I am going to have to bite here. Yes, Allardyce was sacked in 11th place, and we finished 12th.
What that doesn't consider however was that game by game the team had overall picked up worse results than the corresponding fixtures the previous season (for instance we won away at Blackburn the previous season, but lost under Allardyce), and that previous season we had finished 5 points clear of relegation. Immediately after Allardyce was sacked, we had to play 7 games against the top 7 broken up by just one game in the FA Cup and one game against Bolton. We were already in a bad run and we deserved to lose as opposed to being unlucky. Morale was low, and their was a player mutiny started by less rotten players.
Keegan can't work miracles, and that horrific run continued, but he never lost to any team outside the top 7 (and you may or may not agree that you guys would have been lucky with a draw, let alone a win at SJP). However, once the tricky games were over, Keegan brought back confidence by getting the players to play football and attack the opposition. Allardyce would have shut up shop just like he tried to play for a goalless draw at Derby and lost. That would have done nothing for the players, and we could have quite easily got relegated.