The tolerance for underperforming Premiership teams is low. By comparing the teams whose manager resigned or was sacked with The Guardian's prediction for league placings with a team's final position in the league, we can see the managerial changes occured where those teams underperformed.
Or, maybe in Arsenal's case, where the tolerance for predicted mediocrity had run dry.
Sources: League predictions from The Guardian. Part of the Makeover Monday project.