What is the proper way to estimate the CDF for a distribution from samples taken from that distribution? 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.
 A: To get error bars, you can construct a confidence interval around the entire empirical cumulative distribution function (ECDF). This can be done using the Dvoretzky-Kiefer-Wolfowitz inequality. If you want the ECDF to be within $\epsilon$ of the true CDF with confidence $1- \alpha,$ then choose the sample size $n$ using $$n \ge \left( {{1} \over {2 \epsilon^2}} \right) \mathrm{ln} \left({{2} \over {\alpha}} \right)$$
So, for example, if you want the ECDF to be within $0.01$ of the CDF with 95% confidence, we find by plugging in that $$n \ge 18444.4$$ so we select $n=18445.$
A: You could always use a kernel density estimator (which would also give the c.d.f. as the weighted sum of the component c.d.f.s).  You could then get error bars by bootstrapping the available data.  This would be pretty simple to implement and would give nice, well-behaved smooth c.d.f.s with error bars.
A: In a bayesian approach, you could use a Dirichlet Process (DP) to estimate the PDF and then integrate it. What you are trying to do is to estimate the function based on samples at certain values. The DP approach allows you to incorporate a smoothness assumption, which is useful because you will often prefer a solution that is differentiable than one that looks like a staircase. The outcome of your analysis is then a distribution over functions, which in particular gives you a mean function, and some error bars on it.
The following book has a nice chapter on Dirichlet processes:
O'Hagan, A. and Forster, J. J. (2004). Bayesian Inference, 2nd edition, volume 2B of "Kendall's Advanced Theory of Statistics". Arnold, London. 
A: In statistics there really is no concept of a 'right' estimate, it's just if the estimate you construct has the properties that you are looking for.
Typically if you are trying to estimate a CDF, you will use the ECDF (Empirical CDF) which is just $ Pr(X < x) = \Sigma_{i=1}^n \mathbb{I}_{x_{(i)} \le x}(x) n^{-1}$.  Where $X_{(i)}$ is the $i$th order statistic.
The ECDF has many nice properties such as being strongly consistent (pointwise even) to the CDF.
Since you have a discrete approximation of a continuous distribution you can generate quantiles that can be used for confidence intervals in the usual discrete way.  
$inf_x( x : Pr(X <x) \ge \pi) $
Of course there is no reason why a confidence interval should be symmetric so I'm confused by your last statement that I think should be clarified.
