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I think I know the answer to this, but just wanted to confirm.

If we are given the number of observations in a dataset (n) and the arithmetic mean (m), we can easily solve for the total sum of all observations by multiplying m*n

However, if we are given the number of observations in a dataset (n) and the harmonic mean (h), it is not clear to me that it is possible to solve for the total sum of all observations. I tried to work out the algebra, but it got very messy very quickly, so I wanted to check if there is an established way to do this before spending too much time trying to solve this problem.

NB3
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  • The only thing you can deduce generally is that the sum of $n$ observations is no less than $n$ times their harmonic mean. See https://en.wikipedia.org/wiki/HM-GM-AM-QM_inequalities. – whuber Mar 04 '22 at 14:22

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Harmonic means : 1/(1/5+1/3) and 1/(1/2+1/30) ... so you can't get the sum !

MrSmithGoesToWashington
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  • Thank you! I was fairly confident there was no way to get the sum due to the harmonic mean taking a sum of reciprocals. I guess we can still recover the total sum of all reciprocals, but this isn't very meaningful for what I was hoping to do (as there isn't a way to transform the sum of reciprocals into the total sum) – NB3 Mar 04 '22 at 11:47