I've never seen AND and OR work in an array formula. You can get around this
by using multiplication for AND and addition for OR:
=AVERAGE(IF(($E$2:$E$6<I6)*($E$2:$E$6<I5),$D$2:$D$ 6,""))

although I think you could write the formula as
=AVERAGE(IF($E$2:$E$6<MIN(I6,I5),$D$2:$D$6,""))


"PAL" wrote:

I am trying to get the average of several numbers using arrays. I want
average only if the date with the values are associated with a date range.

{=AVERAGE(IF(AND($E$2:$E$6<I6,$E$2:$E$6<I5),$D$2:$ D$6,""))}