Here's an attempt (without select)

    lst = {1, 2, 2, 1, 2, 5, 2, 4};
    n = 4;
    p = 1;
    Flatten[If[Abs[Take[#, -1] - Mean[Drop[#, -1]]][[1]] > p *StandardDeviation[Drop[#, -1]],
     Take[#, -1], {}] & /@ Partition[lst, n + 1, 1, 1]] // First

Alternatively with Select.

    Flatten@Select[Partition[lst, n + 1, 1, 1], 
       Abs[Take[#, -1] - Mean[Drop[#, -1]]][[1]] > p*StandardDeviation[Drop[#, -1]] &] // Last