data = Transpose@{{44, 42, 42, 41, 41, 41, 39}, {"Solomon", "Jason",
     "Errol", "Garry", "Bernard", "Barry", "Stephen"}};

rank[data_, type_] :=
 Module[{t = Transpose@{Sort@data, Range[Length@data, 1, -1]}},
  Switch[type,
   "standard", data/.Rule@@@First/@SplitBy[t, First],
   "modified", data/.Rule@@@Last/@SplitBy[t, First],
   "dense", data/.Thread[#->Range[Length@#]]&@SplitBy[t, First][[All, 1, 1]],
   "ordinal", Reverse@Ordering[data],
   "fractional", data/.Rule@@@(Mean[#]/.{a_Rational:>N[a]}&)/@ SplitBy[t, First]]]

fmtRankedData[data_, type_] :=
 Labeled[Grid[
   SortBy[ArrayFlatten@{{Transpose@{rank[data[[All, 1]], type]},
       data}}, First], Alignment->Left], type<>" ranking:", Top]

Grid@{fmtRankedData[data, #] & /@ {"standard", "modified", "dense",
    "ordinal", "fractional"}}