Beliefs about polarization have significant social consequences, whether accurate or not [1][2][3][4] . They also complicate the study of social issues as reported attitudes might be impacted by inaccurate out-group perceptions and thus make conflicts around a specific issue or policy appear more severe than they actually are 5 . This can result in significant distortions in numerous behaviours, including health, voting and financial choices, each of which has consequences on population wellbeing 6 .There is growing interest in the origins of polarization across populations as well as its features and impacts across communities and society 7,8 . This interest is global and includes both scientific research as well as general public interest 9,10 . As polarization seemingly permeates a growing number of personal and public domains, there is some sense of renewed urgency to understand it and its effects more deeply 11,12 . This includes extending study to understand the extent of polarization on community and global levels.However, with increasing interest in polarization itself, broadening evidence indicates that inaccuracies in perceptions of how the out-group feels about the in-group can be harmful 13 . The origin of these 'meta-perceptions' may be rooted in negative stereotypes that individuals feel have been applied to them, often incorrectly 14 . This results in an inaccurate perception of differences in beliefs and attitudes between groups 15 , which can have negative results for individuals 16,17 . On a population level, such misperceptions can even result in overstated reactions that exacerbate existing biases 18 .To investigate roots and moderators of polarization across groups, Lees and Cikara 13 ran a series of experiments with US participants identifying as Republican or Democrat. In what they refer

Economic inequality is associated with preferences for smaller, immediate gains over larger, delayed ones. Such temporal discounting may feed into rising global inequality, yet it is unclear whether it is a function of choice preferences or norms, or rather the absence of sufficient resources for immediate needs. It is also not clear whether these reflect true differences in choice patterns between income groups. We tested temporal discounting and five intertemporal choice anomalies using local currencies and value standards in 61 countries (N = 13,629). Across a diverse sample, we found consistent, robust rates of choice anomalies. Lower-income groups were not significantly different, but economic inequality and broader financial circumstances were clearly correlated with population choice patterns.

A pervading global narrative suggests that political polarisation is increasing in the US and around the world. Beliefs in increased polarisation impact individual and group behaviours regardless of whether they are accurate or not. One driver of polarisation are beliefs about how members of the out-group perceive us, known as group meta-perceptions. A 2020 study by Lees and Cikara in US samples suggests that not only are out-group meta-perceptions highly inaccurate, but informing people of this inaccuracy reduces negative beliefs about the out-group. Given the importance of these findings for understanding and mitigating polarisation, it is essential to test to what extent they generalise to other countries. We assess that generalisability by replicating two of the original experiments in 10,207 participants from 26 countries in the first experiment and 10 in the second. We do this by studying local group divisions, which we refer to as fault lines. In line with our hypotheses, results show that the pattern found in the US broadly generalises, with greater heterogeneity explained by specific policies rather than between-country differences. The replication of a simple disclosure intervention in the second experiment yielded a modest reduction in negative motive attributions to the out-group, similar to the original study. These findings indicate first that inaccurate and negative group meta-perceptions are exhibited in a large number of countries, not only the US, and that informing individuals of their misperceptions can yield positive benefits for intergroup relations. The generalisability of these findings highlights a robust phenomenon with major implications for political discourse worldwide.

Modelling an empirical distribution by means of a simple theoretical distribution is an interesting issue in applied statistics. A reasonable first step in this modelling process is to demand that measures for location, dispersion, skewness and kurtosis for the two distributions coincide. Up to now, the four measures used hereby were based on moments.In this paper measures are considered which are based on quantiles. Of course, the four values of these quantile measures do not uniquely determine the modelling distribution. They do, however, within specific systems of distributions, like Pearson's or Johnson's; they share this property with the four moment-based measures.This opens the possibility of modelling an empirical distribution-within a specific system-by means of quantile measures. Since moment-based measures are sensitive to outliers, this approach may lead to a better fit. Further, tests of fit-e.g. a test for normality-may be constructed based on quantile measures. In view of the robustness property, these tests may achieve higher power than the classical moment-based tests.For both applications the limiting joint distribution of quantile measures will be needed; they are derived here as well. 1Consider a random variable x with mean p = E(x) and central moments A quantile measure for kurtosisThe (very familiar) moment-based measures for location, dispersion, skewness and kurtosis now are 0 the mean p 0 the variance p2 0 the third standardized moment PI = p3/p;'* 0 the fourth standardized moment pZ = p4/& They all exist provided E(x4) < 00. (Note that the symbol PI is used instead of the usual fi.) ~ ~~~ ~ ~ * moors@kub.nl QVVS. 1996 Published by Bluckwell Publishen, 108 Cowley Road. Oxford OX4 IJF. U K and 238 Main Sllec~. Cambndp. MA 02142. USA For the first three measures quantile-based alternatives are well-known. Defining quartiles Qi by P ( x < Q J 5 i/4, P(x > QJ 5 1i/4 for i = 1,2,3, they are given by 0 the median Q = Q2 0 the half interquartile range R = (Q3 -Q l ) / 2 0 Bowley's skewness measure S = (Q3 -2Q2 + Q I ) / ( Q~ -Q I )provided that Q3 # QI. MOORS (1986, 1988) presented a new interpretation of kurtosis as well as a quantile-based alternative for p2. Define octiles Ei by ' P(x < Ei) 5 i/8, P(x > Ei) 5 1i/8 for i = 1,2,. . . ,7. Then the quantile measure T for kurtosis readsprovided that €6 # E2. Note that Tis much less sensitive to outliers than a; it can be found even by graphical means. Furthermore, T exists even for distributions without finite moments; e.g. T = 2 for the Cauchy distribution.The quartet ( Q , R, S, T) can be seen as an alternative to ( p , p2, 81, P,). Like PI and p2, S and Tremain unchanged under linear transformations: these four quantities are location-scale-invariant. This is the main reason why in the sequel attention is focused on the pair (S, T). The rest of the paper is organized as follows. Sections 2 and 3 consider the Pearson system of distributions. Section 2 reviews its properties with emphasis on the moment measures 01 and P,; in Section 3 the quanti...

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