Editing Geometric mean (section)

===Applications in the social sciences===

Although the geometric mean has been relatively rare in computing social statistics, starting from 2010 the United Nations Human Development Index did switch to this mode of calculation, on the grounds that it better reflected the non-substitutable nature of the statistics being compiled and compared:
: The geometric mean decreases the level of substitutability between dimensions [being compared] and at the same time ensures that a 1 percent decline in say life expectancy at birth has the same impact on the HDI as a 1 percent decline in education or income. Thus, as a basis for comparisons of achievements, this method is also more respectful of the intrinsic differences across the dimensions than a simple average.<ref>{{cite web|url=http://hdr.undp.org/en/statistics/faq/|title=Frequently Asked Questions - Human Development Reports|website=hdr.undp.org|url-status=live|archive-url=https://web.archive.org/web/20110302103418/http://hdr.undp.org/en/statistics/faq/|archive-date=2011-03-02}}</ref>

Not all values used to compute the [[Human Development Index|HDI (Human Development Index)]] are normalized; some of them instead have the form <math>\left(X - X_\text{min}\right) / \left(X_\text{norm} - X_\text{min}\right)</math>. This makes the choice of the geometric mean less obvious than one would expect from the "Properties" section above.

The equally distributed welfare equivalent income associated with an [[Atkinson Index]] with an inequality aversion parameter of 1.0 is simply the geometric mean of incomes. For values other than one, the equivalent value is an [[Lp space|Lp norm]] divided by the number of elements, with p equal to one minus the inequality aversion parameter.