Why are the arithmetic and geometric means equal when all members of the data set are equal?

Question

The following statement is excerpted from http://en.wikipedia.org/wiki/Geometric_mean:

The geometric mean of a data set is smaller than or equal to the data set's arithmetic mean (the two means are equal if and only if all members of the data set are equal).

This means if we have a data set [10,10,10].

The arithmetic mean and the geometric mean are really both equal to 10.
But what is the theory behind it, making it such a coincidence?

Thanks.

Answer 1

No real theory, that's just the way the formulas work.



Let's assume we have a set of numbers all the same:



(n, n, n, ,n . . . ) and we have y of them.



For the arithmetic mean, we add up all y n's and divide by y



ny/y = n



For the geometric mean, we multiply the numbers together and take the yth root of it. multiplying n over and over again y times is n^y



the yth root of (n^y) is n

Answer 2

Geometric recommend: the sq. root of four and 12 (sixteen) is 4. arithmetic recommend: upload 4 and 12(sixteen) and divide via 2 = 8 The geometric recommend isn't the arithmetic recommend and this is not an basic primary. this is the nth root of the manufactured from n numbers. which ability you multiply a gaggle of numbers mutually, and then take the nth root, the place n is the form of values you purely stronger.

Answer 3

Say you have some number of the same number. Let's say you have 11 seven times. Obviously the arithmetic mean will be 11.



The geometric mean will be the seventh root of 11 to the seventh power. That should also be 11. The nth root of N to the nth power will be always be N.



The average of n N's will also always be N.

Answer 4

Every conceivable definition of "average" finds a value somewhere between the minimum and the maximum. If all of your data points are the same, then any average, regardless of which of dozens of definitions you're using, has to be that exact value.