Does central tendency (Geometric Mean specifically) always lie in between the two numbers or can it lie outside the interval of the two numbers?

Question

Does central tendency necessarily have to lie between the two numbers?For eg. I understand $2,4,8$ are in GP with $4$ being the GM of $2,4,8$. But what about $2,-4,8$ if we consider common ratio to be $-2$? $-4$ does not lie between $2,8$.Still is it acceptable? If we say $a,b,c$ are in GP,should be say $b=\pm\sqrt {a×c}$ or only $b=+\sqrt {a×c}$?

Can someone provide me an answer with proper citations preferably?

Thanks for any help!!

@StubbornAtom So ,$-4$ is not accecpted as the GM of $2$ and $8$?Only $+4$ is accepted? — Soham, Jul 09 '17 at 17:48
What is the gm if you take -4? Your answer cannot be a complex number. — StubbornAtom, Jul 09 '17 at 18:05
@StubbornAtom I am saying GM of $2$ and $8$ is $-4$ with common ratio $-2$... — Soham, Jul 10 '17 at 02:31

score 1 · Answer 1 · answered Jul 08 '17 at 18:03

According to wikipedia

the geometric mean applies only to numbers of the same sign.

If we were to try to take the GM of -4,2,8, we would get some imaginary number. By definition, central tendencies always lie between the minimum and maximum. They are supposed to measure 'averages' of values, so if they gave an answer not within the range of values they would not be doing their job. Also, you seem to be confusing geometric mean and geometric progression. They are not the same thing. A GM applies to a set of values. If we did it on the numbers 2,8, we will always get 4.

Does central tendency (Geometric Mean specifically) always lie in between the two numbers or can it lie outside the interval of the two numbers?

1 Answers1