Suppose I have a population of $N\ge 1$ real numbers, all known to lie in the real interval $[a,b]$, where $a,b\in\mathbb{R}$, $a\le b$, and $a$ and $b$ are known values. I know nothing else about the population distribution.
I take a sample $x_1,\ldots,x_n$ of the population, $1\le n\le N$. What is an unbiased estimate (or otherwise "good" estimate) of the population mean? My first guess would be the sample mean $$\overline{X}=\frac{1}{n}\sum_{k=1}^nx_k,$$ but my intuition tells me that if I have just one sample point $x_1=b$, then $\overline{X}=b$ is not a very "intelligent" estimate, because it seems extremely unlikely that the entire population is equal to $b$.
The sample mean is always an unbiased estimator of the population mean. Moreover, without knowing more about the distribution I doubt you can get a better unbiased estimator.