Editing Highest averages method (section)

=== Generalized average ===
In general, it is possible to construct an apportionment method from any generalized [[average]] function, by defining the signpost function as {{Math|1=post(''k'') = avg(''k'', ''k''+1)}}.<ref name="Pukelsheim-2017-1" />

==== {{Anchor|Imperiali method|Danish method}}Stationary family ====
A divisor method is called '''stationary'''<ref name="Pukelsheim-2017-4" />{{Rp|68}} if for some real number <math>r\in[0,1]</math>, its signposts are of the form <math>d(k) = k+r</math>. The Adams, Webster, and d'Hondt methods are stationary, while Dean and Huntington-Hill are not. A stationary method corresponds to rounding numbers up if they exceed the [[weighted arithmetic mean]] of {{Math|''k''}} and {{Math|''k''+1}}.<ref name="Pukelsheim-2017-1" /> Smaller values of {{Math|''r''}} are friendlier to smaller parties.<ref name="Pukelsheim-2017-2" />

[[Elections in Denmark|Danish elections]] allocate [[leveling seat]]s at the province level using-member constituencies. It divides the number of votes received by a party in a multi-member constituency by 0.33, 1.33, 2.33, 3.33 etc. The fencepost sequence is given by {{Math|1=post(''k'') = ''k''+{{frac|1|3}}}}; this aims to allocate seats closer to equally, rather than exactly proportionally.<ref name="Denmark-2016" />

==== {{Anchor|Dean's method}}Power mean family ====
The '''[[power mean]] family''' of divisor methods includes the Adams, Huntington-Hill, Webster, Dean, and Jefferson methods (either directly or as limits). For a given constant {{Math|''p''}}, the power mean method has signpost function {{Math|1=post(''k'') = {{radic|''k''{{sup|''p''}} + (''k''+1){{sup|''p''}}|{{sub|''p''}}}}}}. The Huntington-Hill method corresponds to the limit as {{Math|''p''}} tends to 0, while Adams and Jefferson represent the limits as {{Math|''p''}} tends to negative or positive infinity.<ref name="Pukelsheim-2017-1" />

The family also includes the less-common '''Dean's method''' for {{Math|1=''p''=-1}}, which corresponds to the [[harmonic mean]]. Dean's method is equivalent to ''rounding to the nearest average''—every state has its seat count rounded in a way that minimizes the difference between the average district size and the ideal district size. For example:<ref name="Balinski-1982" />{{Rp||page=29}}<blockquote>The 1830 representative population of Massachusetts was 610,408: if it received 12 seats its average constituency size would be 50,867; if it received 13 it would be 46,954. So, if the divisor were 47,700 as Polk proposed, Massachusetts should receive 13 seats because 46,954 is closer to 47,700 than is 50,867.</blockquote>Rounding to the vote average with the smallest relative error once again yields the Huntington-Hill method because {{Math|1={{abs|log({{frac|x|y}})}} = {{abs|log({{frac|y|x}})}}}}, i.e. relative differences are reversible. This fact was central to [[Edward V. Huntington]]'s use of relative (instead of absolute) errors in measuring misrepresentation, and to his advocacy for Hill's rule:<ref name="Lauwers-2008" /> Huntington argued the choice of apportionment method should not depend on how the equation for equal representation is rearranged, and only the relative error (minimized by Hill's rule) satisfies this property.<ref name="Balinski-1982" />{{Rp||page=53}}

==== Stolarsky mean family ====
Similarly, the [[Stolarsky mean]] can be used to define a family of divisor methods that minimizes the [[generalized entropy index]] of misrepresentation.<ref name="Wada-2012" /> This family includes the [[logarithmic mean]], the [[geometric mean]], the [[identric mean]] and the [[arithmetic mean]]. The Stolarsky means can be justified as minimizing these misrepresentation metrics, which are of major importance in the study of [[information theory]].<ref name="Agnew-2008" />