Yaw of Averages

Copyright 2005  by Paul Niquette. All rights reserved.

For people in a hurry, the solution to the puzzle is an Okie named...

Upon leaving the Okies and joining the Calies, Nollwyn, who is 135 centimeters tall, increases the average heights of both Okies and Calies.  Only Nollwyn can accomplish that, as you would eventually figure out if you take the trouble to move students one by one, each time re-averaging both lists.  Sophisticated solvers need not do that.

ou probably observe that out of the 26 students in the class, there are nine Okies who are each taller than the average of the Calies (Nollyn, Pailwyn, Qualyn, Ravyn, Soulyn, Ulewyn, Wailyn, Xelwyn, Zalyn) and three Calies who are each taller than the average of the Okies (Irwyn, Oldewyn, Terryn). Any or all of them moving over to the opposite group would increase the latter's average height.  The name must have been selected by the teacher from just those twelve.  Is it necessary to move each of those names one at a time, re-calculating both averages fully a dozen times?  Maybe not.  Let's figure out what the teacher might have done while the students were at recess...

Suppose there to be two groups of individuals named Group 1 and Group 2, each having some number of members, respectively N1 and N2.  Assume that every member of both groups can be characterized by some numerical parameter of interest, whether height in centimeters, as in the Yaw of Averages puzzle, or their weight in pounds or their respective test scores or the currency in their pockets or -- well, their intelligence, as in what has become known as the "Will Rogers Phenomenon."  To determine group averages, one simply divides the sums of the parameters in each group by the number of members in that group.  The sums might be represented by the symbols S1 and S2.   The respective averages are then given as S1 / N1 and S2 / N2.
Inspired by Will Rogers, we must set about to show that the averages in both groups can be increased by moving a member from one group to the other.  To do that, we simply postulate the movement of a member, whose parameter is X, from Group 1 to Group 2. The new averages will be given as (S1-X)/(N1-1) and (S2+X)/(N2+1).  What we hope to make true are the following two inequalities:
[1]  (S1-X)/(N1-1) > S1/N1 and
[2]  (S2+X)/(N2+1) > S2/N2
A little algebra music please.  From [1] we find that  X < S1/N1, and from [2] we find that X > S2/N2. Taken together these expressions say that to increase the average of Group 1, we most remove a member whose value is lower than the average of Group 1, and to increase the average for Group 2 at the same time, we must add a member whose value is higher than the average for Group 2.  We can chain the expressions to read...
S2/N2 < X < S1/N1
...which tells us that if we move a member whose parametric value lies between the averages of the two groups, the averages of both groups will increase.  It says something else, too.  For both averages to increase,  S1 / N1 > S2 / N2.  In words, the starting average for the from-group must be higher than the starting average for the to-group.  Kind of obvious, when you think about it.

If so, then the to-group must grow in number, and the from-group must shrink.  Indeed, here is how the inequalities would look in a swap...

[3]  (S1-X+Y)/N1 > S1/N1 and
[4]  (S2+X-Y)/N2 > S2/N2
From [3] we find that X > Y, and from [4] we find that X < Y.  Both cannot be simultaneously true, which mandates what might be called the Yaw of Averages (a pretentious pun invoking a nautical term, which has been appropriated by aviation).

h, but wait. Throughout our analysis, we have been making a secret assumption -- that X is the parametric value for one solitary member.  Same thing for Y.  We have learned elsewhere not to do that but instead to make our assumptions explicit.  Thus, we might consider the proposition that X represents, say, the average of parametric values for a set of members.  Thus, we have X = Sx/Nx such that S2/N2 < Sx/Nx < S1/N1.

Back in our classroom, suppose the teacher had crossed out two names of Okies -- Ravyn and Jenwyn -- and moved them both to the Calies list.  The average height of those two Okies just happens to be 135 centimeters, same as Nollyn alone.  Accordingly, the averages of both groups would increase by moving Ravyn and Jenwyn instead of -- or in addition to -- Nollyn.  In general, other combinations will be possible, each calling for the movement of various numbers of names.
Intuition seems to insist that, in moving sets between groups, there must exist some combination which will allow one-for-one swaps, thereby setting aside the Yaw of Averages.  Here is how the inequalities would look in swaps of averages...
[5]  (S1-Sx/Nx+Sy/Ny)/(N1-Nx+Ny) > S1/N1 and
[6]  (S2+Sx/Nx-Sy/Ny)/(N2+Nx-Ny) > S2/N2
...where Sx/Nx and Sy/Ny represent the respective parametric averages of the sets being swapped.  For constant group sizes, we must set Nx = Ny.  Alas, from [5] we find that Sx > Sy, and from [6] we find that Sx < Sy, and since both cannot be simultaneously true, we have confirmed the Yaw of Averages for all time.  Exclamation point optional.
The classroom exercise was obviously cooked up for Puzzles with a Purpose, the only purpose being mere aesthetics (for large values of "mere").  Sophisticated solvers may want to experiment with the Yaw of Averages by cooking up groups with parametric distributions that illustrate other features, answering questions like...
  • What is the minimum size group for which the Yaw of Averages applies?
  • What is the maximum increase that can be created for both groups?
  • What is required for the averages of both groups to decrease?
  • What is required for the Yaw of Averages to apply with more than two groups?
You are invited to send your comments and ideas to solvers@niquette.com.
Scientific Importance

s we have seen in this puzzle, the movement of one or more members of a given group into another can increase the average for some common parameter in both groups.  In another puzzle, Life in a Bathtub, we will find out that the Yaw of Averages has considerable scientific value -- mostly in avoiding trouble.

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