I would bet you 50BTC (engaged through an escrow) that the analysis of ALL my hands played through to showdown on SWC shows abnormal distribution.
Well, OF COURSE that selection of hands would show an abnormal distribution.
Any sampling that is a result of choice, is going to be abnormal.
All those hands that you chose not to run to the river and show would be missing from the sample, so any kind of analysis for finding a normal distribution is impossible.
The fact that you would even offer that as the term of the bet, just tells me you don't have a clue about how random sampling for statistical analysis even works, for determining a normal distribution of a random number generator, and the cards that are being dealt.
You need every card dealt, for every hand, for every table, for a crapload of rounds that were played. Not just the ones that you thought would have been good enough to win, regardless of the other players' choices to hold or fold.
-- Smoov
Actually you would see skew no matter what if you look at board cards. It would be relatively small and I can't guess the direction without more thought.
In some games a flop is almost always seen. In some games a preflop raise often takes the pot, in those games there will tend to only be a flop when multiple players have higher than average ranks in their hands and you will see lower than average cards on the board. There will also be some flops that tend to make the hand end early and deal no rivers. Without looking at any data I would guess than flushes come in a little bit less often than simple analysis would predict because the turn and river are more likely to be dealt if two or more players are on a flush draw, partially blocking each other.
edit: If you just looked at a tally of every card that ever appeared that would be correctly distributed. It's just if you look at "river cards" or similar specifically that there will be skew.