A question on the mean and expected value $M$ is the possible mean (average) score of a class of $n$ students and $E$ is the expected score of a student. Can someone please tell me the relationship between these two?
 A: More context and definitions are needed here but I guess here what is meant by mean is the empirical mean or average. (just the sum of the $n$ scores divided by the size of the class. This is a random variable because the sample is supposedly random). The expected sore is the true mean or the theoretical mean and is a constant. The empirical mean is an estimator of the expectation (expected score) and by the law of large numbers, as the sample size goes to infinity, it should converge to the expected value (expected score).

Let $X_1,...,X_n$ be the scores. (I am assuming here they are independent and identically distributed. The identical distribution means that $V[X_i]$ is constant for all $i$. Let's write it $V[X]$. The independence implies that all $cov[X_i,X_j]=0$ for all distint pairs $i, j$. Therefore, the variance of the sample mean is given by
$$\begin{array}{lll}
  V \left[ \frac{1}{n} \sum_{i = 1}^n X_i \right] & = & \frac{1}{n^2} V \left[
  \sum_{i = 1}^n X_i \right]\\
  & = & \frac{1}{n^2} \sum_{i = 1}^n V \left[ X_i \right]\\
  & = & \frac{1}{n^2} n V \left[ X \right]
\end{array}$$
This is $V[X]/n$ as mentioned in my comment.
