What is the hardest statistical concept to grasp? This is a similar question to the one here, but different enough I think to be worthwhile asking.
I thought I'd put as a starter, what I think one of the hardest to grasp is.  
Mine is the difference between probability and frequency.  One is at the level of "knowledge of reality" (probability), while the other is at the level "reality itself" (frequency).  This almost always makes me confused if I think about it too much.
Edwin Jaynes Coined a term called the "mind projection fallacy" to describe getting these things mixed up.
Any thoughts on any other tough concepts to grasp?
 A: I think that very few scientists understand this basic point: It is only possible to interpret results of statistical analyses at face value, if every step was planned in advance. Specifically:


*

*Sample size has to be picked in advance. It is not ok to keep analyzing the data as more subjects are added, stopping when the results looks good. 

*Any methods used to normalize the data or exclude outliers must also be decided in advance. It isn't ok to analyze various subsets of the data until you find results you like.

*And finally, of course, the statistical methods must be decided in advance. Is it not ok to analyze the data via parametric and nonparametric methods, and pick the results you like. 


Exploratory methods can be useful to, well, explore. But then you can't turn around and run regular statistical tests and interpret the results in the usual way.
A: Tongue firmly in cheek: For frequentists, the Bayesian concept of probability; for Bayesians, the frequentist concept of probability.  ;o)
Both have merit of course, but it can be very difficult to understand why one framework is interesting/useful/valid if your grasp of the other is too firm.  Cross-validated is a good remedy as asking questions and listening to answers is a good way to learn.
A: From my personal experience the concept of likelihood can also cause quite a lot of stir, especially for non-statisticians. As wikipedia says, it is very often mixed up with the concept of probability, which is not exactly correct.
A: Fiducial inference. Even Fisher admitted he didn't understand what it does, and he invented it.
A: What do the different distributions really represent, besides than how they are used.
A: I think the question is interpretable in two ways, which will give very different answers:
1) For people studying statistics, particularly at a relatively advanced level, what is the hardest concept to grasp?  
2) Which statistical concept is misunderstood by the most people?  
For 1) I don't know the answer at all.  Something from measure theory, maybe?  Some type of integration?  I don't know.
For 2) p-value, hands down.
A: Confidence interval in non-Bayesian tradition is a difficult one. 
A: I think people miss the boat on pretty much everything the first time around.  I think what most students don't understand is that they're usually estimating parameters based on samples.  They don't know the difference between a sample statistic and a population parameter.  If you beat these ideas into their head, the other stuff should follow a little bit easier.  I'm sure most students don't understand the crux of the CLT either.
A: for some reason, people have difficulty grasping what a p-value really is.
A: Similar to shabbychef's answer, it is difficult to understand the meaning of a confidence interval in frequentist statistics. I think the biggest obstacle is that a confidence interval doesn't answer the question that we would like to answer. We'd like to know, "what's the chance that the true value is inside this particular interval?" Instead, we can only answer, "what's the chance that a randomly chosen interval created in this way contains the true parameter?" The latter is obviously less satisfying.
A: What is the meaning of "degrees of freedom"? How about df that are not whole numbers?
A: Conditional probability probably leads to most mistakes in everyday experience. There are many harder concepts to grasp, of course, but people usually don't have to worry about them--this one they can't get away from & is a source of rampant misadventure. 
