Problem Statement

A particular activity of bats occurs at these times of the day: Using the idea that there are twenty-four hours in a day, which is analogous to there being 360 degrees in a circle, map times of day to and from angles; and using the ideas of Averages/Mean angle compute and show the average time of the nocturnal activity to an accuracy of one second of time.

Let's start with the solution:

(load "@lib/math.l")

(de meanTime (Lst)
   (let Tim
      (*/
         (atan2
            (sum '((S) (sin (*/ ($tim S) pi 43200))) Lst)
            (sum '((S) (cos (*/ ($tim S) pi 43200))) Lst) )
         43200 pi )
      (tim$ (% (+ Tim 86400) 86400) T) ) )

: (meanTime '("23:00:17" "23:40:20" "00:12:45" "00:17:19"))
-> "23:47:43"