# Did Arsenal deserve to win the 1988-89 First Division?

Yesterday was the 22nd anniversary of the most dramatic league decider in the history of English football, and perhaps the most dramatic match of its kind anywhere: Liverpool vs Arsenal in 1989.  Arsenal had led for most of the season until the closing weeks when Liverpool charged past them into first place with a match to play.  The match was originally to be played on 23 April but rescheduled because of the Hillsborough disaster, so it became the final match of the First Division season.  Arsenal needed to win by two clear goals to win the First Division, and they did it — with a goal in the final minute of the game by Michael Thomas.

But were Arsenal deserved winners of the Football League that season?  I developed a Pythagorean table for that season to find out.

Before we start, let's look at the Pythagorean exponents for the First Division teams from 1988-89.  I fitted the three-parameter Weibull distribution simultaneously to the offensive and defensive goal distributions.  This post describes the procedure in more detail; it's a least-squares approach.  The relevant parameter is the gamma parameter which describes the skew of the distribution and serves as the Pythagorean exponent.  Below is a table of the teams and their Pythagorean exponents:

 Club Exponent Arsenal 1.83 Aston Villa 2.08 Charlton Athletic 1.39 Coventry City 1.76 Derby County 1.67 Everton 1.85 Liverpool 1.82 Luton Town 1.40 Manchester United 1.57 Middlesbrough 1.33 Millwall 1.52 Newcastle United 1.39 Norwich City 1.84 Nottingham Forest 1.69 Queens Park Rangers 1.39 Sheffield Wednesday 1.85 Southampton 2.07 Tottenham Hotspur 2.05 West Ham United 1.83 Wimbledon 1.89 Average 1.71 Standard Deviation 0.24

The league Pythagorean exponent is 1.71 ± 0.24, which is right in line with the league Pythagorean exponent of 1.70 that I've been using.  So even going back twenty years that league exponent still holds well.

So let's use that value to generate the Pythagorean table.  Below is the table for the 1988-89 First Division.

League Table Pythagorean
Team GP W D L GF GA GD Pts W D L Pts Δ
Arsenal 38 22 10 6 73 36 +37 76 22 9 7 75 +1
Liverpool 38 22 10 6 65 28 +37 76 22 9 7 75 +1
Nottingham Forest 38 17 13 8 64 43 +21 64 19 9 10 66 -2
Norwich City 38 17 11 10 48 45 +3 62 14 11 13 53 +9
Derby County 38 17 7 14 40 38 +2 58 14 12 12 54 +4
Tottenham Hotspur 38 15 12 11 60 46 +14 57 17 10 11 61 -4
Coventry City 38 14 13 11 47 42 +5 55 15 11 12 56 -1
Everton 38 14 12 12 50 45 +5 54 15 10 13 55 -1
Queens Park Rangers 38 14 11 13 43 37 +6 53 15 12 11 57 -4
Millwall 38 14 11 13 47 52 -5 53 13 10 15 49 +4
Manchester United 38 13 12 13 45 35 +10 51 16 11 11 59 -8
Wimbledon 38 14 9 15 50 46 +4 51 15 10 13 55 -4
Southampton 38 10 15 13 52 66 -14 45 12 9 17 45 +0
Charlton Athletic 38 10 12 16 44 58 -14 42 11 10 17 43 -1
Sheffield Wednesday 38 10 12 16 34 51 -17 42 9 11 18 38 +4
Luton Town 38 10 11 17 42 52 -10 41 12 10 16 46 -5
Aston Villa 38 9 13 16 45 56 -11 40 12 10 16 46 -6
Middlesbrough 38 9 12 17 44 61 -17 39 11 10 17 43 -4
West Ham United 38 10 8 20 37 62 -25 38 9 10 19 37 +1
Newcastle United 38 7 10 21 32 63 -31 31 7 10 21 31 +0

This is such an amazing table for so many reasons.  Even without the Pythagorean expectation, it captures the thin margin between first and second place in the league and shows just how unrecognizable the football world is between 1989 and today.

First of all, the most overachieving side in the First Division was Norwich City with a Pythagorean residual of +9.  Their defensive record should have been enough to make them a mid-table side, but they managed to win enough close matches to finish in the top six.  Next, the most underachieving side in the top flight was Manchester United with a Pythagorean residual of -8.  They were good enough to finish in the top four but ended up in mid-table in Alex Ferguson's second season with the club.  If Manchester United had fired him at the time, the club directors would have been justified by the results, and the history of English football and the Premier League would have been very different today.  It's something to consider as Manchester United go for a fourth European Cup tomorrow.

Finally, look at Arsenal and Liverpool.  Not only did both clubs finish with identical Pythagorean expectations, they also finished with identical expected records!  Both teams played almost in line with their expectations over the course of the season, but mid-season and 3/4-season tables would have been very interesting (Arsenal had at least a 10-point lead at the start of 1989 and faltered down the stretch, while Liverpool were undefeated from New Year's Day until the final match of the season).

So did Arsenal deserve to win the Football League in 1988-89?  As the Pythagorean table from that season showed, the title truly was up for grabs.