Probably the following two rules can be collapsed into one. Anyway:
rulTimes = Sum[p__ x[a_]^i_, {a_, n}] -> p Symbol["s" <> ToString@i];
rulNaked = Sum[ x[a_]^i_, {a_, n}] -> Symbol["s" <> ToString@i];
a = Sum[x[r] ((Sum[x[i], {i, n}] - x[r])^2 - Sum[x[i]^2, {i, n}]), {r, n}];
a /. Sum[a__, l_List] :> (Sum[#, l] & /@ Expand@a) /. rulNaked /. rulTimes
(*s1^3 - 3 s1 s2 + s3*)