Given $n$ samples from a (continuous) distribution X, the obvious thing to do is sort them, and distribute them equally across $[0,1]$ by taking $(x_{(k)}, (k-1/2)/n)$ as estimates of particular points on the CDF, and doing some sort of interpolation between points, as necessary.
Is this the "right" way to make this estimate? How do I get error bars for the estimated points? It doesn't seem like they'd necessarily be symmetric.