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The special case I'm considering is the following. Suppose that from a population of size n, we add together only the non-zero values. Now, to find the average, suppose we divide the sum of the non-zero values by the number of non-zero values, instead of n. From my elementary understanding of statistics, I believe this is a special case of averaging, because we should normally divide by n. So my question is: what is the term for describing this type of averaging?

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    $\begingroup$ That's just the average of the non-zero values. If you have a good reason for leaving out zeros, then it's the average you want. For example, average coffee cups per day is the average for coffee drinkers in the sample if you leave out zeros and the average for everyone in the sample if you don't. That example could be more complicated, but the principle may be clear. If values can be negative, the story is more complicated, but correspondingly you need a better reason to leave out zeros. $\endgroup$
    – Nick Cox
    Commented Mar 1, 2015 at 12:41
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    $\begingroup$ Small point: The data you have are a sample and the number in the sample is often denoted $n$. Only exceptionally are they also the complete population. $\endgroup$
    – Nick Cox
    Commented Mar 1, 2015 at 12:43

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This is the average after removing the zero values. No definition for that.

But, if you do so, you must explain why did you remove those values. Are those values correct/possible values? Are they outliers? Do you expect to have more on other experiments or real world?...

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