We address a difficulty in appreciating that the pair-difference chi-squared-like statistic (Steele et al 2002 Metrologia 39 269-77, Douglas and Steele 2006 Metrologia 43 89-97) is truly unmediated (i.e. independent of any reference value), namely the incorrect thought that this aggregating statistic might merely be hiding a reference value that has the same chi-squared. We show that the equation for this reference value commonly has no real solution. Regarding pair-difference statistics as unmediated statistics conveniently avoids any complex reference value, and offers a stark simplicity that exactly captures the simplest way in which measurements are really used in practical chains of metrological inference.
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