By now, we all know what happened. Chelsea were attempting to extend a narrow, Didier Drogba-powered lead over Tottenham Hotspur in the FA Cup when they were handed a completely illegitimate goal by referee Martin Atkinson in the 49th minute. Spurs responded quickly, pegging the Blues back to 2-1, but ultimately couldn't complete the comeback and ended up at the wrong end of a 5-1 thrashing.
The reaction to this is pretty predictable. Spurs fans and those who wanted Chelsea to lose are saying it was a travesty (which it was) that changed the course of the game, while everyone else is pointing to the final score as evidence that one goal didn't matter that much. Every argument here goes into hypotheticals - we don't know what would have happened had the goal not been given, and so any subjective discussion is incredibly pointless.
So let's not do this subjectively. As humans, we're blessed with the ability to solve problems like this, so we can at least get an estimation of how much Atkinson's call changed the game. Time to use the power of statistics!
Y'all should be fairly familiar with the concept of win probability - it's a simple statistical tool that you can use to determine how likely a team is to win, draw or lose in a given moment because on score, yellow cards, venue, etc. I've used it extensively here, and it matches up to historical records impressively well.
Some modifications to make it apply to cup games (where you can't draw) are in order, but they're pretty trivial to make, and once those are done we have the ability to figure out how much an individual goal actually matters. Here's what we end up with, assuming Wembley is a neutral ground:
Spurs, then, were cheated out of 14.3 percent of a win by Martin Atkinson, and a football match determined the other 85 percent of the game. But... what if some of the assumptions in the model are wrong? That's a reasonable suggestion, and one I'm open to. So l played around with the parameters a little bit. The first assumes that Spurs are the home side:
17.4 percent. That's a little bit more. What if we go crazy, though, and say that Tottenham are literally twice as good as Chelsea, capable of scoring two goals for every one that the Blues notch?
20.8 percent. That's not helping much. In the most favourable possible scenario to Spurs, at most Atkinson stole one fifth of a win. Obviously, that's unfortunate, and I think everyone wishes that the match could have been refereed competently, but the second goal didn't decide the match, and most of the scenarios at 1-0 have Chelsea winning as well.
Sorry, but the phantom goal just is not as big a deal as it's being made out to be. Any argument otherwise is speculative to the point of being worthless. We've established the order of magnitude of the effect, so feel free to argue within that area, but outside it, you're being silly.