Intersecting counts

Cross-tabulate two characteristics and the sample beneath each cell shrinks fast — sometimes below the threshold at which any estimate is publishable.

When a population divides twice

A national survey that interviews around 100,000 adults in a year looks, on first reading, like more than enough to support detailed analysis. It is — until the question asks not just "disabled" but "disabled and from a particular ethnic group" or "disabled women in part-time employment". Each split divides the sample; two splits multiply the divisions. A disabled population that represents roughly a fifth of adults becomes, after crossing with a ten-category ethnicity classification and a handful of employment categories, a set of cells in which some hold fewer than fifty observations. At that point, the confidence interval around any estimate is so wide that the figure cannot reliably be distinguished from its neighbours.

The Office for National Statistics and the Department for Work and Pensions both publish guidance on the minimum cell sizes below which estimates are suppressed or flagged. The Annual Population Survey — the principal source for regional and demographic breakdowns of disability employment — typically suppresses figures where the unweighted base falls below a threshold in the low tens. The exact threshold varies by release; the point is that suppression is not a failure of the statistical system but a signal that the question being asked is harder to answer than the headline sample size implies.

Crossing disability with ethnicity is the intersection that most consistently produces suppressed cells in published bulletins. This is not because disabled people from ethnic minority groups are absent from surveys; it is because no single continuous survey is large enough to distribute that population across all the sub-groups the analysis demands while keeping each cell above the publishable minimum. The UK Biobank and linked administrative datasets offer larger denominators but cover different populations and carry their own comparability over time problems when the definition of disability used in the source does not match the one used in the headline count.

The practical consequence is a pattern of named gaps and unmeasured gaps sitting alongside each other in the same release. The employment gap for disabled people overall is reported annually. The employment gap for disabled people of a specific ethnic background, in a specific region, may carry a suppression marker — or may simply not appear in the table. The missing cells are rarely addressed through larger commissioned samples, so the gaps accumulate release by release.

Intersecting counts, in short, expose a structural limit: the sample size that justifies a headline figure is rarely the sample size that justifies the breakdown a policy question actually needs. Reading any bulletin that reports a gap without checking whether the underlying cells are published, suppressed or absent is reading only part of the release.