But taking the whole population doesn't obviate sample size issues. If I flipped a coin three times, and it came up heads every time, then you could say you're not sampling that the coins has a 100% chance of coming up heads; you're measuring the entire population!
But that doesn't mean the coin can only come up heads. It just happened to in your small sample size (even if it's the size of the entire population). If you bet your life savings that the next throw of the coin was heads… oi.
So it's not that you're sampling a small chunk of Fryer's matches and leaving the rest unaccounted for. It's that Fryers simply hasn't done enough (the population from which to draw data isn't big enough) for us to tell what his effect actually is. It takes many tosses of a coin (MoE = 1/sqrt(n)) to determine that it's 50/50 heads/tails, not just three (or four!). Similarly, Fryers has to make many more appearances before we can tell what role he plays in terms of influenced match outcomes.
I can't precisely tell what you mean when you say (paraphrasing) "but the statistics show!" Statistics never
show. They
suggest, since it's impossible to account for all the variables, etc. That's why it's called statistical
inference and not statistical
proof.
While admiring what you've been up to here, I have to say that it remains the case that Fryer's record is inflated by strength of opposition (4? of 10? matches are Europa League group stage), for example. I can't tell if you're seriously suggesting that if we started Fryers we would be at in a better position to win any given match. Because that's not a sample size problem (which
VirginiaSpur
already highlighted), that's a "correlation does not equal causation" problem. Just because we've had a good record under Fryers doesn't mean that it's
because of Fryers that we've had the good record. Teasing out Fryers's specific contribution to that record is far more complicated, despite what 100% of the statistics show.
Your 100% accounts for the matches that have
already happened, and have little or no predictive value for the future. Returning to the coin tosses from the top, you can say with 100% certainty that in those three tosses, it will come up heads every time, but that doesn't prepare you at all for talking about what the fourth throw will yield.