What Implications Can be Drawn from a Binomial Distribution? Hello everyone I understand how to calculate a binomial distribution or how to identify when it has occurred in a data set. My question is what does it imply when this type of distribution occurs?
Lets say for example you are a student in a physics class and the professor states that the distribution of grades on the first exam throughout all sections was a binomial distribution. With typical class averages of around 40 to 50 percent. How would you interpret that statement?


Lets say for example you are a student in a physics class and the professor states that the distribution of grades on the first exam throughout all sections was a binomial distribution. With typical class averages of around 40 to 50 percent. How would you interpret that statement?

Most likely the professor was talking loosely and his statement
means that the histogram of percentage scores resembled the 
bell-shaped curve of a normal density function with average or 
mean value of $40\%$ to $50\%$.  Let us assume for convenience
that the professor said the average was exactly $50\%$.  The 
standard deviation of scores would have to be at most $16\%$
or so to ensure that only a truly exceptional over-achiever 
would have scored more than $100\%$.

As an aside, in the US, raw scores on the GRE and SAT are
  processed through a (possibly nonlinear) transformation so
  that the histogram of reported scores is roughly bell-shaped
  with mean $500$ and standard deviation $100$.  The highest
  reported score is $800$, and the smallest $200$.  As the saying
  goes, you get $200$ for filling in your name on the answer sheet.
  At the high end, on the Quantitative GRE, a score of $800$ ranks
  only at the $97$-th percentile.

What if the professor had said that there were no scores that were
a fraction of a percentage point, and that the histogram of 
percentage scores matched a binomial distribution with mean $50$
exactly?  Well, the possible percentage scores are $0\%$, 
$1\%, \ldots, 100\%$ and so the binomial distribution in question
has parameters $(100, \frac{1}{2})$ with $P\{X = k\} = \binom{100}{k}/2^{100}$.
So, if $N$ denotes the number of students in the course, then
for each $k, 0 \leq k \leq 100$, $N\cdot P\{X = k\}$ students had a percentage score of $k\%$.
Since $N\cdot P\{X = k\}$ must be an integer, and 
$P\{X = 0\} = 1/2^{100}$, we conclude that $N$ is an integer 
multiple of $2^{100}$.  I am aware that physics classes are often 
large these days,
but having $2^{100}$ in one class, even if it is subdivided
into sections, seems beyond the bounds of 
plausibility!  So I would say that your professor had his tongue
firmly embedded in his cheek when he made the statement.
