# Estimating Standard Deviation in On-ice Shooting Percentage Talent

I have tackled the subject of on-ice shooting percentage a number of times here but I think it is a subject that has been under researched in hockey analytics. Historically people have done some split half comparisons found weak correlations and written it off as a significant or useful factor in hockey analytics. While some of the research has merit, a lot of the research deals with too small of a sample size to get any really useful correlations. Split-half season correlations with majority of the players is including players that might have 3 goals int he first half and 7 in the second half and that is just not enough to draw any conclusions from. Even year over year correlations have their issues and in addition to smallish sample sizes it suffers problems related to roster changes and how roster changes impact on-ice shooting percentages. Ideally we’d want to eliminate all these factors and get down to actual on-ice shooting percentage talent factoring out both luck/randomness and roster changes.

Today @MimicoHero posted an article discussing shooting percentage (and save percentage) by looking at multi-year vs multi-year comparisons. It’s a good article so have a read and I have written many articles like this in the past. This is important research but as I eluded to above, year over year comparisons suffer from issues related to roster change which potentially limit what we can actually learn from the data. People often look at even/odd games to eliminate these roster issues and that is a pretty good methodology. Once in the past I took this idea to the extreme and even used even/odd seconds in order to attempt to isolate true talent from other factors (note that subsequent to that article I found a bug in my code that may have impacted the results so I don’t have 100% confidence in them. I hope to revisit this in a future post to confirm the results.). This pretty much assures that the teammates a player plays with and the opponents they play against and the situations they play in will be almost identical in both halves of the data. I hope to revisit the even/odd second work in a future post to confirm and extend on that research but for this post I am going to take another approach. For this post I am going to focus solely on shooting percentage and use an even/odd shot methodology which should do a pretty good job of removing roster change effects as well.

I took all 5v5 shot data from 2007-08 through 2013-14 and for each forward I took their first 200, 400, 600, 800, 1000, 1200, 1400, 1600, 1800 and 2000 shots for that they were on the ice for. This allowed me to do 100 vs 100 shot, 200 vs 200 shot, … 1000 vs 1000 shot comparisons. For comparison sake, in addition to even/odd shots I am also going to look at first half vs second half comparisons to get an idea of how different the correlations are (i.e. what the impact of roster changes is on a players on-ice shooting percentage). Here are the resulting correlation coefficients.

Scenario | SplitHalf | Even vs Odd | NPlayers |

100v100 | 0.186 | 0.159 | 723 |

200v200 | 0.229 | 0.268 | 590 |

300v300 | 0.296 | 0.330 | 502 |

400v400 | 0.368 | 0.375 | 443 |

500v500 | 0.379 | 0.440 | 399 |

600v600 | 0.431 | 0.481 | 350 |

700v700 | 0.421 | 0.463 | 319 |

800v800 | 0.451 | 0.486 | 285 |

900v900 | 0.440 | 0.454 | 261 |

1000v1000 | 0.415 | 0.498 | 222 |

And here is the table in graphical form.

Let’s start with the good news. As expected even vs odd correlations are better than first half vs second half correlations though it really isn’t as significant of a difference as I might have expected. This is especially true with the larger sample sizes where the spread should theoretically get larger.

What I did find a bit troubling is that correlations seem to max out at 600 shots vs 600 shots and even those correlations aren’t all that great (0.45-0.50). In theory as sample size increases one should get better and better correlations and as they approach infinity they should approach 1.00. Instead, they seem to approach 0.5 which had me questioning my data.

After some thought though I realized the problem was likely due to the decreasing number of players within the larger shot total groups. What this does is it restricts the spread in talent as only the top level players remain in those larger groups. As you increase the shot requirements you start weeding out the lesser players that are on the ice for less ice time and fewer shots. So, while randomness decreases with increased number of shots so does the spread in talent. My theory is the signal (talent) to noise (randomness) ratio is not actually improving enough to see improving results.

To test this theory I looked at the standard deviations within each even/odd group. Since we also have a definitive N value for each group (100, 200, 300, etc.) and I can calculate the average shooting percentage it is possible to estimate the standard deviation due to randomness. With the overall standard deviation and an estimated standard deviation of randomness it is possible to calculate the standard deviation in on-ice shooting percentage talent. Here are the results of that math.

Scenario | SD(EvenSh%) | SD(OddSh%) | SD(Randomness) | SD(Talent) |

100v100 | 2.98% | 2.84% | 2.67% | 1.15% |

200v200 | 2.22% | 2.08% | 1.91% | 1.00% |

300v300 | 1.99% | 1.87% | 1.56% | 1.14% |

400v400 | 1.71% | 1.70% | 1.35% | 1.04% |

500v500 | 1.56% | 1.57% | 1.21% | 1.00% |

600v600 | 1.50% | 1.50% | 1.11% | 1.01% |

700v700 | 1.35% | 1.39% | 1.03% | 0.90% |

800v800 | 1.35% | 1.33% | 0.96% | 0.93% |

900v900 | 1.24% | 1.26% | 0.91% | 0.86% |

1000v1000 | 1.14% | 1.23% | 0.86% | 0.81% |

And again, the chart in graphical format.

The grey line is the randomness standard deviation and it flows as expected, decreasing in a nice manner. This is a significant driver of the even and odd standard deviations but the talent standard deviation slowly falls off as well. If we call SD(Talent) the signal and SD(Randomness) as the noise then we can plot a signal to noise ratio calculated as ST(Talent) / SD(Randomness).

What is interesting is that the signal to noise ration improves significantly up to 600v600 then it sort of levels off. This is pretty much in line with what we saw earlier in the first table and chart. After 600v600 we start dropping out the majority of the fourth liners who don’t get enough ice time to be on the ice for 1400+ shots at 5v5. Later we start dropping out the 3rd liners too. The result is the signal to noise ratio flattens out.

With that said, there is probably enough information in the above charts to determine what a reasonable spread in on-ice shooting percentage talent actually is. Specifically, the yellow SD(Talent) line does give us a pretty good indication of what the spread in on-ice shooting percentage talent really is. Based on this analysis a reasonable estimate for one standard deviation in shooting percentage talent in a typical NHL season is probably around 1.0% or maybe slightly above.

What does that mean in real terms (i.e. goal production)? Well, the average NHL forward is on the ice for ~400 5v5 shots per season. Thus, a player with an average amount of ice time that shoots one standard deviation (I’ll use 1.0% as standard deviation to be conservative) above average would be on the ice for 4 extra goals due solely to their on-ice shooting percentage. Conversely an average ice time player with an on-ice shooting percentage one standard deviation below average would be on the ice for about 4 fewer goals.

Now of course if you are an elite player getting big minutes the benefit is far greater. Let’s take Sidney Crosby for example. Over the past 7 seasons his on-ice shooting percentage is about 3.33 standard deviations above average and last year he was on the ice for just over 700 shots. That equates to an extra 23 goals due to his extremely good on-ice shooting percentage. That’s pretty impressive if you think about it.

Now compare that to Scott Gomez whose 7-year shooting percentage is about 1.6 standard deviations below average. In 2010-11 he was on the ice for 667 shots for. That year his lagging shooting percentage talent an estimated 10.6 goals. Imagine, Crosby vs Gomez is a 33+ goal swing in just 5v5 offensive output.

(Yes, I am taking some liberties in those last few paragraphs with assumptions relating to luck/randomness, quality of team mates and what not so not all good or bad can necessarily be attributed to a single player or to the extent described but I think it drives the point, a single player can have a significant impact through on-ice shooting percentage talent alone).

In conclusion, even after you factor out luck and randomness, on-ice shooting percentage can player a significant role in goal production at the player level and, as I have been saying for years, must be taken into consideration in player evaluation. If you aren’t considering that a particular player might be particularly good or particularly bad at driving on-ice shooting percentage you may not be getting the full story.

(In a related post, there was an interesting article on Hockey Prospectus yesterday looking at how passing affects shooting percentage which supports some earlier findings that showed that good passers are often good at boosting teammates on-ice shooting percentage. Of course I have also shown that shots on the rush also result in higher shooting percentage so to the extent that players are good at generating rush shots they should be good at boosting their on-ice shooting percentages).

I just looked at that mimicohero article you linked to and I wonder if one of the flaws would be the choice of seasons he used in his comparisons. He picked the 4-years from 2008-11 vs the past 3 years 2011-14. However, in going over figures myself not too long ago I learned that the league-wide averages saw a shift in recent years.

There were more 5-on-5 goals per 60 from 2008-11 (2.31/2.34/2.29) than there were from 2011-14 (2.25/2.24/2.25). There were also more 5-on-5 shots per 60, although not quite by as wide a margin, from 2008-11 (29.2/29.3/29.5) than there were from 2011-14 (28.9/28.5/29.1). Conversely Sv% was lower (and as such Sh% was higher) from 2007-10 (.920/.921/.920) than it was from 2010-14 (.922/.922/.921/.923).

So when you are basing your results on comparing a period in which the league average scoring was higher (2007-11) to one in which the numbers were down on the whole (2011-14) then it stands to reason you are going to see a decline, which he attributes simply to being a “regression to the mean.” It’s not a huge difference (especially in the percentages themselves), but it could very well be enough to influence the results.

Your method of instead looking at even/odd allows a balance so that each data set has some information including the high scoring years as well as some information from the low years.

Personally I try to stick just the past 3-4 seasons in which league-wide numbers have remained more or less steady (at the time I researched it the 2014-15 numbers were in line with recent years as well), but then that does severely limit your samples size, so it probably wouldn’t be helpful to a project like this.

I’m not overly surprised to see that you found a point in which the signal and noise were at odds, as I remember reading somewhere (it may even have been from you) that after you hit 500-750 minutes you have enough data to offset the effect of noise on your results.

Anyway, thanks once again for providing us with an entertaining and informative look at the data.