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What is the geometric mean of $-1$ and $-16$? Should it give $-4$ or $+4$?I think it should be $-4$ because a mean should always be greater than the least number and less than the greatest number. But square root always gives positive value, so what should it be?

msinghal
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Pritam Ghoshal
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    For this reason, we only define the geometric mean when all the terms are positive. – Cataline Jul 23 '15 at 12:27
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    $-1\times-16=16$. $16$ cannot "remember" that it was the product of two negative numbers. –  Jul 23 '15 at 12:34

2 Answers2

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I wouldn't define the geometric mean on non-positive numbers. There are several reasons for this:

1) If you have a set $\{a_1,\ldots,a_n\}$ of numbers, where $n$ is even, and an odd amount of these numbers are negative, then the geometric mean of the $a_i$ is not defined.

2) If you include $0$ in the set of numbers for which you take the geometric mean, you always get $0$ (do you want this?)

3) If you have a mix of positive and (an even number of) negative numbers, their geometric mean wouldn't make much sense, because you would completely ignore the sign of each individual number in the calculation of the geometric mean.

But anyway, if you insist, then I would define the geometric mean of a set of strictly negative numbers as the negative of the geometric mean of their absolute values, so that if $a_1,\ldots,a_n<0$, then

$\text{geo.mean}(a_1,\ldots,a_n) = -(|a_1\ldots a_n|)^{1/n}.$

Mankind
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I saw a method for applied in negative numbers. I call it "Transformed geometric mean".

1)Change numbers to percentage and then add +1, so all data are positive and multiply by 100(transform the data)

2)Calculate the geometric average of transformed data

3)finally add -1 to all data (inverse transformation)

 > x<-c(-0.43,-1.31,-1.87)
 > y<-1+x/100
 > (-1+prod(y)^(1/length(x)))*100
 [1] -1.205109

For more detail see geometric-mean-calculator

enter image description here

Masoud
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