  For any physical observable in statistical systems, the most frequently
studied quantities are its average and standard deviation. Yet, its full
distribution often carries extremely interesting information and can be invoked
to put any surprising properties of the individual moments into perspective. As
an example, we consider a problem concerning simple random walks which was
posed in a recent text. When a drunk is observed over L nights, taking N steps
per night, and the number of steps to the right is recorded for each night, an
average and a variance based on these data can be computed. When the variance
is used to estimate p, the probability for the drunk to step right, complex
values for p are frequently found. To put such obviously nonsensical results
into context, we study the full probability distribution for the variance of
the data string. We discuss the connection of our results to the problem of
data binning and provide two other brief examples to demonstrate the importance
of full distributions.
