What is your favorite layman's explanation for a difficult statistical concept? I really enjoy hearing simple explanations to complex problems. What is your favorite analogy or anecdote that explains a difficult statistical concept?
My favorite is Murray's explanation of cointegration using a drunkard and her dog. Murray explains how two random processes (a wandering drunk and her dog, Oliver) can have unit roots but still be related (cointegrated) since their joint first differences are stationary.

The drunk sets out from the bar, about to wander aimlessly in random-walk fashion. But
  periodically she intones "Oliver, where are you?", and Oliver interrupts his aimless
  wandering to bark. He hears her; she hears him. He thinks, "Oh, I can't let her get too far
  off; she'll lock me out." She thinks, "Oh, I can't let him get too far off; he'll wake
  me up in the middle of the night with his barking." Each assesses how far
  away the other is and moves to partially close that gap.

 A: Sam Savage's book Flaw of Averages is filled with good layman explanations of statistical concepts.  In particular, he has a good explanation of Jensen's inequality.  If the graph of your return on an investment is convex, i.e. it "smiles at you", then randomness is in your favor: your average return is greater than your return at the average.
A: Along the lines of the mean as balance point, I like this view of the median as a balance point:


*

*A Pearl: a Balanced Median Necklace
A: Behar et al have a collection of 25 analogies for teaching statistics.
Here are two examples:

2.9 All Models are Theoretical:
  There Are No Perfect Spheres in the Universe It appears that the most common geometric form in the
  universe is the sphere. But how many mathematically perfect spheres
  are there in the universe? The answer is none. Neither the Earth, nor
  the Sun, nor a billiard ball is a perfect sphere. So, if there are no
  true spheres, what good are the formulas for ascertaining the area or
  volume of a sphere? So it is with statistical models in general and,
  in particular, with a normal distribution. Although one of the most
  commonplace examples is height distribution, if we were to have at our
  disposal the height of every adult on the planet, the histogram proﬁle
  would not correspond to a Gaussian bell curve, not even if the data
  were stratiﬁed by gender, race, or any other characteristic. But the
  normal distribution model still provides approximate results that are
  good enough for practical purposes.
2.25 Residuals Should Not Contain Information: A Trash Bag Residuals are what remain after removing all the information from the data.
  Since they should carry no information, we consider them as “trash.”
  It is necessary to make sure that we do not throw out any trash that
  has value (information) and that can be exploited to better explain
  the behavior of the dependent variable.

Other examples include


*

*"Effect of Sample Size on the Comparison of Treatments: Magniﬁcation of Binoculars"

*"The Sample Size Versus the Size of the Population: A Spoon for Tasting the Soup"


References


*

*Behar, R., Grima, P., & Marco-Almagro, L. (2012). Twenty Five Analogies for Explaining Statistical Concepts. The American Statistician, (just-accepted).

A: Fun question.
Someone found out I work in biostatistics, and they asked me (basically) "Isn't statistics just a way of lying?"
(Which brings back the Mark Twain quote about Lies, Damn Lies, and Statistics.)
I tried to explain that statistics allows us to say with 100 percent precision that, given assumptions, and given data, that the probability of such-and-so was exactly such-and-such.
She wasn't impressed.
A: A p value is a measure of how embarrassing the data are to the null hypothesis
Nicholas Maxwell, Data Matters: Conceptual Statistics for a Random World Emeryville CA: Key College Publishing, 2004.
A: *

*If you carved your distribution (histogram) out
of wood, and tried to balance it on
your finger, the balance point would
be the mean, no matter the shape of the distribution.

*If you put a stick in the middle of
your scatter plot, and attached the
stick to each data point with a
spring, the resting point of the
stick would be your regression line. [1]
[1] this would technically be principal components regression. you would have to force the springs to move only "vertically" to be least squares, but the example is illustrative either way. 
A: I have used the drunkard's walk before for random walk, and the drunk and her dog for cointegration; they're very helpful (partially because they're amusing).
One of my favorite common examples is the Birthday Paradox (wikipedia entry), which illustrates some important concepts of probability.  You can simulate this with a room full of people.
Incidentally, I strongly recommend Andrew Gelman's "Teaching Statistics: A Bag of Tricks" for some examples of creative ways to teach statistical concepts (see the table of contents).  Also look at his paper about the course that he teaches on teaching statistics: "A Course on Teaching Statistics at the University Level".  And on "Teaching Bayes to Graduate Students in Political Science, Sociology,
Public Health, Education, Economics, ...".
For describing Bayesian methods, using an unfair coin and flipping it multiple times is a pretty common/effective approach.
A: I like to demonstrate sampling variation and essentially the Central Limit Theorem through an "in-class" exercise. Everybody in the class of say 100 students writes their age on a piece of paper. All pieces of paper are the same size and folded in the same fashion after I've calculated the average. This is the population and I calculate the average age. Then each student randomly selects 10 pieces of paper, writes down the ages and returns them to the bag. (S)he calculates the mean and passes the bag along to the next student. Eventually we have 100 samples of 10 students each estimating the population mean which we can describe through a histogram and some descriptive statistics. 
We then repeat the demonstration this time using a set of 100 "opinions" that replicate some Yes/No question from recent polls e.g. If the (British General) election were called tomorrow would you consider voting for the British National Party. Students them sample 10 of these opinions.
At the end we've demonstrated sampling variation, the Central Limit Theorem, etc with both continuous and binary data.
A: Definitely the Monty Hall Problem. http://en.wikipedia.org/wiki/Monty_Hall_problem
A: 1) A good demonstration of how "random" needs to be defined in order to work out probability of certain events:
What is the chance that a random line drawn across a circle will be longer than the radius?
The question totally depends how you draw your line. Possibilities which you can describe in a real-world way for a circle drawn on the ground might include:
Draw two random points inside the circle and draw a line through those. (See where two flies / stones fall...)
Choose a fixed point on the circumference, then a random one elsewhere in the circle and join those. (In effect this is laying a stick across the circle at a variable angle through a given point and a random one e.g. where a stone falls.)
Draw a diameter. Randomly choose a point along it and draw a perpendicular through that. (Roll a stick along in a straight line so it rests across the circle.)
It is relatively easy to show someone who can do some geometry (but not necessarily stats) the answer to the question can vary quite widely (from about 2/3 to about 0.866 or so).
2) A reverse-engineered coin-toss: toss it (say) ten times and write down the result. Work out the probability of this exact sequence $\left(\frac{1}{2^{10}}\right)$. A tiny chance, but you just saw it happen with your own eyes!... Every sequence might come up, including ten heads in a row, but it is hard for lay people to get their head round it. As an encore, try to convince them they have just as good a chance of winning the lottery with the numbers 1 through 6 as any other combination.
3) Explaining why medical diagnosis may seem really flawed. A test for disease foo which is 99.9% accurate at identifying those who have it but .1% false-positively diagnoses those who don't really have it may seem to be wrong really so often when the prevalence of the disease is really low (e.g. 1 in 1000) but many patients are tested for it. 
This is one that is best explained with real numbers - imagine 1 million people are tested, so 1000 have the disease, 999 are correctly identified, but 0.1% of 999,000 is 999 who are told they have it but don't. So half those who are told they have it actually do not, despite the high level of accuracy (99.9%) and low level of false positives (0.1%). A second (ideally different) test will then separate these groups out. 
[Incidentally, I chose the numbers because they are easy to work with, of course they do not have to add up to 100% as the accuracy / false positive rates are independent factors in the test.]
