Excuse my naiveness about this, but it seems most arbitrations seem like they just find the middle ground.
Is there an example where the arbitrator takes the side noticeably closer to one side (team) than the other (player) or vice versa?
Like let's say one side brings in 5 and the other says three. It feels like the arbitrators (at least from what I've been reading typically tend to pick 4, give or take small margin of difference)? Or is this just a complete oversimplification, but same result?
That's the point I've been making.
The vast majority of cases never make it to arbitration. Those that do, the team and the player are coming off of a long negotiation process so they know very well where the other side stands. To use your numbers, if both sides begrudgingly know it's going to come in around $4 million based in how negotiations have gone, the team will come in with a lowball offer of $3 million and the player $5 million knowing full well the arbitrator is coming in somewhere in the middle.
To answer your question about an arbitrator leaning more to one side or the other: Off the top of my head Shea Weber comes to mind. In 2011, Nashville offered $4.75 million while his agent was seeking $8.5 million. The arbitrator awarded $7.5 million. In 2006, the Devils went in with $3.5 million for Scott Gomez while the player was asking $6.5 million. The arbitrator came in at $5 million.
One of the weirdest was PK Subban and the Canadiens. The team had offered $5.25 million and the player was asking $8.5 million. They went to arbitration where both sides admitted it didn't go well for them but before the arbitrator handed down the award they agreed to a 8-year $9 million per year deal. Unlike today, in those days you could negotiate up until the arbitrator makes their decision.
Those were a long time ago in hockey terms.