1.1 The actuarial profession has for decades, indeed centuries, used age-related tables, particularly (although not exclusively) of mortality, which in preparation have been subjected to the process known as graduation. Five purposes of such graduation have recently been set out by Vallin, Pollard and Heligman and these may be summarized as follows:

(1) to smooth the data, make them easier to handle, remove irregularities and inconsistencies;

(2) to make the result more precise on the reasonable assumption that the real mortality underlying the observations is a smooth curve, i.e. to remove sampling and other errors;

(3) to aid inferences from incomplete data;

(4) to facilitate comparisons of mortality;

(5) to aid forecasting.

Of these purposes, (1), (2) and (3) may all be paraphrazed as implying that the purposes are to make the table smooth, and (4) and (5) as indicating uses to which a table may better be put if smoothed. So the purposes of what the profession has come to term ‘graduation’ all boil down to the production of a smooth table, and for the rest of this paper until the last section I shall use the word ‘smoothing’ rather than ‘graduation’. But what is smooth?