Partial sum of a list

      Accumulate@Last@Transpose[A]/Range@Length@A

{b, (b + d)/2, 1/3 (b + d + f), 1/4 (b + d + f + h)}

Clear[f,g, list];
list= {{a,b}, {c, d}, {e, f}, {g, h}}

f[list_,n_]:=Part[FoldList[Plus,Last@Transpose@list] / Range@Length@list, n]

or another version I personally prefer (sadly no operator forms for FoldList and Part)

g[list_, n_]:=list //Transpose
                   //Last
                   //FoldList[Plus, #]& 
                   //Divide[#,Range@Length@list]& 
                   //Extract[n]

I especially like that one can out-comment or partially copy the function definition to find out what the function does step by step

Edit: Instead of //Extract[n] previously //Part[#,n]& was used in the definition of g. Without regard for a potential difference in performance I now like Extract[n] better.

list = {{a, b}, {c, d}, {e, f}, {g, h}};

Mean[Take[Last /@ list, #]] & /@ Range@Length@list