Average

Date: 05/10/2001 at 10:34:45
From: Rick Wright
Subject: Averages

Dr. Math -

My fourth grader tells me that the average of a set of data is always 
computed by adding all the numbers together and dividing by the total 
number. Isn't this a misconception? I need to provide some specific 
discussion or an activity to correct his thinking. Please help.  

Thank you.

Date: 05/10/2001 at 13:17:06
From: Doctor Twe
Subject: Re: Averages

Hi Rick - thanks for writing to Dr. Math.

It depends on how strictly you want to define the term "average." 
Merriam-Webster's OnLine Dictionary at:

   http://www.m-w.com/   

defines "average" as:

   1a: a single value (as a mean, mode, or median) that summarizes 
       or represents the general significance of a set of unequal
       values

    b: MEAN

   2a: an estimation of or approximation to an arithmetic mean

    b: a level (as of intelligence) typical of a group, class, or 
       series <above the average>

   3:  a ratio expressing the average performance especially of an 
       athletic team or an athlete computed according to the number 
       of opportunities for successful performance

   - on average or on the average: taking the typical example of the
     group under consideration <prices have increased on average by 
     five percent>

Later, the dictionary goes on to say:

  "AVERAGE is exactly or approximately the quotient obtained by 
   dividing the sum total of a set of figures by the number of 
   figures (scored an average of 85 on tests)."

This is what mathematicians define as the "mean," and it is the most 
common meaning of the term "average." (Note that this is exactly 
definition 1b, and definition 2a also refers to the mean.)

The mean is one of three common measures of central tendency, the 
others being the median and the mode. Medians and modes are also 
sometimes referred to as averages, as supported by the dictionary's 
definition 1a. (The median is the middle value of the data set when 
arranged in ascending order, and the mode is the most frequently 
occurring value(s) in the data set.)

There are also "weighted averages," where some values are given more 
weight (or counted more often) than other values in the data set.

However, for a fourth grader, understanding what an average is and how 
to compute it (in the common usage of the term) is more important than 
making distinctions among other types of "average" measures. When your 
student is a little older, he or she will learn more about statistical 
measures and tools, and will learn about these other "averages."

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/