# Flaws of AveragesEnglish Mathematics

### CPALMS

This lesson is CPALMS Vetted and Approved

### Instructors

Daniel Livengood
Rhonda Jordan

Massachusetts Institute of Technology
Cambridge, Massachusetts 02139 USA

## Introduction

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.

## Instructor Biography

Rhonda L. Jordan is a Ph.D. student in the Engineering Systems Division at MIT and is interested in sustainable energy development in developing economies.

Daniel J. Livengood is a Ph.D. student in the Engineering Systems Division at MIT and is interested in how the evolution of the electric grid will incorporate responsive electricity demand to help manage the grid most efficiently, especially as weather-dependent sources of electricity supply (such as wind turbines) become larger percentages of the installed capacity.

## For Teachers

See this blog where one can buy 'futures financial contracts' relating to the amount of snow that falls in an area. The discussion uncovers yet another Flaw of Averages.

“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/
http://www.awea.org

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