Daniel Livengood
Rhonda Jordan
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139 USA

Video Summary: This learning video presents an introduction to the Flaws of Averages using three exciting examples: the “crossing of the river” example, the “cookie” example, and the “dance class” example. Averages are often worthwhile representations of a set of data by a single descriptive number. The objective of this module, however, is to simply point out a few pitfalls that could arise if one is not attentive to details when calculating and interpreting averages. Most students at any level in high school can understand the concept of the flaws of averages presented here. The essential prerequisite knowledge for this video lesson is the ability to calculate an average from a set of numbers. Materials needed include: pen and paper for the students; and a blackboard or equivalent. During this video lesson, students will learn about three flaws of averages: (1) The average is not always a good description of the actual situation, (2) The function of the average is not always the same as the average of the function, and (3) The average depends on your perspective. To convey these concepts, the students are presented with the three real world examples mentioned above. The total length of the four in-class video segments is 12 minutes, leaving lots of time in a typical class session for the teacher to work with the students on their own learning examples (such as those from the supplementary notes) to firm up the ideas presented here on the flaws of averages.

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Download Supplement #1 to Video Lesson (download in Word or PDF format)

Download Supplement #2 to Video Lesson (download in Word or PDF format)

Download Supplement #3 to Video Lesson (download in Word or PDF format)

Download Supplement to Closing Segment of Video Lesson (download in Word or PDF format)

Additional Online Resources:

• “The Flaw of Averages” by Sam Savage published 10/08/2000 in the San Jose Mercury News
  http://www.stanford.edu/~savage/flaw/Article.htm

• The Flaw of Averages web page by Sam Savage at Stanford University
  http://www.stanford.edu/~savage/flaw/

• Jensen’s Inequality (i.e. the function of the average is not always the same as the average of the function)
  http://en.wikipedia.org/wiki/Jensen%27s_inequality

• Lake Wobegon, where “…all the children are above average” (see the Teacher’s Segment for ‘the average depends on your perspective’)
  http://en.wikipedia.org/wiki/Lake_Wobegon

• The following three sites relate to Power Generated by a Wind Turbine
  http://www.windpower.org/EN/tour/wres/enrspeed.htm
  http://www.awea.org/faq/windpower.html
  http://www.awea.org/faq/wwt_basics.html

• The following two sites relate to Expected Value & Arithmetic Mean
  http://mathworld.wolfram.com/ArithmeticMean.html
  http://en.wikipedia.org/wiki/Expected_value

• The following two sites relate to Random Variables
  http://mathworld.wolfram.com/RandomVariable.html
  http://en.wikipedia.org/wiki/Random_variable

• The following two sites relate to Probability Distributions
  http://mathworld.wolfram.com/ProbabilityFunction.html
  http://en.wikipedia.org/wiki/Probability_distribution

• The following two sites relate to Discrete Distributions
  http://mathworld.wolfram.com/DiscreteDistribution.html
  http://en.wikipedia.org/wiki/Discrete_probability_distribution

• The following two sites relate to Normal Distribution
  http://mathworld.wolfram.com/NormalDistribution.html
  http://en.wikipedia.org/wiki/Normal_distribution

• The following two sites relate to Exponential Distribution
  http://mathworld.wolfram.com/ExponentialDistribution.html
  http://en.wikipedia.org/wiki/Exponential_distribution