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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).

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    $\begingroup$ The term empirical distribution refers to the sample distribution for the discrete set of n observations. If you have n observed values Xi i=1,...n then the sum is just a single number. So what do you want from it? Maybe I could say its empirical distribution is a point mass at the observe point. But that can't possibly be what you want to know. $\endgroup$
    – Michael R. Chernick
    Commented May 18, 2012 at 12:47
  • $\begingroup$ @MichaelChernick, wouldn't it make sense to use the empirical distribution of the $X_{i}$'s, assume iid, and then use that to estimate the distribution of the sum? Of course, in that case you're right - empirical distribution would be the wrong terminology. $\endgroup$
    – Macro
    Commented May 18, 2012 at 12:54
  • $\begingroup$ I have way more than n data points. I need to know n for roughly n<= 20, I have about 6x10^7 data points. $\endgroup$
    – cammil
    Commented May 18, 2012 at 13:06
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    $\begingroup$ Now I'm confused about what the data actually is @cammil - can you clarify by editing the question? $\endgroup$
    – Macro
    Commented May 18, 2012 at 13:28
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    $\begingroup$ Cammil, a possible interpretation of the question is that you have a large batch of $N$ data values $(x_1,x_2,\ldots,x_N)$; you stipulate a smallish number $n \le N$; and you wish to determine (or estimate) the distribution of $(x_{i_1}+x_{i_2}+\cdots+x_{i_n})$ as the indexes $(i_1,i_2,\ldots,i_n)$ range over all $n$-element subsets of $1,2,\ldots,N$. Am I on the mark? $\endgroup$
    – whuber ♦
    Commented May 18, 2012 at 14:50

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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.

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