Revenge of the DAR

Copyright 2007 by Paul Niquette. All rights reserved.

ost people like to characterize statistics using one of the three m-words...
  1. mean ~~~~ Add up all the numbers and divide by the number of numbers.
  2. median ~~ Put the numbers in numerical order and pick the middle number.
  3. mode ~~~ Find the numbers that appear most frequently.
Statisticians describe the m-words as mere measures of central tendency, and often all three come out about the same.  The really important characterizations have more to do with deviations and variances -- how the numbers are distributed within a range.  Thus do statisticians deal with numbers based on  peakedness (using platykurtic for numbers that are flattened out over the whole range and leptokurtic for numbers that are squeezed together) and skew (for numbers that are pushed toward the high or low ends of the range).

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).    

An anecdote for distinguishing mean from median takes place at a meeting of a ladies literary club.  Their mean financial resources can be estimated by averaging the amount of money in their respective checkbooks.  Chances are the median would not be much more or less than the mean

The door opens and in walks, say, Oprah Winfrey, who takes a seat.  The average wealth of the women in the room -- the mean -- suddenly takes a jump.  But the median probably will not change much.  Indeed, the median is most famous for disregarding an outlier.

We might reasonably suppose that the blue-book 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 middle-most numbers to calculate the median.  That might result in a non-integer 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...

  • To have such a pronounced effect on the mean, the DAR's score must be as high as possible.  That would be no more than 100% by our assumption. 
  • All the other scores on the blue-book test must be low enough, so that taken together they will not elevate the median and prevent the DAR from having the effect on the mean specified in the Revenge of the DAR
  • The lowest possible grade, of course, is zero, and only one student can have it.
  • Whatever the lowest grade in the class, though, the rest of the grades must be integral amounts higher, making each score unique. 
  • The smallest separation, of course, is one
  • If the number of students is even, a median formed by averaging the two middle-most numbers will not be an integer.  So we conclude that the number of students is odd. 
  • Computed as exactly twice the median, the mean is likewise an integer.

Let n = the number of students, and s[i] = the score of the ith student in ascending order, such that s[n] = s[DAR]. 

From what is known, we write...

mean = {s[1] + s[2] + ... + s[n-1] + s[DAR]} / n 
median = s[(n + 1) / 2]

{s[1] + s[2] + ... +s[n-1] + s[DAR]} / n = 2 s[(n + 1) / 2]

s[DAR] = 2 n s[(n + 1) / 2] - {s[1] + s[2] + ... + s[n-1]}
We 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.
 
n =
3min
3max
5min
5max
7min
7max
9min
9max
11min
11max
13
lowest
0
23
0
14
0
9
0
5
0
2
0
median
1
24
2
16
3
13
4
9
5
7
6
mean
2
48
4
32
6
26
8
18
10
14
12
DAR
5
97
14
98
27
99
44
94
65
89
90

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...
 
 
Dynamic Alternative Routing, Dynamic Address Resolution (Cisco), Dynamic Adaptive Routing (Cisco) Dryer, Compressed Air, Refrigerated, Dry Adiabatic Rate (meteorology), Driving After Revocation (citation), Drawing Analysis Record, Double Address Rate, Dominion Atlantic Railway (Nova Scotia), Domestic Animals Registries Inc, Document Approval and Release, Division of Animal Resources, Distributed Array Radar, Distortion Adaptive Receiver, Display Aspect Ratio, Disc Area Ratio, Disabled Aircraft Recovery, Direct Assistance Request, Direct and Alternate Route, Digital Autopilot Requirements, Digital Audio Recorder, Digital Audio Radio, Digital Assets Repository, Digital Aids Recorder (Airbus Industrie), Dielectric Absorption Rate, Dial-A-Ride (senior program), Diagnostic Assessment of Reading, Deviation Authorization Request, Deviation Approval Request, Detector-Array Receiver, Designated Area of Recovery (combat search and rescue), Designated Airworthiness Representative, Design Assessment Report, Department of Agrarian Reform (Philippines), Defense Acquisition Regulation, Defense Acquisition Radar, Defect Analysis Report (software development), Declassification Areas, Decision Analysis and Resolution, Day-After Recall, Daughters of the American Revolution, Data Automation Requirement, Data Automation Request, Data At Rest, Data Analysis Report, Data Aided Receiver, Data Acquisition Requirement, Data Acquisition Request, Data Acquisition Recording, Data Access Register, Darier-White Disease, Dar Es Salaam, Tanzania - International (Airport Code), Damned Average Raiser (testing), Damage Assessment Routines, Daily Activity Report

As 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?
 


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