Biased Rectangles?

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Participants from the Data and Probability Working Group developed a Fathom (Key Curriculum Press) lesson to support the first activity in Navigating Through Data Analysis: Grades 9 - 12 (Burrill, et al, 2003). The activity entitled Biased Rectangles? is a simulation illustrating bias. The original scenario comes from Random Rectangles found in Activity-Based Statistics (Scheaffer, et al, 1996).

Objective:
The intention of the activity is to demonstrate to students how bias affects mean. Students are asked to select 5 representative rectangles from the 100 rectangles shown on the 100 rectangles sheet. Each member of the class is asked to calculate the mean of their sample. The information is then collected and summarized for the class. The mean of the samples is compared to the actual mean of all 100 rectangles. Note: the tendency is for the mean to be higher than the actual mean, illustrating bias.
 
Student Background:
Students need to know and understand how to calculate mean, as well as how to take a sample.
 
Equipment needed:
100 rectangles sheet, computer with Fathom software.
 
Lesson:
Provide each student with a copy of the 100 rectangles sheet. Students should look at the sheet for an equal amount of time and only for a few seconds. Have them select 5 representative rectangles recording the number. Find the area of each selected rectangle. Compute the mean of each sample. Collect the class results. Find the mean for the class and compare to the actual mean for all rectangles.
 
The handout created, to use with the Fathom software, can now be used as a demonstration with the class or as an individual student exploration. The activity is very detailed and can be done by a relatively new user of Fathom. (Fathom Activity Guide)
 
References:
Burrill, Gail, Frankin, Christine A., Godbold, Landy, and Young, Linda J. Navigating Through Data Analysis: Grades 9 - 12, Reston, Va.: National Council of Teachers of Mathematics, 2003.
 
Scheaffer, Richard L., Gnanadesikan, Mrudulla, Watkins, Ann, and Witmer, Jeffrey A. Activity-Based Statistics, New York, NY: Springer-Verlag, 1996.

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This material is based upon work supported by the National Science Foundation under Grant No. 0314808 and Grant No. ESI-0554309. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.