(Math) Mean, Median, Mode: Dropping Lowest Scores
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#1: (Math) Mean, Median, Mode: Dropping Lowest Scores Author: adediosLocation: Angel C. de Dios PostPosted: Tue Jul 18, 2006 10:31 am
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Week of June 10, 2006; Vol. 169, No. 23

Dropping Lowest Scores
Ivars Peterson

As a school year draws to a close, grades are inevitably on the minds of both students and teachers.

When computing final grades, teachers sometimes allow students to drop the lowest score—or even several of their lowest scores—from a sequence of quizzes, tests, homework assignments, or exams. Usually, it's the teacher who makes the decision on which scores to drop in any individual case. If the goal is to maximize a student's score, that's sometimes easy to do. But, in certain situations, it can be quite tricky to tell which score or scores to drop from the total.

In the June Mathematics Magazine, Daniel M. Kane of the Massachusetts Institute of Technology and Jonathan M. Kane of the University of Wisconsin-Whitewater describe some of the intricacies of this problem and present a simple, efficient algorithm for finding the best set of grades to drop. Jonathan is a math professor, and his son Daniel is a student at MIT. Daniel was a member of the U.S.A. International Mathematical Olympiad team in 2002 and 2003.

Suppose that Alan earned the quiz scores shown below and can drop two scores.

Alan's Quiz Scores
Quiz 1 2 3 4 5
Score 2 6 24 3 6
Possible Score 8 12 40 4 24
Percentage 25 % 50 % 60 % 75 % 25 %


If his teacher computes Alan's final grade by simply adding up all the quiz scores, Alan does best if the teacher drops the scores for quizzes 1 and 4—even though quiz 4 has the highest percentage.

However, if a teacher bases a student's final grade on the ratio of total points earned to total points possible, the problem is a little more complicated.

"Intuitively, one might suspect that a way to obtain an optimal solution would be to drop those quiz grades where the student performed the worst either by obtaining the smallest number of points or by obtaining the smallest percentage grade," Daniel and Jonathan King note. "However, this is not always the case."

Consider Beth's quiz scores.

Beth's Quiz Scores
Quiz 1 2 3
Score 80 20 1
Possible Score 100 100 20
Percentage 80 % 20 % 5 %


If Beth drops quiz 1, her final grade is 17.5 percent. If she drops quiz 2, it's 67.5 percent. If she drops quiz 3, it's 50 percent.

Note that dropping the quiz 2 score leaves quiz 3, which has both the lowest raw score and the lowest percentage. This result reflects the fact that quiz 3 is worth so few points that its impact is small compared with that of quiz 2.

Carl's quiz scores demonstrate another quirk.

Carl's Quiz Scores
Quiz 1 2 3 4
Score 100 42 14 3
Possible Score 100 91 55 38
Percentage 100 % 46 % 25 % 8 %


If Carl is allowed to drop just one score, the best one to omit is quiz 4. That would give him a grade of 63.4 percent.

However, if Carl were allowed to drop two scores, it's best to omit quiz 2 and 3, and he would end up with a grade of 74.6 percent.

"Notice that the optimal deletion set of two grades does not include the best single grade to drop," Daniel and Jonathan Kane conclude. "Also surprising is how slight changes to a problem can result in radically different results."

One way to solve the problem of determining which grades to drop is by trying all the possibilities. But for large datasets, that's a time-consuming task. To drop 10 out of 100 grades, for example, you would have to consider 17,310,309,456,440 possibilities! At a rate of 1 million possibilities per second, it would take you about 200 days to go through them all.

To see the ingenious, efficient algorithm that Daniel and Jonathan Kane developed for solving the problem, check their paper (http://web.mit.edu/dankane/www/droplowest.pdf). The method appears to be particularly efficient when applied in a computer gradebook program.

Of course, another answer for teachers is to give all quizzes the same possible value.

*************************************************************

Questions to explore further this topic:

Organizing Data

http://www.bbc.co.uk/schools/r....._act.shtml
http://bobhall.tamu.edu/Finite.....ity_a.html

Histograms

http://bobhall.tamu.edu/Finite.....ity_b.html

Measures of Central Tendency

http://www.bbc.co.uk/schools/r....._act.shtml



Measures of Spread


Averages:
Mean, Mode, Median, or Midrange?


http://www.andrews.edu/~calkin.....stat03.htm
http://bdaugherty.tripod.com/K.....rages.html
http://www.explorelearning.com.....urceID=262

Arithmetic, Geometric, Harmonic, Quadratic, Trimmed, Weighted, etc. are all means!

http://www.andrews.edu/~calkin.....stat04.htm

Measures of dispersion

http://www.andrews.edu/~calkin.....stat05.htm

Introduction to Statistics (Lesson)

http://www.shodor.org/interact.....s/sm1.html
http://teachhealthk-12.uthscsa.....a-mean.htm
http://teachhealthk-12.uthscsa.....median.htm
http://teachhealthk-12.uthscsa.....c-mode.htm
http://teachhealthk-12.uthscsa.....-range.htm
http://teachhealthk-12.uthscsa.....tistic.htm
http://icp.giss.nasa.gov/education/statistics/
http://ctap295.borderlink.org/cdenton/
http://www.georgetown.edu/depa...../begin.htm
http://www.pubmedcentral.nih.g.....tid=137399

Measures: Exploration Questions

http://www.shodor.org/interact.....sheet.html

Mean, Median, Mode Discussion

http://www.shodor.org/interact.....s/sd1.html

Sample problems on mean, median mode

http://www.shodor.org/interact.....s/st4.html

Introduction to Descriptive Statistics

http://web.mit.edu/17.871/www/.....s_2006.ppt
http://www.fao.org/DOCREP/X5684E/x5684e03.htm

The Bell Curve

http://www.shodor.org/interact...../sm1a.html
http://www.andrews.edu/~calkin.....stat06.htm

GAMES

http://www.bbc.co.uk/education.....train.html
http://www.aaamath.com/B/sta.htm


Last edited by adedios on Sat Jan 27, 2007 3:08 pm; edited 1 time in total

#2: Lake Wobegon Averages Author: adediosLocation: Angel C. de Dios PostPosted: Sat Aug 12, 2006 7:19 am
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Week of Aug. 12, 2006; Vol. 170, No. 7

Lake Wobegon Averages
Ivars Peterson

On the radio program A Prairie Home Companion, host Garrison Keillor starts off each show with the words: "Welcome to Lake Wobegon, where all the women are strong, all the men are good-looking, and all the children are above average."

The tendency to treat all members of a group as above average, especially with respect to numerical values such as test scores, is now often called the Lake Wobegon effect (Word Spy). It can reflect a human propensity to overestimate one's achievements and capabilities in relation to others (Wikipedia).

But there's more to averages than you might think. Although computing averages is straightforward, the results may depend on your point of view, as mathematician Allen Schwenk of Western Michigan University demonstrates in the September College Mathematics Journal.

For the full article:

http://www.sciencenews.org/art.....thtrek.asp



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