Some pretty striking cross-local-authority correlations have been reported in the media since the referendum result was announced. For example, the percentage voting Leave appears to have a strong negative relationship with both average education and average income across local authorities.
Among the data available on the referendum from the Electoral Commision are the percentage of ballot papers that were rejected in each local authority. Ballot papers may be rejected because the voter in question crossed both boxes, didn’t cross either box, or chose to make a political point by defacing the ballot paper in some other way (e.g., scrawling something unflattering about the Prime Minister).
I linked the referendum results to data from the 2011 Census on the demographic characteristics of local authorities. Interestingly, as the two charts below indicate, the percentage of ballot papers rejected has a strong positive association with both percentage non-white (r = .73) and percentage foreign-born (r = .71). In other words, a higher percentage of ballot papers were rejected in local authorities with more non-white residents and foreign-born residents.
I checked to see whether these associations were attributable to confounding by age or education. One might have expected percentage of ballot papers rejected to be correlated with old age or lack of qualifications, since elderly or uneducated people might be more likely to make mistakes when filling out their ballot papers. However, as the table below indicates, neither association is noticeably reduced when controlling for percentage aged 65+ or percentage with no qualifications. An obvious alternative explanation, then, is that language difficulties led some non-white/foreign-born people to make mistakes when filling out their ballot papers.
Is the effect big enough to have affected the referendum result? My back-of-the-envelope calculation suggests almost certainly not. Linear regression implies that a hypothetical local authority with 100% non-white people would have had 0.22% of ballot papers rejected, compared to 0.057% in a hypothetical local authority with 100% whites. 46 million people were eligible to vote in the referendum, of whom roughly 13% were non-white. Turnout was 72%. And the non-white vote was split about 70:30 for Remain. So if non-white people had spoiled ballot papers at the same rate as whites, there would have been approximately 46,000,000*0.13*0.72*0.7*(0.0022 – 0.00057) = ~5,000 more votes for Remain, and 46,000,000*0.13*0.72*0.3*(0.0022 – 0.00057) = ~2,100 more votes for Leave.