The Roto Grotto: z-scores applied

Z-scores based on the target stats for various roto tiers can be applied to specific team circumstances. Previously, the zTotal columns which were driving my overall value order were context-neutral. They could tell which players to select with your first pick in a fantasy draft based only on their final 2012 stats and league scoring, but overall value may not be what you need depending on your team composition, your accumulated statistics, and your distance from specific roto point tiers.

First, you need to have all of the means and standard deviations I use to calculate the Z-scores for each tier, so you can apply them given your situation in this and future seasons. These numbers are per game:

Points uHR devHR uSB devSB uRBI devRBI uRuns devRuns uAvg devAvg
10 0.1275 0.0660 0.0911 0.0742 0.4653 0.1667 0.4780 0.1440 0.2856 0.0364
9 0.1275 0.0660 0.0824 0.0742 0.4491 0.1667 0.4624 0.1440 0.2813 0.0364
8 0.1154 0.0660 0.0770 0.0742 0.4379 0.1667 0.4522 0.1440 0.2784 0.0364
7 0.1116 0.0660 0.0725 0.0742 0.4284 0.1667 0.4432 0.1440 0.2762 0.0364
6 0.1079 0.0660 0.0684 0.0742 0.4196 0.1667 0.4348 0.1440 0.2743 0.0364
5 0.1042 0.0660 0.0643 0.0742 0.4104 0.1667 0.4261 0.1440 0.2724 0.0364
4 0.1004 0.0660 0.0602 0.0742 0.4003 0.1667 0.4166 0.1440 0.2704 0.0364
3 0.0960 0.0660 0.0556 0.0742 0.3886 0.1667 0.4051 0.1440 0.2682 0.0364
2 0.0906 0.0660 0.0500 0.0742 0.3734 0.1667 0.3903 0.1440 0.2655 0.0364

With that, let’s run through a couple of examples.

From my first one of these articles, I mentioned a hypothetical league where Team B needed steals and Team C needed batting average (both were chasing Team A in their respective categories). Let me flesh out the details. Theirs is a 10-team league—which has been my assumption for all of the Z-scores I’ve used. I’ll assume that Team B currently has five roto points in steals and could jump to six if he passed Team A. I’ll also assume that Team C currently has seven roto points in average and could jump to eight if he passed Team A.

For Team B, here is a table of player values based on the six roto points benchmark for stolen bases:

Player Season Points zSB
Emilio Bonifacio 2012 6 5.19
Dee Gordon 2012 6 4.04
Everth Cabrera 2012 6 4.00
Mike Trout 2012 6 3.83
Tony Campana 2012 6 3.62
Coco Crisp 2012 6 3.46
Ben Revere 2012 6 3.43
Rajai Davis 2012 6 3.35
Jarrod Dyson 2012 6 3.04
Juan Pierre 2012 6 2.91
Darin Mastroianni 2012 6 2.75
Michael Bourn 2012 6 2.73
Anthony Gose 2012 6 2.69
Carlos Gomez 2012 6 2.62
Starling Marte 2012 6 2.52
Shane Victorino 2012 6 2.49
Jose Reyes 2012 6 2.45
Jordan Schafer 2012 6 2.26
Desmond Jennings 2012 6 2.24
Alcides Escobar 2012 6 2.12
Jose Altuve 2012 6 2.10
Quintin Berry 2012 6 2.09
Drew Stubbs 2012 6 2.05
B.J. Upton 2012 6 1.94
Pedro Ciriaco 2012 6 1.92
Jason Kipnis 2012 6 1.83
Norichika Aoki 2012 6 1.76
Alejandro de Aza 2012 6 1.75
Alexi Casilla 2012 6 1.75
Ryan Braun 2012 6 1.70
Jimmy Rollins 2012 6 1.67
Dewayne Wise 2012 6 1.64
Jacoby Ellsbury 2012 6 1.63
Angel Pagan 2012 6 1.62
Gregor Blanco 2012 6 1.56
Ichiro Suzuki 2012 6 1.49
Maicer Izturis 2012 6 1.37
Ezequiel Carrera 2012 6 1.32
Lorenzo Cain 2012 6 1.29
Cameron Maybin 2012 6 1.28
Will Venable 2012 6 1.26
Ian Desmond 2012 6 1.26
Sam Fuld 2012 6 1.22
Starlin Castro 2012 6 1.16
Jon Jay 2012 6 1.15
Justin Ruggiano 2012 6 1.15
Michael Saunders 2012 6 1.11
Carlos Gonzalez 2012 6 1.07
Alex Rios 2012 6 1.05
Elliot Johnson 2012 6 1.05

For Team C, here is a table of player values based on the eight roto points benchmark for batting average (which I left unscaled for at-bats because of the proximity to the end of the season in the hypothetical example. I’m just assuming that all listed players are receiving playing time):

Player Season Points zAvg
Melky Cabrera 2012 8 1.87
Joey Votto 2012 8 1.61
Buster Posey 2012 8 1.58
Miguel Cabrera 2012 8 1.41
Andrew McCutchen 2012 8 1.34
Mike Trout 2012 8 1.30
Carlos Ruiz 2012 8 1.29
Jeff Keppinger 2012 8 1.27
Andy Dirks 2012 8 1.19
Adrian Beltre 2012 8 1.18

The first thing that stands out is just how much more dispersed stolen bases are than batting average. It makes sense. A player that can’t hit won’t last in the majors for long, but a player than can’t steal can still be a great player.

So, what trade should Team B propose? Well, any positive Z-score player is one that will help him improve in a needed category. In other words, trading Joey Votto for Sam Fuld is a win for him in the sense that Fuld has a higher zSB than Votto does, and that is the only category that can make an impact for him.

Most fantasy players will probably never find themselves in a situation so idealized. However, that is why Z-scores can really become useful. Since these Z-scores are built around a target mean based on expected points needed to reach a certain roto tier, a trade where each side’s total Z-score based on his context—his ability to gain and lose roto points in various categories—is equal is a fair-value trade.

In the example, Joey Votto has a zAvg of 1.61. For Team B to break even in terms of contextual value, he needs to trade Votto for a player with a zSB of 1.61 or greater. Since stolen base is such a dispersed category, that should not be hard to do. Players like Angel Pagan and Dewayne Wise are close advantageous players he could target on Team C, if Team C has either player.

Again, a situation where an owner should want to trade Joey Votto for Angel Pagan is probably unrealistic. One that is more plausible is between owners—who I will call Team X and Team Y—at the start of the season and where Team X has decided to punt batting average.

Since the season is just starting, I’ll assume Team Y is targeting eight roto points in all categories. His top-25 looks the same as the generic one for his point benchmark (this time I am scaling average for at-bats):

Player Season Points zHR zSB zRBI zRun zAvgScl zTotal
Mike Trout 2012 8.00 1.52 3.71 0.95 3.30 1.06 10.55
Ryan Braun 2012 8.00 2.28 1.59 1.74 1.73 0.99 8.32
Miguel Cabrera 2012 8.00 2.39 -0.70 2.55 1.56 1.28 7.08
Josh Hamilton 2012 8.00 2.65 -0.40 2.56 1.69 0.14 6.65
Andrew McCutchen 2012 8.00 1.24 0.68 1.04 1.59 1.16 5.72
Edwin Encarnacion 2012 8.00 2.47 0.12 1.74 1.14 0.04 5.51
Mike Stanton 2012 8.00 2.81 -0.38 1.57 1.09 0.20 5.29
Jose Bautista 2012 8.00 2.70 -0.31 1.61 1.69 -0.50 5.19
Matt Kemp 2012 8.00 1.54 0.11 1.28 1.71 0.39 5.02
David Ortiz 2012 8.00 2.12 -1.04 1.37 1.88 0.51 4.85
Carlos Gonzalez 2012 8.00 0.72 0.96 1.15 1.44 0.51 4.78
Chase Headley 2012 8.00 1.17 0.38 1.66 0.96 0.20 4.36
Adrian Beltre 2012 8.00 1.75 -0.95 1.29 1.09 1.04 4.22
Allen Craig 2012 8.00 1.05 -0.81 2.01 1.29 0.54 4.09
Melky Cabrera 2012 8.00 -0.27 0.51 0.56 2.02 1.26 4.07
Alex Rios 2012 8.00 0.66 0.94 0.85 0.97 0.63 4.05
Ian Desmond 2012 8.00 1.16 1.14 0.74 0.71 0.29 4.04
Aramis Ramirez 2012 8.00 1.00 -0.22 1.60 1.15 0.50 4.02
Robinson Cano 2012 8.00 1.36 -0.79 0.88 1.39 0.86 3.70
Yoenis Cespedes 2012 8.00 0.95 0.63 1.19 0.63 0.26 3.66
Curtis Granderson 2012 8.00 2.32 -0.20 1.35 1.29 -1.12 3.64
Josh Willingham 2012 8.00 1.91 -0.76 1.92 0.93 -0.38 3.62
Evan Longoria 2012 8.00 1.73 -0.67 1.83 0.52 0.12 3.53
Adam Jones 2012 8.00 1.24 0.29 0.41 1.27 0.23 3.45
B.J. Upton 2012 8.00 1.16 1.82 0.58 0.62 -0.75 3.43

The numbers are a little different than in previous articles because of a small code fix.

Meanwhile, Team X knows he will get one point in batting average, so he has to try to win every other category. His top-25 looks a bit different:

Player Season Points zHR zSB zRBI zRun zTotal
Mike Trout 2012 10.00 1.34 3.52 0.79 3.13 8.78
Ryan Braun 2012 10.00 2.10 1.40 1.57 1.55 6.62
Josh Hamilton 2012 10.00 2.47 -0.59 2.40 1.51 5.79
Miguel Cabrera 2012 10.00 2.21 -0.89 2.39 1.38 5.08
Jose Bautista 2012 10.00 2.51 -0.50 1.45 1.51 4.98
Edwin Encarnacion 2012 10.00 2.28 -0.07 1.58 0.96 4.75
Mike Stanton 2012 10.00 2.63 -0.57 1.40 0.91 4.37
Curtis Granderson 2012 10.00 2.14 -0.39 1.18 1.11 4.04
Matt Kemp 2012 10.00 1.36 -0.08 1.11 1.53 3.91
Andrew McCutchen 2012 10.00 1.06 0.49 0.88 1.41 3.84
David Ortiz 2012 10.00 1.94 -1.23 1.21 1.70 3.62
Carlos Gonzalez 2012 10.00 0.54 0.77 0.99 1.26 3.55
B.J. Upton 2012 10.00 0.97 1.63 0.41 0.44 3.46
Chase Headley 2012 10.00 0.99 0.20 1.49 0.78 3.45
Josh Willingham 2012 10.00 1.72 -0.95 1.76 0.75 3.29
Ian Desmond 2012 10.00 0.98 0.95 0.58 0.53 3.03
Adam Dunn 2012 10.00 2.18 -1.05 1.02 0.68 2.84
Allen Craig 2012 10.00 0.87 -1.00 1.85 1.12 2.83
Aramis Ramirez 2012 10.00 0.81 -0.41 1.44 0.97 2.80
Carlos Beltran 2012 10.00 1.28 -0.07 1.06 0.50 2.77
Coco Crisp 2012 10.00 -0.54 3.15 -0.49 0.62 2.73
Jimmy Rollins 2012 10.00 0.30 1.36 -0.18 1.22 2.71
Alex Rios 2012 10.00 0.48 0.75 0.69 0.79 2.71
Evan Longoria 2012 10.00 1.55 -0.86 1.67 0.34 2.69
Yoenis Cespedes 2012 10.00 0.77 0.44 1.02 0.45 2.68

The top-10s look pretty similar because those players are major contributors in all categories. I expanded the lists to 25 players so you could see the biggest moves, B.J. Upton and Curtis Granderson. Removing the one category that hurts them makes each player a top-15 values and a likely trade target for Team X.

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