Well, I don't know if this is exactly what you want: (since you haven't provided me AllData)
Say I generate a number of Spec lists of two variables:
Spec[j_] = Table[Partition[RandomSample[Range[40], 20], 2], {i, 1, 10}][[j]] // Quiet;
and any element say,
Spec[1]
(*{{2, 1}, {24, 14}, {17, 28}, {34, 20}, {29, 5}, {8, 15}, {12, 39}, {21, 6}, {10, 35}, {22, 4}}*)
Now, I want to normalize all y's to 0,1 and Thread them together. So, I get, say, Newspec:
NewSpec[i_] := Thread[{Spec[i][[All, 1]], Normalize[Spec[i][[All, 2]]]}] // N
Your new list with normalized y values is:
NewSpec[1]
(*{{2., 0.0150261}, {24., 0.210366}, {17., 0.420732}, {34., 0.300523}, {29., 0.0751307}, {8., 0.225392}, {12., 0.586019}, {21., 0.0901568}, {10., 0.525915}, {22., 0.0601045}}*)