I believe it was Iain Fyffe who did a statistical analysis conclusively demonstrating that the ‘new coach bump’ is a myth. Coaches get fired when their teams are at rock bottom and have nowhere to go but up. As soon as the team comes out of the slump and strings a few wins together, people look to the coaching change as an explanation. But it happens just as often when a team goes through a losing streak and doesn’t fire the coach – as the analysis showed.
Unfortunately, if it was indeed Iain Fyffe who did that study, it's no longer available online, as his site was taken down some years ago.
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