Editing Geometric mean (section)

===Normalized values===
The fundamental property of the geometric mean, which does not hold for any other mean, is that for two sequences <math>X</math> and <math>Y</math> of equal length,

: <math>\operatorname{GM}\left(\frac{X_i}{Y_i}\right) = \frac{\operatorname{GM}(X_i)}{\operatorname{GM}(Y_i)}</math>.

This makes the geometric mean the only correct mean when averaging ''normalized'' results; that is, results that are presented as ratios to reference values.<ref>{{cite journal |first1=Philip J. |last1=Fleming |first2=John J. |last2=Wallace |title=How not to lie with statistics: the correct way to summarize benchmark results |journal=Communications of the ACM |volume=29 |issue=3 |pages=218–221 |year=1986 |doi=10.1145/5666.5673 |s2cid=1047380 |doi-access=free }}</ref>  This is the case when presenting computer performance with respect to a reference computer, or when computing a single average index from several heterogeneous sources (for example, life expectancy, education years, and infant mortality).  In this scenario, using the arithmetic or harmonic mean would change the ranking of the results depending on what is used as a reference.  For example, take the following comparison of execution time of computer programs:

'''Table 1'''
{| class="wikitable"
|-
! &nbsp; !! Computer A !! Computer B !! Computer C
|-
| '''Program 1''' || 1 || 10 || 20
|-
| '''Program 2''' || 1000 || 100 || 20
|-
| '''Arithmetic mean''' || 500.5 || 55 || '''20'''
|-
| '''Geometric mean''' || 31.622... || 31.622... || '''20'''
|-
| '''Harmonic mean''' || '''1.998...''' || 18.182... || 20
|}

The arithmetic and geometric means "agree" that computer C is the fastest.  However, by presenting appropriately normalized values ''and'' using the arithmetic mean, we can show either of the other two computers to be the fastest.  Normalizing by A's result gives A as the fastest computer according to the arithmetic mean:

'''Table 2'''
{| class="wikitable"
|-
! &nbsp; !! Computer A !! Computer B !! Computer C
|-
| '''Program 1''' || 1 || 10 || 20
|-
| '''Program 2''' || 1 || 0.1 || 0.02
|-
| '''Arithmetic mean''' || '''1''' || 5.05 || 10.01
|-
| '''Geometric mean''' || 1 || 1 || '''0.632...'''
|-
| '''Harmonic mean''' || 1 || 0.198... || '''0.039...'''
|}

while normalizing by B's result gives B as the fastest computer according to the arithmetic mean but A as the fastest according to the harmonic mean:

'''Table 3'''

{| class="wikitable"
|-
! &nbsp; !! Computer A !! Computer B !! Computer C
|-
| '''Program 1''' || 0.1 || 1 || 2
|-
| '''Program 2''' || 10 || 1 || 0.2
|-
| '''Arithmetic mean''' || 5.05 || '''1''' || 1.1
|-
| '''Geometric mean''' || 1 || 1 || '''0.632'''
|-
| '''Harmonic mean''' || '''0.198...''' || 1 || 0.363...
|}

and normalizing by C's result gives C as the fastest computer according to the arithmetic mean but A as the fastest according to the harmonic mean:

'''Table 4'''

{| class="wikitable"
|-
! &nbsp; !! Computer A !! Computer B !! Computer C
|-
| '''Program 1''' || 0.05 || 0.5 || 1
|-
| '''Program 2''' || 50 || 5 || 1
|-
| '''Arithmetic mean''' || 25.025 || 2.75 || '''1'''
|-
| '''Geometric mean''' || 1.581... || 1.581... || '''1'''
|-
| '''Harmonic mean''' || '''0.099...''' || 0.909... || 1
|}

In all cases, the ranking given by the geometric mean stays the same as the one obtained with unnormalized values.

However, this reasoning has been questioned.<ref>{{cite journal |first=James E. |last=Smith |title=Characterizing computer performance with a single number |journal=Communications of the ACM |volume=31 |issue=10 |pages=1202–1206 |year=1988 |doi=10.1145/63039.63043|s2cid=10805363 |doi-access=free }}</ref>
Giving consistent results is not always equal to giving the correct results. In general, it is more rigorous to assign weights to each of the programs, calculate the average weighted execution time (using the arithmetic mean), and then normalize that result to one of the computers. The three tables above just give a different weight to each of the programs, explaining the inconsistent results of the arithmetic and harmonic means (Table 4 gives equal weight to both programs, the Table 2 gives a weight of 1/1000 to the second program, and the Table 3 gives a weight of 1/100 to the second program and 1/10 to the first one). The use of the geometric mean for aggregating performance numbers should be avoided if possible, because multiplying execution times has no physical meaning, in contrast to adding times as in the arithmetic mean. Metrics that are inversely proportional to time (speedup, [[Instructions per cycle|IPC]]) should be averaged using the harmonic mean.

The geometric mean can be derived from the [[generalized mean]] as its limit as <math>p</math> goes to zero. Similarly, this is possible for the weighted geometric mean.