything into one and shows who is truly more erratic.

jeff hemlick
10 years ago

Huh?

Adam Franssen
10 years ago

Thanks for the article! Comparing QBs to WRs is a bit of an apples to oranges comparison, though. This method is probably better served within groups rather than between groups.

10 years ago

I love math.

Basically this is the stats argument against the likes of V.Jax and D.Jax. Right?

Sensei John Kreese
10 years ago

A few years back we referred to this as “The Lee Evans argument”.

It might be more aptly characterized as the “Torrey Smith argument” nowadays.

Keith Fortier
10 years ago

I’d love to see the numbers crunched on Gronk and Graham. Great article Ghost. Consistency is king. And yes, Torrey Smith is the poster boy for gbux analysis, but I am buying because I feel he will become more consistent…

Skinny Elvis
10 years ago

Took me 2 minutes in Excel: PPR/4 points per TD
Gronk – mean = 20.7 ppg; standard deviation = 9.2
Graham – mean = 18.5 ppg; standard deviation = 5.5

Gronk had stinker game in Week 4 (2.5 points) and had the higher max game (34.0 points) which increased the variability. Graham’s min/max was 7.9 & 29.2. On a per week basis (not counting byes), Gronk out scored Graham for 10 of 15 weeks.

Discuss!

Skinny Elvis
Reply to  Skinny Elvis
10 years ago

Correction: 6 per rush/receiving td!

bmdassit
10 years ago

Good thought-provoking article.

I totally agree that the Coefficient of Variation is better when comparing means that are not the same. Because the closer the SD is to the mean, the more inconsistent the player is. If both player A and B have a SD of 10, you might think they are equally consistent. But if player A has a mean of 20 PPG, and player B has a mean of 15 PPG, then player A is actually WAY more consistent. The data in this case means that:

68% of the time, Player A scored between 10-30 points
68% of the time, Player B scored between 5-25 points

Yet, the CV would be (SD divided by the mean):
Player A: .5 (50%)
Player B: .66 (66%)

Smaller = more consistent. So you can immediately see that A is more consistent, even though the SD’s are the same. The CV basically means:

Player A’s score any given Sunday will be within 50% of his average score most of the time. While Player B’s will be within 66% of his average most of the time.

But in reality, you would not make the comparison above. Comparing players with vastly different means (that is, 4+ points apart) is dicey. Even if the lower scoring player is more consistent, you are probably better off with the higher scoring one.

It’s best to use these stats to compare guys with similar PPG.

Example 1:

Vincent Jackson:
Average 14PPG.
SD = 12
CV = .86 (86%)

Boldin:
Average 12PPG.
SD = 6.
CV = .5 (50%)

Boldin was much more consistent, and indeed he outscored V-Jax more often than not.

Example 2:

Mike Wallace vs. Percy Harvin – both averaged 16PPG, but Wallace was more consistent.
Wallace: SD = 6. CV = .4 (40%)
Harvin: SD = 9. CV = .6 (60%)

This stuff could also be useful to compare RB/WR/TE for a flex spot.

Danny Hall
10 years ago

Hey Ghost, great article. I was thinking more about this and was wondering if this standard Deviation has even more value when looking at Player rankings and season totals from year to year. Could this also help us solve the looming and mysterious affects of “Regression”. This really only helps with players with a body of work but I wanted to see if this would be helpful tool in player analysis.

Calvin Johnson 2007 2008 2009 2010 2011 Avg STD Dev 2012 Avg Projections
Receptions 48 78 67 77 96 73.2 16 93
Yards 756 1331 984 1129 1681 1176.2 314 1453
tds 4 12 5 12 16 9.8 5 11
total FP 147.6 283.1 195.4 261.9 360.1 249.62 73 325

Larry Fits 2004 2005 2006 2007 2008 2009 2010 2011 Avg Std Dev 2012 avg Projections
Recetptions 58 103 69 100 96 97 90 80 86.625 15 92
Yards 780 1409 946 1409 1431 1092 1137 1411 1201.875 235 1329
Tds 8 10 6 10 12 13 6 8 9.125 2 9
Total FP 184 303.9 199.6 300.9 311.1 284.2 239.7 269.1 261.5625 46 282

Observations: So Based on this information not only does larry have better averages in almost every area than Calvin he also has lower marks in standard deviation. Calvin’s season last year was clearly a career year and may be a bit of an outlier when compared to his standard deviation. Then looking at the avg projections for 2012(stats gathered from PFF, fantasy points an avg of espn,Pff, FFtoolbox) it looks likes the general consensus that people are projecting calvin to have another that would push the high side of his standard deviation. I’m guessing the odds of this happening again are not in his favor, which would lead be to believe that some regression is ahead. However, we have to take into consideration that Stafford is playing at a high level and the offense firing on all cylinders so it is very possible that he can meet and exceed those projections. It seems like larry’s projections are much safer and that there’s a much higher probability that he meets and exceeds those marks. Any one agree? disagree? thoughts on my approach?

Again, great article.

bmdassit
Reply to  Danny Hall
10 years ago

I like the way you’re thinking. But I don’t think we can make any reliable predictions based on year-to-year Standard Deviations. You’d have to statistically control for all the variables like age, QB, offensive system, etc.

I guess you could say Fitz is more likely to meet his modest expectations, but Calvin has much more upside.

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