Given a sample of random variables, and n, how do I find the ecdf of the sum of n Xs? I can't fit $X$ to a common distribution so currently I just have $X \sim \text{ecdf(sample_data)}$.
How do I calculate the empirical distribution of $\sum(X_1 + ... + X_n)$, given $n$? $X_1$ to $X_n$ are iid.
My sample size is around $6\times10^7$. $n$ needs to be around 10-20.
To give an example of what I am doing: Suppose $X$ is the age of a person. If I put 10-20 people in a room, what is the distribution of the sum of ages in the room? I have lots of peoples ages, and no reason to fit the distribution of an age to any of the common distributions.
(previously on StackOverflow - suggested to move here).
 A: I will interpret your question to be: How can we estimate the sampling distribution for the sum (or equivalently, the mean) from a iid sample $X_1, \dotsc, X_n$ in a nonparametric way, without assuming anything about the common distribution of the $X_i$?
One approach would be bootstrapping, see Explaining to laypeople why bootstrapping works 
Another (related) approach is using the saddlepoint approximation, there is an example in How does saddlepoint approximation work?
(these methods can also be used if the goal is sum/mean of a future iid sample of size $m$ possibly different from $n$, from the same population).  The saddlepoint approximation can be seen as approximating the bootstrap solution, but without simulation.
EDIT after you clarified the Q in an edit

You have a very big sample, and want to estimate the cdf for subsampling a (much smaller) sample from that large sample. The ecdf from the big sample, as in the answer above, is still the right answer, assuming you are subsampling by simple random sampling (with or without replacement). If you have some weighted or stratified sample, the answer will be different.
