One thing we can look at with z-scores is the improvement of teams year on year relative to the average team. I wanted to see which teams improved/declined from last season to this season, so I looked at the difference in the z-scores of points between last season and this season (this obviously excludes the teams who were promoted this season, as they were not in the league last year). The results are in the table below. Positive scores designate improvement over the past year, negative scores designate decline.
Man City +.72
Man United +.47
West Ham -.55
Aston Villa -.92
The top of the list is pretty straightforward. Since I spent a lot of time effusively praising United in my earlier post, it is no surprise to see them high on this list. It is also not surprising to see my beloved Manchester City up at the top, as the club spent a lot of money during the offseason to improve the team. Bolton also makes sense as they were poor last season and over-performed this season. Fulham is a bit unexpected, seeing as how they were quite good last year (particularly in Europe) under Hodgson.
The bottom of the list seems to make sense as well. Villa were top 8 last year, but tumbled down the table this year. Birmingham and West Ham were relegated and Blackburn struggled after firing Sam Allardyce.
Another interesting point is that if you add all of the above changes up, you get a sum of -2.30, meaning these 17 teams declined on average. Obviously, if we include all 20 teams the mean should be zero. This indicates that the three promoted teams from this year (Newcastle, West Brom, and Blackpool) had higher z-scores than the relegated teams from the year before (Burnley, Hull City, Portsmouth). Having some very good promoted teams clearly made it more difficult for teams to improve relative to the average team.