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I have a sample of $N$ iid random variables and I would like to get an error estimate for the sum of the variables of my sample. However I have very limited knowledge about my sample: I only have $k$ sums of $n_i$ variables such that $i \in \{1, ..., k\}$ and:

$\sum_{i=1}^k n_i = N$

In terms of an oversimplified example say my sample is:

$\{1, 2, 3, 4, 5, 6, 7\}$

But the only things I know are that I have a sum of 4 sample elements being equal to 11 (=1+2+3+5) and a sum of 3 elements equal to 17 (=4+6+7).

What would be a good error estimate for the sum of all my random variables and how could I calculate it from the knowledge I have?

I am interested in the general case, the example is there just to illustrate the situation I have. Statistics in not my strongest field of expertise, but I hope I got the terminology about right and the question makes some sense.

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  • $\begingroup$ It looks to me your subsums cover every element once? $\endgroup$
    – jf328
    Jun 23, 2015 at 8:11
  • $\begingroup$ Yes that's the case $\endgroup$
    – user3605424
    Jun 23, 2015 at 8:18
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    $\begingroup$ Recasting this question in terms of the means (which is done by dividing the sums by their counts) makes it similar to one that has been asked before. The first half of my answer at stats.stackexchange.com/questions/24936/… then answers this question. $\endgroup$
    – whuber ♦
    Jun 23, 2015 at 16:50

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