It wasn't the level of st deviation I was wondering about, but whether the st. dev is different between different players, and if so, does it tend to correlate with the average or not.
But... that's neat to hear they've done stuff on basketball.
I just wonder if sometimes consistency is a narrative that sometimes gets forced by media types more times than it exists.
Yup, the basketball research showed that within, for example, a group of players that average 20 (+/- 1) points a game, there is wide distribution of standard deviations for game by game point total and FG%. There is also considerable evidence that players with smaller sigma are 'better' (their wins over replacement are much higher).
I mean, it is a bit chicken and egg to be honest though, guys with large sigma tend to be volume shooters that most good teams wouldn't even play.
In hockey it would be nearly impossible to quantify such things because the typical output statistics are sparse. Although it may work for shots per game. You know a fair bit about advanced statistics, has anyone ever looked at the second moment of players' shots per game distribution, for example?
Regarding the narrative question I cannot say anything about hockey but I can give an example from baseball. Using pitch tracking and classification some people have been able to show that players that have a significant weakness against certain pitches and locations tend to experience larger fluctuations in (for example) batting average on different time scales (5, 10, 20 games etc.). Basically in quantitative finance lingo, players with greater strengths and weaknesses have a higher volatility.
Obviously this is very difficult to map on to hockey, but there does exist some evidence that in competitive systems the mean is relevant in addition to the mean.