graduation of a mortality table

Raw, observed death rates jiggle from age to age in a way that nobody believes is real: q-71 might come out higher than both q-70 and q-72 simply because, by chance, a couple of extra people died at 71 in your sample. Yet we are confident that true mortality changes smoothly with age. Graduation is the disciplined process of smoothing the jagged raw rates into a clean, gradually changing curve that is fit to publish and to price from.

The art of graduation is a tug-of-war between two goals. We want the smoothed rates to be regular and well-behaved (smoothness), but we also want them to stay faithful to the data we actually observed (fit, or adherence). Push too hard on smoothness and you erase a genuine feature, like the real bump in young-adult accident mortality; push too hard on fit and you keep the random noise. Techniques range from simple moving-weighted-average methods, through fitting a parametric law like Gompertz-Makeham, to modern spline and Bayesian methods, but all balance the same trade-off.

After graduating, an actuary checks the result with statistical tests: does it still fit the data acceptably (a chi-squared or similar goodness-of-fit test), and is it free of runs of rates all above or all below the data (a sign of over-smoothing)? Graduation matters because the published table feeds directly into premiums, reserves, and reported profits; a lumpy ungraduated table would make adjacent ages behave erratically, while an over-graduated one hides real risk. The honest aim is to remove sampling noise without inventing or destroying signal.