Appendix 1: Improved IAH by the Application of Skew Projection
Let in Fig. 1 the point M be the centre of the semicircle with radius LM = r. If L is adopted as the origin, a nation N can be represented by its mean happiness value m = LG as its abscissa and its internal standard deviation s = NG as its ordinate.
Let W represent the compromise for the worst conceivable situation and WZ be the tangent through W to the semicircle. The skew projection U of N onto the IAH-axis WH is obtained as the intersection of WH with the line segment ND through N parallel to WZ.
In that case, the IAH-value of N equals the ratio (UW/HW) × 100.
From the parallelism of WZ and UD follows the proportionality
$$ \underline{UW} /\underline{HW} = \underline{ZD} /\underline{ZH} . $$
The angles ZMW and DNG are equal; let their value be 2φ, where φ: = angle(WHL). The value of 2φ equals [w
E/(w
E + w
U)](π/2), where w
E and w
U are the weights assigned to the egalitarian and utilitarian views respectively. Since
$$ \begin{aligned} \underline{ZD} & = \underline{ZG} -
\underline{DG} = \underline{ZL} + \underline{LG} - \underline{DG}
= \underline{ZM} - \underline{ML} + \underline{LG} -
\underline{DG} \\ & = r/\cos 2\varphi - r + m - s\,\tan
2\varphi \quad{\text{and}} \\ \underline{ZH} & =
\underline{ZM} + \underline{MH} = r/\cos 2\varphi + r, \\
\end{aligned} $$
the IAH-value of the nation represented by the point N (m,s) equals
$$ \frac{{\frac{r}{\cos \,2\varphi } - r + m - s\,\tan \,2\varphi }}{{\frac{r}{\cos \,2\varphi } + r}} \times 100 $$
This result can also be written as
$$ IAH = \frac{(m - r)\cos \,2\varphi + r - s\,\sin \,2\varphi }{r + r\,\cos \,2\varphi } \times 100 $$
In the case of equal weights w
E = w
U and 2φ = π/4; when happiness is quantified on a [0, 10] scale, then r = 5 and in this particular case this formula can be simplified to IAH ≈ 8.28(m − s) +17.2.
Appendix 2: Inequality-Adjusted Happiness (IAH) in 15 Nations 2003–2009
IAH-values in modified and previous version.
.
|
Nation
|
Happiness
|
Inequality-Adjusted Happiness
|
|---|
|
Average
|
SD
|
IAH (new)
|
IAH (old)
|
|---|
|
Denmark
|
8.03
|
1.53
|
71
|
(75)
|
|
Iceland
|
7.87
|
1.66
|
69
|
(73)
|
|
Switzerland
|
7.74
|
1.58
|
68
|
(72)
|
|
Finland
|
7.61
|
1.56
|
67
|
(71)
|
|
Netherlands
|
7.33
|
1.37
|
67
|
(69)
|
|
Japan
|
6.35
|
1.91
|
54
|
(57)
|
|
France
|
6.45
|
2.11
|
53
|
(58)
|
|
Indonesia
|
6.16
|
2.05
|
51
|
(55)
|
|
Poland
|
6.26
|
2.29
|
50
|
(55)
|
|
China
|
6.14
|
2.45
|
48
|
(53)
|
|
Macedonia
|
4.68
|
2.57
|
35
|
(39)
|
|
Bulgaria
|
4.46
|
2.41
|
34
|
(37)
|
|
Mali
|
4.73
|
2.77
|
33
|
(38)
|
|
Zimbabwe
|
3.23
|
2.28
|
25
|
(26)
|
|
Tanzania
|
3.03
|
2.76
|
19
|
(22)
|