Editing Geometric mean (section)

==Related concepts==

===Iterative means===
The geometric mean of a data set [[inequality of arithmetic and geometric means|is less than]] the data set's [[arithmetic mean]] unless all members of the data set are equal, in which case the geometric and arithmetic means are equal. This allows the definition of the [[arithmetic-geometric mean]], an intersection of the two which always lies in between.

The geometric mean is also the '''arithmetic-harmonic mean''' in the sense that if two [[sequence]]s (<math display="inline">a_n</math>) and (<math display="inline">h_n</math>) are defined:
:<math>a_{n+1} = \frac{a_n + h_n}{2}, \quad a_0 = x</math>

and
:<math>h_{n+1} = \frac{2{a_n}{h_n}}{a_n + h_n}, \quad h_0 = y</math>

where <math display="inline">h_{n+1}</math> is the [[harmonic mean]] of the previous values of the two sequences, then <math display="inline">a_n</math> and <math display="inline">h_n</math> will converge to the geometric mean of <math display="inline">x</math> and <math display="inline">y</math>. The sequences converge to a common limit, and the geometric mean is preserved:
:<math>\sqrt{a_{i+1} h_{i+1}} =
  \sqrt{\frac{a_i + h_i}{2}\frac{2{a_i}{h_i}}{a_i + h_i}} =
  \sqrt{{a_i}{h_i}}
</math>

Replacing the arithmetic and harmonic mean by a pair of [[generalized mean]]s of opposite, finite exponents yields the same result.

===Comparison to arithmetic mean===
{{AM_GM_inequality_visual_proof.svg}}
{{QM_AM_GM_HM_inequality_visual_proof.svg}}
{{main|Inequality of arithmetic and geometric means}}
The geometric mean of a non-empty data set of positive numbers is always at most their arithmetic mean. Equality is only obtained when all numbers in the data set are equal; otherwise, the geometric mean is smaller. For example, the geometric mean of 2 and 3 is 2.45, while their arithmetic mean is 2.5. In particular, this means that when a set of non-identical numbers is subjected to a [[mean-preserving spread]] — that is, the elements of the set are "spread apart" more from each other while leaving the arithmetic mean unchanged — their geometric mean decreases.<ref>{{cite journal |last=Mitchell |first=Douglas W. |title=More on spreads and non-arithmetic means |journal=[[The Mathematical Gazette]] |volume=88 |year=2004 |issue=511 |pages=142–144 |doi=10.1017/S0025557200174534 |s2cid=168239991 }}</ref>

===Geometric mean of a continuous function===
If <math>f:[a,b]\to(0, \infty)</math> is a positive continuous real-valued function, its geometric mean over this interval is

:<math>\text{GM}[f] = \exp\left(\frac{1}{b-a}\int_a^b\ln f(x)dx\right)</math>

For instance, taking the identity function  <math>f(x) = x</math> over the unit interval shows that the geometric mean of the positive numbers between 0 and 1 is equal to <math>\frac{1}{e}</math>.