It's common to look at offence and defence as two sides of the same coin. If you're not pushing the puck up the ice, then it's being pushed down into your end. Our common phrases rely on this assumption often; after all, the best defence is a good offence, so we say. The obvious geographical fact that the puck can't be in both zones at once is powerfully forms our intuitions.

Our simplest and most versatile measure for quantity (if not quality) of offence and defence is
shots--that is, blocks, misses, saves, and goals. Let's look at how individual skaters did at 5v5
in individual games this past year. First, a scatter plot:

Here, each dot represents the 5v5 results of a single home skater in a single game. We restrict to
home skaters because the results of the away skaters are, in aggregate, determined by the home
skater results and we don't want to introduce any bias. We use score-and-venue adjusted shots,
which adjusts out how home teams get more of the shots and also smoothes over some distracting
striation effects that aren't interesting today. Each dot is the same colour, the dark patches
are areas where there are lots of overlapping results. Even with the axes zoomed out this much,
a handful of points are still cut off; even more worryingly, some of these points represent
25-30 minutes of 5v5 ice time for top-pairing defenders, and some of them represent only a handful
of shifts for the likes of 13th forwards.

To account for this, I made a kernel density estimate of the data. It is much easier to look at
and not nearly as complicated as it sounds at first. Notice also that we've zoomed our axes in
a little bit.

The idea behind the graph is this: for every point in the scatter plot, imagine dropping a little
clump of sand at the indicated point. Points representing 25 minutes of 5v5 play get 25 grams of sand,
points representing 3 minutes of 5v5 play get 3 grams. As the sand hits the plot it slides off in all
directions, but the little pile is tallest at the point in question. As the thousands of piles of
sand fall on top of one another, a mountain-shaped sand-castle emerges, and we can make a contour
plot of where the sand is. The above chart shows deciles (tenths) of icetime: if you imagine slicing
the sand-mountain horizontally like a layer cake so that each layer has a tenth of the total, then
each layer gets a different shade of blue. The little piles of sand are called "kernels" because
they are little things that grow up into a full picture; and in fact they are standard gaussians for
those who want all the details. I am very fond of these kernel density pictures, so I explain this
one in some detail.

The most striking thing about this picture of single game shot results is how un-striking it
is---the point of this post is how the dog doesn't bark. If shot generation (the x axis) and
shot suppression (the y-axis) were two sides of the same coin, we'd expect to see a distribution
that tracked more tightly from "BAD" to "GOOD" without as much sand/blue colour/on-ice results
out by "DULL" or "FUN". The (weighted) correlation is -0.26, which is negative, as intuition
suggests, although still fairly small. For visual comparison, the plot below is what this
measurement *would* look like if shot generation and suppression were completely uncorrelated:

The difference, as you can compare by eye, is that there isn't quite as much really "FUN" hockey
as you would expect if shot generation and suppression were totally independent. This suggests
that the effective "hockey" upper-limit for NHL hockey is around 160 shots per sixty minutes, or about
one shot every 22.5 seconds. This is still quite a bit longer than the *physical* upper limit
of how long it takes to get the puck from one end of the ice to the other, by about a factor of
five or six, so that gives you some indication of just how much of the game is played in the
contest to get the puck up the ice, into a position where you can shoot the puck.

Incidentally, this last fact is part of why shot totals are essentially impossible to game. By
the time you've won a faceoff, recovered a lost faceoff, taken a few hits, eventually won a
sequence of puck battles,
made a set of good passes, kept structure on the breakout, moved with speed through the neutral zone,
gained the zone with control, successfully forechecked to recover the puck you dumped in, sealed
the the boards to prevent the breakout, found a teammate open in a spot worth shooting from, and got
the puck there---by the time you've done all the things, as a team, that *permit* a shot of any
kind---you've done the enormous bulk of what shot-measures measure. You can game the last few steps,
perhaps, but it won't move the needle any.