When only a count of people present on certain days is available – not a record per person – the person-days over the period are the mean of those counts scaled to the period's length.
Details
Lakner's \(X_t = (\frac{1}{C}\sum N_i) t\) (1976, p.21). The counts need not cover every day: counting on weekdays only is the usual case, and the mean of the days counted stands in for the days not counted. That substitution is an assumption, and it fails if the days counted differ systematically from the days missed – weekday-only counting where weekend admissions are released before Monday, for instance.
Examples
# Lakner (p.21): counts on 255 days summing to 34,935 imply just over
# fifty thousand detention days across the year.
adp_from_counts(rep(34935 / 255, 255))
#> [1] 50005