## Abstract

Average levels of well-being across countries and time periods have been reported widely over the years as one of the measures that go beyond the Gross Domestic Product. It is relevant for policy makers to also have information on well-being dispersion to identify groups that lag behind with respect to their ability to pursue happiness. In addition, there is an inherent moral appeal to not only maximize wellbeing but also equalize well-being among people. In this paper, we try to answer the following research question: *which measure(s) should be used to gain insight into wellbeing dispersion?* We review sixteen measures and their properties, study their behavior in over 92 thousand simulated distributions, and apply them to 4 years of the Dutch Social Cohesion and Well-Being Survey, using bootstrapping to quantify their precision. Our inventory shows that when applied to a discrete ordinal rating scale, common measures such as the standard deviation and Gini coefficient do not show any advantage over the less restrictive index of ordinal variation (IOV). Only the generalized entropy and Atkinson index adhere to additional principles, most notably of diminishing transfers, at the expense of full scale invariance. The simulation study illustrates that dispersion measures are positively correlated but do not rank distributions the same. The field study shows that only the Atkinson index with high inequality aversion provides additional insight. We recommend using the index of ordinal variation, supplemented if needed with the Atkinson index using a high value for the inequality aversion parameter.

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## Acknowledgements

We thank Kees Zeelenberg and Linda Moonen for discussion and useful comments on an earlier version of this paper. We also thank the anonymous reviewers for their helpful comments. The views expressed in this paper are those of the authors and do not necessarily reflect the policies of Statistics Netherlands.

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## Appendices

### Appendix 1 dispersion measures defined

The dispersion measures are defined below (Table 2). Let \(Y\) be a discrete random variable with possible values \(\left\{ {1, \ldots ,K} \right\}\), probability mass \(p_{k} = P\left( {Y = k} \right)\), \(\hat{p}_{k} = \frac{{\hat{N}_{k} }}{N}\) and cumulative distribution function \(F_{k} = P\left( {Y \le k} \right) = \sum\nolimits_{l = 1}^{k} {p_{l} }\), where \(N_{k}\) is the number of population units in category \(k\) and \(N = \sum\nolimits_{k = 1}^{K} {N_{k} }\) the total population size. For instance, \(Y\) is subjective well-being on a ten point scale with possible scores \(\left\{ {1, \ldots ,K = 10} \right\}\). In a sample survey, the distribution can be estimated using \(\hat{N}_{k} = \sum\nolimits_{i = 1}^{{r_{k} }} {w_{i} }\), where \(w_{i}\) is the calibration weight of respondent \(i\) and \(r_{k}\) the number of respondents in category \(y_{i} = k\). In a homogeneous population or extreme unimodal distribution (no dispersion or a maximally concentrated distribution in which case everyone has the same score), \(p_{k} = 1\) for one category and 0 for the other categories; in a uniform distribution (maximum dispersion or the scores are equally distributed across answer categories), \(p_{k} = \frac{1}{K}\) for all categories; in an extreme bimodal distribution (maximum dispersion with scores on the two extreme categories), \(p_{1} = p_{K} = 0.5\) and 0 for the other categories.

### Appendix 2 convergence

See Fig. 7.

### Appendix 3 performance of bootstrapped confidence intervals

The interpretation of a frequentist 95% confidence interval is that if we would repeat the sample survey an infinite number of times, 95% of the confidence intervals would contain the true population parameter. To verify whether this holds for our bootstrapped approximation of the confidence interval, a simulation study was conducted. An artificial population of about 13 million persons was created from the Social Cohesion and Well-Being Survey, using the rounded calibration weight of each 18 + respondent as the number of population units it represents. Happiness dispersion in this population was taken as the true population value. From the artificial population, we drew 1000 simple random samples of a size equal to the number of respondents in the survey (about 7 thousand). For each sample, happiness dispersion was estimated, including the 95% confidence interval estimated from 1000 bootstrap resamples. For each sample we checked whether the 95% bootstrapped confidence interval contained the true population value. The proportion of intervals containing the true population value is shown in Fig. 8. The simulation was performed independently for 4 years. Six measures were used to quantify happiness dispersion. Happiness dispersion was estimated for the total population and for two age classes. See e.g. Berger and Muñoz (2015) for a more elaborate simulation study on coverage rates of a bootstrap variance estimator.

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Burger, J., van Beuningen, J. Measuring well-being dispersion on discrete rating scales.
*Soc Indic Res* (2020). https://doi.org/10.1007/s11205-020-02275-1

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### Keywords

- Beyond GDP
- Income inequality axioms
- Ordinal rating scale
- Probability mass distribution
- Bootstrap