Irrational old man conducted test involving medium (10)
I believe the answer is:
paranormal
'irrational' is the definition.
'old man conducted test involving medium' is the wordplay.
'old man' becomes 'pa' (both can mean father).
'conducted' becomes 'ran'.
'test' becomes 'oral' (oral is a type of examination**).
'involving' means one lot of letters goes inside another.
'medium' becomes 'm' (as in UK clothing sizes).
'oral' enclosing 'm' is 'ormal'.
'pa'+'ran'+'ormal'='PARANORMAL'
(Other definitions for paranormal that I've seen before include "Beyond scientific understanding" , "Supernatural" , "Scientifically inexplicable" , "mystic" , "on a higher plane?" .)