Copyright ©2005 by Paul Niquette. All rights reserved. 
For people in a hurry,
the solution to the
puzzle is an Okie named...

Nollyn 
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, recalculating 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 N_{1} and N_{2}. 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 S_{1} and S_{2}. The respective averages are then given as S_{1 }/ N_{1} and S_{2 }/ N_{2}.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 (S_{1}X)/(N_{1}1) and (S_{2}+X)/(N_{2}+1). What we hope to make true are the following two inequalities: [1] (S_{1}X)/(N_{1}1) > S_{1}/N_{1} andA little algebra music please. From [1] we find that _{ }X < S_{1}/N_{1}, and from [2] we find that X > S_{2}/N_{2}. 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... S_{2}/N_{2 }< X < S_{1}/N_{1}...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 fromgroup must be higher than the starting average for the togroup. Kind of obvious, when you think about it. If so, then the togroup must grow in number, and the fromgroup must shrink. Indeed, here is how the inequalities would look in a swap... [3] (S_{1}X+Y)/N_{1} > S_{1}/N_{1} andFrom [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 = S_{x}/N_{x} such that S_{2}/N_{2 }< S_{x}/N_{x} < S_{1}/N_{1}. 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 oneforone swaps, thereby setting aside the Yaw of Averages. Here is how the inequalities would look in swaps of averages... [5] (S_{1}S_{x}/N_{x}+S_{y}/N_{y})/(N_{1}N_{x}+N_{y}) > S_{1}/N_{1} and...where S_{x}/N_{x }and S_{y}/N_{y} represent the respective parametric averages of the sets being swapped. For constant group sizes, we must set N_{x} = N_{y}. Alas, from [5] we find that S_{x }> S_{y}, and from [6] we find that S_{x }< S_{y}, 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... 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.




