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We are taught that geometric mean (GM) should only be applied to a dataset of positive numbers, and some insist that it should be strictly positive numbers.

However, I have seen people discussing the calculation of the GM of a dataset which contains $0$s. And there seem to be at least two ways to deal with such situation:

  1. If there exists a $0$ in the dataset, then the GM is also $0$

  2. Substitute the $0$s with some other number (e.g. $1$), then work out the GM as usual

Can someone please share their opinion on when we should apply method 1) instead of method 2) and why (i.e. the justifications), and vice versa? In addition, intuitively, why we want to do 1) as it essentially throws away all the other non-zero values in the dataset? Thanks.

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    For what purpose are you computing the geometric mean of your data at all? This may well affect how you should go about it.2011-12-14
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    I can't imagine a situation where you'd ever want to use (2). You might want to discount 0s entirely, but that is not the same thing as treating them as 1. For example, if your dataset is $(0,4,16)$, if you treat zero as $1$, you'd be taking the geometric mean of $(1,4,16)$ which is $4$. But if you "ignore" the zeros, you'd take the geometric mean of $(4,16)$, which is $8$.2011-12-14
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    @ChrisEagle, I want to see the population consensus of their choices. For example, if each value in the dataset represents a choice out of $1$-$10$, then I want to get an estimate of the population consensus.2011-12-14
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    @ballib Why would you use the geometric mean to compute that? I can think of good arguments for using the mean, the median or the mode, but I can't think of any good reason for using the geometric mean. What does a zero value represent in this dataset? Is it a code for a question that wasn't answered?2011-12-14
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    @ChrisTaylor, sorry, a typo in my comment above, should be "a choice out of $0$-$10$". In addition, another context I am interested in using geometric mean is that if I have 10 distributions over the same random variable, then I wonder if I can use GM to work out the 'average' probability of each outcome, and potentially some distribution(s) may have $0$ for certain outcome.2011-12-14
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    @ballib, what does that give you that the ordinary arithmetic mean doesn't?2011-12-14

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