Animation 3: FVR Contamination

Focal value rounding creates a complex, sign-changing bias in SD and other inequality measures — an effect GHM (2018) do not consider

GHM (2018) do not consider this effect. Focal value rounding (FVR): a fraction λ of respondents round their answer to the nearest focal value — 0, 5, or 10. (Based on the Barrington-Leigh 2024 mixture model, this fraction varies from ~10% to ~60% across countries, strongly predicted by education.)

The FVR bias in SD is sign-changing and non-trivial to predict: near μ ≈ 5 (where the distribution centres on focal value 5), FVR concentrates mass and reduces observed SD. Away from μ = 5 in either direction, FVR creates a bimodal spread across two focal values (e.g. 0 and 5, or 5 and 10), which inflates SD. With σtrue=1 the sign change occurs near μ ≈ 4.2 and μ ≈ 5.8; the deflation window narrows as σtrue increases. Both Gini and P80–P20 are also distorted in complex, non-monotone ways.

Use both sliders. The left panels show the latent distribution (with FVR regions) and the true vs observed PMF. The right panels trace how SD, Gini, and P80–P20 depend on country mean SWL for both measures.
Mean SWL (μ): country mean on the 0–10 scale 7.00
FVR fraction (λ): proportion of respondents who report a focal value 0.00
μ (slider)7.00
λ (slider)0.00
Mean (true)—
Mean (obs)—
SD (true)—
SD (obs)—
SD bias—
Gini (true)—
Gini (obs)—
True/latent PMF / trace (no FVR, λ=0) — dashed line / stair outline
Observed PMF / trace (with FVR at current λ) — solid bars / solid line
true at current μ
observed at current μ

Focal values: 0 (responses 0–2 round here), 5 (responses 3–7), 10 (responses 8–10). σtrue = 1.0 (fixed). Sign of SD bias changes near μ ≈ 4.2 and μ ≈ 5.8: FVR deflates SD within that window (distribution centred on focal 5) and inflates outside it. At fixed μ, σobs is monotone in λ — direction set by μ.