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Assessing Mass Opinion Polarization in the US Using Relative Distribution Method

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Through an analysis of the cumulative data of the American National Election Studies between 1984 and 2008, this study presents evidence of growing mass polarization in terms of standard ANES measures of ideological orientation using the public policy issue dimensions. The empirical findings here suggest that the degree of polarization among US citizens increased as the distributional center of measures of political ideology have progressively declined, though the opinion distribution of the later periods do not dramatically exhibit a text-book style polarized distribution (e.g., bimodal distribution). According to the findings, attitudes toward government guarantees have shifted back and forth between more liberal and more conservative positions while public opinion on cultural issues has generally moved more liberal positions over years.

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  1. To avoid any confusion in concepts used here, partisan polarization is understood as the party sorting which basically indicates the increased correlation between individual partisanship and ideology, while popular or mass polarization represents ideological polarization or radicalization of mass opinion. Actually, the party sorting and the opinion radicalization are the first and the second kinds of polarization in Gelman (2008)’s explanation.

  2. In this article, ‘elite’ refers elected officials such as congressmen or governor.

  3. Namely, the extent of polarization can be different even when the group means are identical, if groups maintain different levels of intergroup homogeneity.

  4. More details about relative distribution method are provided in the appendix C at the end.

  5. For more details on the derivations of LRP and URP indices, see Handcock and Morris (1999, 72).

  6. Namely, \(MRP\left( {F;F_{0} } \right) = \frac{1}{2}LRP\left( {F;F_{0} } \right) + \frac{1}{2}URP(F;F_{0} )\).

  7. Thus, 40 % (25 % + 15 %) of the center population diverge toward respective tails of the distribution.

  8. More specifically, the ANES started to include self-placement of ideology scale (1972), government aid to blacks scale (1970), government guarantee of job security and living standards scale (1972), government responsibility for health insurance scale (1970), a level of government defense spending scale (1980), government spending vs. services scale (1982), and attitude toward the abortion policy (1980) according to 2008 ANES cumulative dataset. In general, most of the ideological scale variables included in the ANES employs 7-point scales. Notable exceptions are attitudes toward abortion policy and homosexuals which stand on the 4-point scale. So, some scholars have recoded the 4-point scale into 7-point scale (e.g., Fiorina and Levendusky 2006; Levendusky 2010). For instance, 1-2-3-4 scales are relocated to 1-3-5-7 scales.

  9. The items include liberal-conservative identification, policy attitudes on abortion policy, governmental aid to blacks, health insurance, jobs and living standards, defense spending, and government services vs. spending.

  10. There are some exceptional issue items that does not rely on 7-point scale, yet use different metric such as 4-point scale (e.g., abortion [VCF0837 and VCF0838], and gay protecting law [VCF0876a]) or 5-point scale (newer life style [VCF0851], tolerance scale [VCF0854]). These scales that have different coding systems are recoded to have the same metric as the majority of issue items loaded in the same issue dimension. For example, if a factor 1 consists of 5 items of 7-point scale and 1 issue item of 5-point scale, the one issue with 5-point scale is recoded to the 7-point scale. This recoding policy to maintain consistency over different issue items follows an approach of the previous literature (e.g., Fiorina and Levendusky 2006; Levendusky 2010).

  11. Although this study employs ANES cumulative data for 1948–2008, questions for ideological preferences are only available from 1970s.

  12. Among those listed 11 issue items, six questions were included to the survey in the early 1970s, the two items (VCF0843 and VCF0839) were added later in the early 1980s, and the rest of the items (the law protecting homosexuals, newer life styles, and tolerance) are only available from the late 1980s survey (1986 and 1988).

  13. Seven issue questions used in Abramowitz (2011) excludes four among those 11 questions mentioned above; those excluded items are “equal role of the women (VCF0834),” “law protecting homosexual from discrimination (VCF0876a),” “attitude toward newer life styles (VCF0851),” and “tolerance of different moral standards (VCF0854).

  14. In general, non-ideologues (responses with “don’t know”) are usually classified together with “moderates” in the previous literature of mass polarization. Namely, it is conventional to recode the “don’t know” answers to four on the 7-point scale (see, for example, Abramowitz and Saunders 2008, 544; Campbell 2006, 157; Fiorina and Abrams 2011).

  15. While Fiorina and Levendusky (2006) classify the policy areas without reporting a dimension reduction analysis, a recent study performs a principal component analysis using issue preference items to determine their dimensionality though the authors do not use ANES and analyzes 2006 CCES (Cooperative Congressional Election Study) data (Levendusky and Pope 2011).

  16. This trend of the two principal components is persistent across years as shown in the Table 1. Exceptions to this trend of the two factor loading are the results from 2000 and 2008. In the presidential year of 2000, three distinct dimensions are suggested according to the principal component analysis, but the problem with this inconsistent result is the very small number of cases unlike the other years. In the ANES survey of 2000, only half among the total respondents were asked to report their preferred policy-issue positions for 7-point scales. As a consequence, only 386 observations are available for the principal component analysis in 2000. If I intentionally drop the question with many missing values (e.g., VCF0803, “self-placement of ideology”) to increase the number of observations, items loaded on the two factors consistent to the other years’ outcome. When it comes to the 2008 analysis, three factors emerged as principal components, and the story is more complicated. Two cultural issue items (newer life and tolerance) comprise the third component together with the size of defense spending (VCF0843). I suspect that these seemingly unrelated issue items have played salient roles in characterizing the late 2000s, thus those three issue items occupy a separate issue dimension by departing from the other two factors. Despite the distinct trend in the principal component analysis of 2008, the reliability test reveals that an exclusion of the defense spending scale from the first scale does not create significant changes in Cronbach’s alpha coefficients (0.711 with the variable vs. 0.716 without the variable). Further, inclusion of all 5 issue items in the cultural scale presents a higher alpha value than when either of the two issues (newer life and tolerance) is removed from the scale. So, given these facts, I decided to adopt the two scales (government guarantee and cultural issues) consistently throughout every presidential year since 1984.

  17. Unlike here, Fiorina and Levendusky (2006) constructed a separate dimension of racial issues, and the government aid to blacks and minorities item is added to the racial items (p. 59). According to the principal component analysis here however, this issue clusters together with other social welfare spending scales since 1980 s. The racial issue dimension has merged largely into the social welfare issue dimension both in the elite and mass attitudes (Highton and Kam 2011, 206–207).

  18. I chose the data of 1984 as a reference time line because most of 7-point ideology scales used here became available in 1984. In particular, values for all 7-point issue items accounting for the primary factor (the first issue dimension or government guarantee) of respondents’ issue positions in the present study have been provided since 1984.

  19. It is, however, controversial among some scholars in recent literature if Cronbach’s alpha is an appropriate measure of internal consistency. Some psychometricians have also raised questions over the ‘standard’ threshold of 0.70 which is often adopted as the acceptable level of the alpha coefficient. Despite this ongoing debate, the Cronbach’s alpha coefficient is still widely employed by many applied researchers as a measure of internal consistency in many subfields of social sciences. Therefore, I decided to use the alpha coefficient as a measure of reliability since this study does not primarily aim at critically evaluating the Cronbach’s alpha. A debate over the appropriateness of the alpha coefficients can be found in a special issue of Psychometrica 74(1), March 2009.

  20. For these figures (kernel plots and RD analyses), a group of respondents who answered “don’t know” is combined to the moderate group following the previous literature (e.g., Abramowitz and Saunders 2008, 544; Campbell 2006, 157, Fiorina and Abrams 2011). Since this potentially non-ideological group of people who answered “haven’t thought much about it” can change the density size of ideological moderate depending on the inclusion and exclusion of the group to the genuinely identified middle of the roaders, I also produce comparable analyses that separate the “don’t know” group from the moderate, and the results are provided in the Appendix 1.

  21. Graphical displays of relative density were produced in STATA 10.0 using the pre-release version of “reldist” package developed by Benn Jann (2008). I appreciate him to allow me use the pre-released version of his package. The relative distribution analysis can also be performed in R using the existing package (“reldist”) developed by Handcock (2011).

  22. Polarization summary statistics were calculated in STATA 10.0 using the pre-release version of “reldist” package developed by Benn Jann (2008).

  23. Remind that MRP is an average value of URP and LRP, thus 0.14 = ½ (0.16 + 0.12). These polarization summary statistics are obtained using the mean absolute deviation from the median of the location-matched relative density. A URP (LRP) value accounts for the contribution to the median relative polarization (MRP) index from above (below) the median. Note that the measure does not explain the questions of whether the distributional upgrading (or downgrading) is more prevalent because such location shifts have been already removed by matching the location. Rather, the summary measures address the issue whether the residual changes (changes in shape or scale) have been more dramatic above or below the median (see Handcock and Morris 1999, 72–73).

  24. Here are values of MRP for each panel: 0.023 (1988:1996 pair), 0.036 (1992:2000 pair), 0.066 (1996:2004 pair), 0.099 (2000:2008 pair), and overall 0.144 (1984:2008 pair).

  25. The ranges of y-axis become loosen in the relative density panel of 2000:2008 than the panel of 1996:2004.

  26. The relational link attempted here between the results of RD analysis and the U.S. election outcomes relies on aggregate voting record, thus does not control for other relevant factors that can potentially influence on those congressional and presidential elections.

  27. It is noteworthy however that the underlying distributional polarization was under the way during the period between 1992 and 2000 as well. Although a majority of population has shifted rightward (the median upshift in the location effect), there was a slight increase of polarization in the lowest deciles in the middle of the prevalent location effect. Actually, 7.5 % of the population shifted away from the median of the distribution to the lower tail (LRP = 0.15) while some population above the median (4 %) converged from the upper tail to the median of the distribution (URP = −0.08). Thus, a positive value of MRP (0.035) gives evidences of an increased distributional polarization during the period between 1992 and 2000, we can also infer that the grown polarization has contributions from the both directions.

  28. Assuming that readers are now more familiar with interpreting relative density display, I do not report the kernel density plots for the second dimension due to the limit of space. The kernel density plots for the cultural issues scale are available upon request from the author.

  29. According to the polarization summary statistics, only two pairs (1988:1996 and 1996:2004) report positive values of MRP among the four different sets of RD pairs, while the first panel covering a comprehensive range of years (1988:2008) exhibits a negative MRP (−0.016) for the cultural issues dimension.

  30. Note that MRP (0.002) = ½ [LRP (0.079) + URP (−0.075)].

  31. There are some exceptional studies that go beyond this mean-based strategy in polarization analysis. To my knowledge, Levendusky and Pope (2011) show a good example of this trend that attempts to analyze the degree of distributional overlap by looking at the entire distribution of mass opinion.

  32. According to Treier (2010) who estimates the president’s ideological positions by analyzing both congressional roll call data (Poole) and signed bills (107th–109th), Clinton’s position ranges from liberal to moderate, while Bush’s position spans from moderate to extremely conservative positions.

  33. The derivative of the inverse function \(Q_{0} (r) = F_{0}^{ - 1} (r)\) is defined as \(\frac{1}{{f_{0} (F_{0}^{ - 1} \left( r \right))}} = \frac{1}{{f_{0} (Q_{0} (r))}}\) where \(\frac{{dF_{0} \left( r \right)}}{dr} = f_{0} \left( r \right)\).

  34. Namely, the overall RD \(\frac{{f\left( {y_{r} } \right)}}{{f_{0} (y_{r} )}}\) = the location RD \(\frac{{f_{0L} \left( {y_{r} } \right)}}{{f_{0} (y_{r} )}}\) × the shape RD \(\frac{{f\left( {y_{r} } \right)}}{{f_{0L} (y_{r} )}}\).

  35. According to Handcock and Morris (1999, 63), summary measures presented by the relative distribution framework are robust because of the two properties inherent in the method. First, by rescaling of the comparison distribution to the reference distribution, the impact of outliers is limited in the relative distribution. Second, the summary measures in the method are fully non-parametric, thus the absence of parametric assumptions means there are fewer assumptions to deviate.

  36. The median-adjustment of location is chosen over the mean-adjustment because the median is more robust to skewed distributions.

  37. The relative data \(R_{0L}\) is continuous on the outcome space [0, 1].

  38. Please note that \(R_{0L}\) takes the uniform distribution defined on [0, 1] if the reference and comparison distributions are identical, and in this case, \(4\mathop \smallint \nolimits_{0}^{1} \left| {r - \frac{1}{2}} \right|dr - 1 = 0, where\;r \in [0,1]\). Another special case occurs when the comparison population is only concentrated on the center (e.g., median) of the distribution. In this case, \(R_{0L}\) is the constant random variable taking the value of ½, thus \(MRP\left( {F;F_{0} } \right) = - 1 since \, E\left[ {\left| {R_{0L} - \frac{1}{2}} \right|} \right] = 0.\) Finally, the most extreme special case of relative polarization occurs when half of the comparison cohort takes a value equal to the minimum of the reference cohort, while the other half takes a value equal to the maximum of the reference. In this case, \(R_{0L}\) will only take either 0 or 1 with the same probability of ½. Therefore, \(MRP\left( {F;F_{0} } \right) = 1 since \, E\left[ {\left| {R_{0L} - \frac{1}{2}} \right|} \right] = \frac{1}{2}\).


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Correspondence to Jae Mook Lee.


Appendix 1: An Exclusion of “Don’t Knows (Haven’t Thought Much about It)” Category from Moderates

As mentioned above, it is conventional to include “don’t knows (haven’t thought much about it)” group to the middle of the roaders in the scholarly literature of mass opinion polarization (e.g., Abramowitz and Saunders 2008, 544; Campbell 2006, 157, Fiorina and Abrams 2011). In this appendix, these “don’t knows (DK)” answers are recoded as missing values, namely these are regarded separately from moderates.

Figure 4 provides two separate sets of kernel density plots comparing between when DKs are considered as the middle of the roaders (1st column) and when those answers are excluded from moderates (2nd column). Focusing on the aspect of growing distributional polarization, density reduction in the middle becomes much clearer on the government guarantee scale comparing ideological distributions between 1984 and 2008, as I remove the DKs from moderates. The density gap on the center areas (around scale 4) between the distribution of 1984 (a dotted line) and the distribution of 2008 (a line) is greater in the second column excluding DKs than in the first column including DKs for the government guarantee issues scale. That is, this increased gap of density differences could be attributed to an exclusion of those non-ideologicals from moderates. In contrast to the primary government guarantee scale, the cultural issues scale does not present dramatic differences in terms of distributional polarization between the two columns in the Fig. 4, as this scale does not show strong evidence of growing ideological polarization over time.

Fig. 4
figure 4

Kernel density plots of the two ideological issue scales

Corresponding to the exploratory kernel density analysis, relative distributional analysis also presents much clearer evidence of distributional polarization on the government guarantee issues scale as I exclude DKs from moderates. As shown in the Fig. 5, the overall RD effect evidences that relative density in distributional center comprising 4th through 6th deciles is dramatically reduced in 2008 compared to the distribution of 1984 while both extremes of distribution of the government guarantee issue scale—especially the liberal end of the distribution—gain significant amount of density during the period. The disappearing center appears to be even deeper as well as more apparent compared to the distributional analysis including DKs into moderates. Both the location and shape effects contribute together to the distributional changes for the period given that entire distribution of the primary scale shift to the left (more liberal side) while polarization is occurring on the distributional shape. Unlike the government guarantee scale however, the cultural issues scale does not exhibit significant differences in the RD analysis as I remove the DKs from moderates. In sum, both kernel density analysis and corresponding RD method indicate that growing polarization of mass ideological distribution over time are more clearly identified when the non-ideologicals (respondents who answered “haven’t thought much about it”) are handled separately from middle of roaders.

Fig. 5
figure 5

Relative density of the two ideological issue scales (1980–2008)

Appendix 2: American National Election Studies (ANES) Questionnaires Used in the Study

As described in the main text, a group of indicators is adopted from a cumulative data file (1948-2008) to construct ideological scales used in this study. In this appendix, I describe each issue item used here and identify the years from which it was available.


  • - [VCF0803] R Placement: Liberal-Conservative Scale (available from 1972)

  • - [VCF0806] R Placement: Government Health Insurance Scale (available from 1970)

  • - [VCF0809] R Placement: Guaranteed Jobs and Income Scale (available from 1972)

  • - [VCF0830] R Placement: Aid to Blacks Scale (available from 1970)

  • - [VCF0839] R Placement: Government Services/Spending Scale (available from 1982)

  • - [VCF0843] R Placement: Defense Spending Scale (available from 1980)


  • - [VCF0834] R Placement: Women Equal Role Scale (available from 1972)

  • - [VCF0837] R Opinion: When Should Abortion Be Allowed (available in 1972, 1976, and 1980)

  • - [VCF0838] R Opinion: By Law, When Should Abortion Be Allowed (available from 1980)

  • - [VCF0876a] R Opinion Strength: Law Against Homosexual Discrimination (available from 1988)

  • - [VCF0851] R Opinion: Newer Lifestyles Contributes to Society Breakdown (available from 1986)

  • - [VCF0854] R Opinion: Tolerance of Different Moral Standards (Available from 1986)

Appendix 3: Relative Distribution Method (Handcock and Morris 1999)

Formally, let Y 0 be a random outcome with its CDF (cumulative distribution function) F 0(Y 0) in the reference population. Suppose we also observe another outcome variable Y from a different (comparison) population of which CDF is given as F(Y). In general, Y is the distribution for a different group or the same group measured in different time periods. The RD (or “relative rank”) of Y to Y 0 is defined as the distribution of a random variable R = F 0(Y) and this implies “the proportion of the reference population whose values are at most Y” (Handcock and Morris 1999, 21; Hao and Naiman 2010, 65–66). R is obtained from Y by transforming the value by the CDF for the reference population F 0. The transformation is also referred to as the “grade transformation” and the resulting data, which is continuous on the outcome space [0,1] is called relative data. As a random variable, the relative data R (and its realization r) has a CDF and a corresponding PDF (probability density function). The CDF of R, G(r) is expressed as follows;

$$G\left( r \right) = P\left[ {R \le r} \right] = P\left[ {F_{0} \left( Y \right) \le r} \right] = P\left[ {Y \le F_{0}^{ - 1} \left( r \right)} \right] = P\left[ {Y \le Q_{0} \left( r \right)} \right] = F(Q_{0} \left( r \right))$$

where \(r \in [0,1]\) is the proportion of values, and \(Q_{0} \left( r \right) = F_{0}^{ - 1} (r)\) is the quantile function of \(F_{0}\) (Handcock and Morris 1999, 22; Hao and Naiman 2010, 66). The corresponding PDF (or relative density of R) is obtained as the derivative of CDF, G(r);Footnote 33

$$g\left( r \right) = \frac{{f(Q_{0} \left( r \right))}}{{f_{0} (Q_{0} \left( r \right))}} = \frac{{f(y_{r} )}}{{f_{0} (y_{r} )}}$$

where y r denotes the rth quantile of R (\({\text{i}} . {\text{e}} . ,Q_{0} (r)\)) using the original measurement scale of the reference population.

According to Handcock and Morris (1999, 22–26; see also Hao and Naiman 2010, 68–69), the RD method is an intuitively appealing strategy for comparing two distributions because the relative data, and its density function have clear interpretation. The PDF \(g(r)\) can be considered as a density ratio: the ratio of the fraction of values in the comparison cohort to the fraction in the reference cohort at a given level of outcome values. So if \(F_{0} (Y_{0} )\) and \(F(Y)\) are almost identical distributions, the display of the relative PDF will converge to the uniform distribution\(g(r) = 1\). On the other hand, if a greater frequency is observed at a certain decile (percentile) of interest in the comparison distribution, the relative density is greater than \(g\left( r \right) = 1.\) In contrast, if less frequency is observed in \(F(Y)\) compared to the frequency observations in \(F_{0} (Y)\), it is less than the horizontal line \(g\left( r \right) = 1\).

Distributional differences can be decomposed into location and shape shifts (Handcock and Morris 1999, 41–47). Formally, if we assume a hypothetical distribution \(Y_{0L}\) which is constructed to have the same location (median) with a comparison cohort (\(Y\)) while the distribution maintains same shape with a reference group (\(Y_{0} )\). This location-adjusted distribution \(Y_{0L}\) can be represented as \(Y_{0} - \delta\) for an additive median shift, where \(\delta\) is the difference between the medians of Y and \(Y_{0} (i.e., median\left( Y \right) - median\left( {Y_{0} } \right)\)). The corresponding CDF and PDF of \(Y_{0L}\) can be denoted as \(F_{0L} \left( Y \right) = F_{0} (Y + \delta )\) and \(f_{0L} = f_{0} (Y + \delta )\) respectively.

Using these three distributions—\(f_{0} \left( {Y_{0} } \right), f\left( Y \right), and f_{0L} (Y)\)—we can derive two relative densities (RDs) that account for the location and shape effects respectively. We define the RD of \(Y to Y_{0}\) as \(R_{0} \equiv F_{0} \left( Y \right),\) where \(F_{0}\) is a CDF of the reference group \(Y_{0}\). Hence, the first RD of \(f_{0L} (Y)\) to \(f_{0} (Y_{0} )\), or \(R_{0}^{0L} = F_{0} \left( {Y_{0L} } \right) = F_{0} (Y_{0} - \delta )\) represents the location shift between the comparison and the reference group. This location effect will display the uniform distribution defined in [0,1] if the reference and the comparison distributions have the same median. In contrast, if they have different medians, the location shift will display monotonically increasing (or decreasing) graphs in r (relative data), if the comparison median is greater (or smaller) than the reference median. Second, the RD of \(f(Y)\) to \(f_{0L} (Y)\), or \(R_{0L} = F_{0L} \left( Y \right) = F_{0} (Y + \delta )\) represents the shape shift isolated from the location effect between the two distributions. Thus, the shape effect will converge to the uniform distribution defined in [0,1] if the comparison distribution and the location-adjusted reference distribution have almost identical distributional shape. Moreover, this shape effect allows researchers to determine if distributional upgrading or downgrading—the shift of distributional mass to the upper or lower tails—occurred, or otherwise, if convergence toward the median happened in the distribution. Finally, these two RDs compose together the overall relative density \(R_{0} = F_{0} (Y)\) between the comparison \(f(y)\) and the reference \(f_{0} (Y)\) distributions.Footnote 34

While graphical analysis is a key component of the RD method, readers might still want to have measures to summarize distributional differences or changes. To meet the needs, the RD method also provides summary measures which are robust to both outliers and to violations of parametric assumptions (Handcock and Morris 1999, Chapter 5).Footnote 35 The median relative polarization (MRP) index evaluates to what extent a comparison distribution is more polarized than a reference distribution (Hao and Naiman 2010, 86). The measure compares a reference and a comparison distribution at their tails to see if the former stretches wider (narrower) or has heavier (lighter) tails than the latter. As we have discussed above, assessing polarization is often relevant to a shape-effect, so locations are adjusted to isolate differences due to the distributional shape. Thus, MRP is defined in terms of the RD of the comparison distribution to the location-matched reference distribution, where the location is adjusted to equalize the median (or mean) between the two distributions (Handcock and Morris 1999, 70; Hao and Naiman 2010, 86).Footnote 36 Intuitively, the polarization statistic measures the deviations of the relative density from the uniform distribution, where the uniform relative density indicates that the reference and comparison distributions coincide. We defined above \(R_{0L}\) as the RD between the comparison distribution (\(Y\)) and the location-matched reference distribution (\(Y_{0L}\)). That is, \(R_{0L} = F_{0L} (Y) = F_{0} \left( {Y + \delta } \right)\) where \(\delta = Q\left( \frac{1}{2} \right) - Q_{0} \left( \frac{1}{2} \right)\) is the difference between the medians of Y and \(Y_{0}\). We then measure the mean absolute deviation from ½ and the relative polarization index is constructed using a linear transformation (four times the mean absolute deviation from ½ minus 1).Footnote 37 Namely, the median relative polarization index of Y relative \(Y_{0}\) is defined as (Handcock and Morris 1999, 70–71);

$$MRP\left( {F;F_{0} } \right) = 4E\left[ {\left| {R_{0L} - \frac{1}{2}} \right|} \right] - 1 = 4\mathop \smallint \limits_{0}^{1} \left| {r - \frac{1}{2}} \right|g_{0L} \left( r \right)dr - 1$$

where \(g_{0L}\) is the density of \(R_{0L}\).

After a linear transformation, the index ranges from −1 to 1. The zero score means there are no differences in distributional shapes, the positive values of MRP (closer to 1) index implies that the comparison distribution is more polarized than the reference distribution, and the negative values (closer to −1) indicates the comparison is less polarized than the reference distribution.Footnote 38 Intuitively, the value of the MRP can be interpreted as a proportional shift of the population in the distribution from more central to less central locations. For example, a MRP value of 0.382 implies 38.2 % of the population shift from the middle of the distribution to the upper and lower percentiles. In contrast, a MRP of −0.287 means that 28.7 % of the population converges from the two tails toward the center of the distribution.

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Lee, J.M. Assessing Mass Opinion Polarization in the US Using Relative Distribution Method. Soc Indic Res 124, 571–598 (2015).

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