From a very informative article about managerial changes in the EPL taking into account data from the past 20 years:
"the bar of expectations is set differently for top-flight clubs and for middling clubs. The five top clubs (Arsenal, Chelsea, Liverpool, Manchester City and Manchester United) tend to win, on average, 60-70% of the maximum points available in a season and their change threshold is when this figure drops to 50-60%. For the five clubs below them, the normal average in a season is about 45% and the change threshold is around 40%."
Also that new manager bounce is very much a fact and usually doesn't mean that the manager was a genius for the first ten games and then became dross:
"There’s another factor at a relative high: the ‘bounce’, or the improvement in a team’s results following a new appointment. We measured the average points per game recorded by a team in the first 10 matches under a new manager and compared it with the last 10 under the previous manager. In the latest five-year block, nearly 85% of new appointments resulted in a bounce."
And then that bounce doesn't last unless the manager is good enough:
"Bounce is one thing. Maintaining that bounce, or ‘persistence’, is quite another. In 54 of the 74 changes in the 20-year period, the first 10 matches under a new manager saw a bounce in team performance. But in only 21 of these 54 instances was this bounce sustained or improved over the next 20 games. Thus, there are many managers who either experience a non-persistent bounce or no bounce at all but only a few can sustain a bounce."
A great example of quality versus plain bouncy luck:
"stellar example of this bounce and persistence is Antonio Conte at Chelsea in 2016-17, his first season there. Under him, Chelsea added an average of 1.1 points in his first 10 games (compared to the 10 games pre-Conte), and he built on this by adding another 0.25 points in the next 20 games and won the league. Conversely, there’s Ole Gunnar Solskjaer, who added an average of 1 point in his first 10 games with Manchester United but shed 0.6 point in the next 20. Top clubs don’t have time for that."
Read the last line of the above paragraph for added spice.
Takeaways:
1. We are way below the threshold performance level which leads to a manager getting sacked in the PL.
2. Ole's bounce stats should be read separately from the full time post-bounce stats.