SortBy

Let's say we want to Sort a List as if a function were applied to the List but return the original elements of the List. That is what distinguishes SortBy from Sort. For example, say you want to sort 10 rolls of a die 3 times by the number of 6s that come up.

In[265]:= rollADie = Table[RandomChoice[Range@6, 3], {10}]

Out[265]= {{3, 5, 4}, {2, 1, 5}, {6, 3, 4}, {6, 6, 6}, {5, 2, 2}, {1, 3, 1}, {5, 4,
  4}, {2, 4, 1}, {2, 2, 3}, {4, 1, 4}}

In[266]:= SortBy[rollADie, Count[#, 6] &] // Reverse

Out[266]= {{6, 6, 6}, {6, 3, 4}, {5, 4, 4}, {5, 2, 2}, {4, 1, 4}, {3, 5, 4}, {2, 4,
  1}, {2, 2, 3}, {2, 1, 5}, {1, 3, 1}}

A way to think about SortBy is to consider that it can only "see" the result of applying the function to the list elements and Sorts by that result. For instance, we can write more compact expressions for sorting by one List position than with Sort. Here SortBy only sees the 2nd element of each List and Sorts by it (in ascending order since that is Sort's default).

In[267]:= SortBy[rollADie, #[[2]] &]

Out[267]= {{2, 1, 5}, {4, 1, 4}, {2, 2, 3}, {5, 2, 2}, {1, 3, 1}, {6, 3, 4}, {2, 4,
  1}, {5, 4, 4}, {3, 5, 4}, {6, 6, 6}}

SortBy can take more than one sorting function and applies them in order to break ties. In this sense SortBy can be used like successive sorts in a database or program like Microsoft Excel used as a database.

In[268]:= SortBy[rollADie, {#[[2]] &, Last}]

Out[268]= {{4, 1, 4}, {2, 1, 5}, {5, 2, 2}, {2, 2, 3}, {1, 3, 1}, {6, 3, 4}, {2, 4,
  1}, {5, 4, 4}, {3, 5, 4}, {6, 6, 6}}

 SortBy[e,f] is equivalent to Sort[{f[#],#}&/@e][[All,-1]].

Last, the Doc Center unveils the function underlying SortBy: SortBy[e,f]is equivalent to Sort[{f[#],#}&/@e][[All,-1]]. Translated into English, this says "make a List of the function f applied to each argument and that argument, then Sort by the first element of the List (i.e. the function applied to the argument), then only return the 2nd element of each List (i.e., the original List elements, but now re-ordered by the Sort.