Does life-satisfaction inequality measure societal inequality? A focal-value-rounding critique

The dispersion of self-reported life satisfaction has been proposed and used as a comprehensive measure of societal inequality. A negative cross-country association between mean life satisfaction and its standard deviation has been read as evidence that this inequality is itself welfare-relevant, but critics have pointed to the nonlinearity and boundedness of response scales. I describe a further problem: a substantial and predictable share of respondents simplify the 0--10 scale to the subset {0, 5, 10} --- "focal-value rounding" (FVR) --- a behaviour that breaks not only linearity of the scale but even its order. I derive a sharp bound on the bias that FVR can induce in the standard-deviation; at empirically-typical FVR fractions the possible bias is roughly half the cross-country range of dispersion. I estimate and correct for the bias by fitting a model of FVR behavior to the Gallup World Poll Cantril ladder and three other multi-country life evaluations. FVR inflates measured standard deviation in 133 of 135 countries, by a median of 0.090 points. However, because the corrections are nearly uniform in sign across countries, rankings of the standard deviation survive essentially intact. The ordinal inequality indices advocated as the theoretically appropriate alternative suffer even worse from FVR. The correlation between mean and standard deviation survives FVR correction with modest attenuation and some caveats. Thus FVR does not, by itself, overturn previously reported correlations; it does, however, add a further reason for caution in interpreting scalar measures of subjective wellbeing inequality: candidate statistics are differently contaminated, the ordinal repairs no less than the cardinal originals, and they disagree with one another. Lastly, I show that FVR-correction can improve the pairwise dominance-comparability of full distributions.