For average of a group in Excel 2016, you can use AVERAGEIF. For the other statistics, you use this pattern:

{=STAT(IF(group_range=group,data_range))}

Where STAT is the chosen statistical function (In your case, one of MEDIAN, MIN or MAX). Note that the braces {...} are to indicate that this is an array formula. They are not typed into the formula bar or the cell. You type the formula, then press Ctrl+Shift+Enter and this converts your formula to an array formula.

So:

The formulas in column E:

=AVERAGEIF($A$2:$A$13,E$1,$B$2:$B$13)
{=MEDIAN(IF($A$2:$A$13=E$1,$B$2:$B$13))}
{=MIN(IF($A$2:$A$13=E$1,$B$2:$B$13))}
{=MAX(IF($A$2:$A$13=E$1,$B$2:$B$13))}

If one has Office 365 with the dynamic array formula use:

=MEDIAN(INDEX(A2:A6,MATCH(SEQUENCE(SUM(B2:B6),,0),SUMIF(OFFSET(B1,0,,ROW(B2:B6)-MIN(ROW(B2:B6))+1,),"<>"))))

and

=MODE.SNGL(INDEX(A2:A6,MATCH(SEQUENCE(SUM(B2:B6),,0),SUMIF(OFFSET(B1,0,,ROW(B2:B6)-MIN(ROW(B2:B6))+1,),"<>"))))

If not then this array formula:

=MEDIAN(INDEX(A2:A6,MATCH(ROW($ZZ1:INDEX($ZZ:$ZZ,SUM(B2:B6)))-1,SUMIF(OFFSET(B1,0,,ROW(B2:B6)-MIN(ROW(B2:B6))+1,),"<>"))))

and

=MODE.SNGL(INDEX(A2:A6,MATCH(ROW($ZZ1:INDEX($ZZ:$ZZ,SUM(B2:B6)))-1,SUMIF(OFFSET(B1,0,,ROW(B2:B6)-MIN(ROW(B2:B6))+1,),"<>"))))

Being array formula, then need to be confirmed with Ctrl-Shift-Enter instead of Enter when exiting edit mode.