## Abstract

Eight years ago we proposed a new measure of happiness in nations, called Inequality-Adjusted Happiness (IAH). This measure indicates how successful nations are in combining a high level of happiness and a low degree of inequality in happiness among citizens. The index gives equal weight to the level of happiness measured using the mean and inequality of happiness measured using the standard deviation. Scores on this index are now available for 139 nations. In this paper we present a technical improvement for the method to calculate IAH.

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## Notes

Questions on happiness can be phrased in more ways, using different key words. An overview of questions that fit the concept of ‘the subjective appreciation of one’s life as-a-whole is available in Veenhoven (2011b). The method described in this paper applies to all measures.

In an earlier paper in this journal we argued that the standard deviation is a proper statistic for measuring inequality in happiness (Kalmijn and Veenhoven 2005). Recently, Delhey and Kohler (2011) proposed an adjustment to that method. We advised against that adjustment in Kalmijn (2012) and Veenhoven (2012) to which Delhey and Kohler (2012) replied.

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## Acknowledgments

The authors wish to thank Dr. Jan Koster (Erasmus University Rotterdam) for his suggestion to apply the skew projection method.

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## Appendices

### 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*=

*as its abscissa and its internal standard deviation*

__LG__*s*=

*as its ordinate.*

__NG__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*/

*) × 100.*

__HW__From the parallelism of WZ and UD follows the proportionality

The angles *ZMW* and *DNG* are equal; let their value be 2*φ*, where *φ*: = angle(*WHL*). The value of 2*φ* 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

the *IAH*-value of the nation represented by the point *N* (*m*,*s*) equals

This result can also be written as

In the case of equal weights *w*
_{E} = *w*
_{U} and 2*φ* = *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) |

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Kalmijn, W., Veenhoven, R. Index of Inequality-Adjusted Happiness (IAH) Improved: A Research Note.
*J Happiness Stud* **15**, 1259–1265 (2014). https://doi.org/10.1007/s10902-013-9474-3

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DOI: https://doi.org/10.1007/s10902-013-9474-3