So I recently got back an exam score and I found myself wondering if I got the lowest score in the class…this led me to formulate an interesting problem:
Suppose an exam is scored from $0$ to $100$. What is the lowest possible score in the class given the number of students $N$, the class average $\mu$, standard deviation $\sigma$ and your grade $x$? What is the highest possible grade?
Obviously for large enough $N$, the lowest possible grade is $0$ and the highest is $100$, but what are the limits for general $N$?
Best Answer
If we call the test scores $y_i$, then we'll begin by noting that in order to find the lowest (or highest) score, that $y_1$ can be our lowest (or highest) score, and $y_2=y_3=...=y_n$ must be all the other scores. Since we know that
$$\frac{1}{n}(y_1+\sum_2^n y_i) = \mu$$
and
$$\frac{1}{n}(y_1^2+\sum_2^ny_i^2)-\mu^2=\sigma^2$$
we can replace our sums to get $\frac{1}{n}(y_1+(n-1)y_2)=\mu$ and $\frac{1}{n}(y_1^2+(n-1)y_2^2)-\mu^2=\sigma^2$. Since we've now got two equations in two variables, we can just solve to find the values for $y_1$ and $y_2$, to get (skipping the intermediary math):
$$y_1=\mu\mp\sigma\sqrt{n-1}$$
and
$$y_2=\mu\pm\frac{\sigma}{\sqrt{n-1}}$$
to get either the minimum score possible, or the maximum score possible.
Edit: To clarify, this is the lowest score possible given $\mu$ and $\sigma$ for the population. It does not necessarily mean that it was the actual lowest score (and in fact it's highly unlikely that it was the lowest score). It just means that given those two parameters, it's not possible for anyone to achieve a lower score.