Copyright © 2007 by Paul Niquette. All rights reserved. 

ost people like to
characterize statistics using one of the three mwords...
The Revenge of
the DAR puzzle
describes an unusual case wherein the mean is
twice the median (while making the mode
moot, since all the test scores are different from one
another, each appearing only once).
We might reasonably suppose that the bluebook grades are percentages. Better still, let's make that an explicit assumption and, while we're at it, that scores are all integers. Now, if the number of students is odd, the median will be one of the scores and therefore an integer. If the number of students is even, it is customary to average the two middlemost numbers to calculate the median. That might result in a noninteger median. We shall put this possibility to the side for the moment. ure enough,
the instructor sees that the grade average is raised
by the student who is described by classmates, without
affection, as a DAR. We note
that a DAR, being the outlier, has no effect
on the median. Observations...
Let n = the number of
students, and s[i] = the score of the i^{th}
student in ascending order, such that s[n]
= s[DAR]. mean = {s[1] + s[2] + ... + s[n1] + s[DAR]} / ne proceed deterministically and ascertain that the total number of students who took the test cannot be more than 13. If n = 13, the lowest score must have been zero, and the others are separated by one. The test scores are as follows: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 90% The instructor must have been
disappointed with those students. Except for
one.
The table above shows the range of possible scores for n < 13. For n > 13, the conditions of the Revenge of the DAR puzzle cannot be met. If n = 15, the next higher odd integer, the DAR would have to be given bonus points to reach 119%. 
Revenge of the DAR derives its title from an acronym. You are invited to figure out its definition from this table of 55 acronyms courtesy of The Free Dictionary by Farlex...
s for the
'revenge' part, suppose that a passing grade requires
a score of at least half the class average. How
many of the 13 students flunk the exam as a
consequence of the DAR's aggressive study habits?




