Let's say we are taking $n$ samples from a uniform distribution, that spans from $0$ to $1$. According to the central limit theorem, the mean of the $n$ samples will follow a normal distribution with the mean at $0.5$.

What I want to figure out is how to get the distribution of the mean of those $n$ samples, without resorting to a central limit theorem; for example the central limit theorem can't be used when $n=2$.

I fruitlessly tried to figure this out on my own. I started with $n=2$, thinking that if I can figure it for this $n$, I will be able to generalize to larger $n$. With sample mean being $S = \frac{X_{1} + X_{2}}{2}$, I realized that the hardest part is figuring out how to get the sum in $S$. So, instead of trying to figure out what distribution $S$ will follow, I decided to find the distribution of $Z=X_{1} + X_{2}$.

Next, I started thinking that Z could be imagined as a 3D space, with $x_{1}$ along one axis, $x_{2}$ along the other, and the value of the distribution being uniformly at $1$.

But what I need is a distribution of $Z$, which will be proportionate to the number of ways a certain $z$ can be attained from a sum of $x_1$ and $x_2$. There are less ways to get $x_1+x_2=0.01$, than there are for $x_1+x_2=1$.

This is were I got stumped.

PS. I'm looking for an answer generalizable to any $n$, not just $n=2$