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Correct; "average" and average can be different things.
Yeah, in particular the "average age of death" might be 51 if the average includes a lot of people who died as children. OTOH, the average person dying at 51 is fundamentally different in how you think of it.
Life expectancy at, is used by academics when relevant. Average at birth, adulthood and even once they're over the hill have utility. Like identifying outliers.
Regardless, the average person is going to use average as a nebulas concept occasionally informed by science but hearsay and superstition on an average day.
*nebulous