What Does "Mean" Mean?

Date: 14 Apr 1995 13:02:13 -0400
From: Richard Brown

Dear Dr. Math,

In Webster's dictionary one of the definitions of average is MEAN and 
another is "being about midway between extremes."

Mean's definition is "a middle point between extremes." Additionally, 
mean also has two calculations defined as 1) "a value computed by 
dividing the sum of a set of terms by the number of terms" and 2) "a value 
computed by dividing the sum of two extremes of a range of values by 
two."

Can the terms average and mean be used interchangeably? 

Definitions 1 and 2 of mean can yield different results with the same 
numbers. What examples can you give that would make it clear when to use 
mean definition 1 and mean definition 2?

Any help in clearing this up for me would be appreciated.

Regards, 

Richard Brown
Phoenix Arizona USA

Date: 14 Apr 1995 14:25:10 -0400
From: Dr. Ken

Hello there!

The word "mean" is actually used to "mean" a whole bunch of things.  In
general, it purports to give a value that is somehow typical or in the
middle of the rest of the values.  Usually when people talk about means,
they say whether they mean the arithmetic mean (the average) or the
geometric mean (this is defined as the nth root of a1 * a2 * a3 * ... * an. )
for example, if you have the numbers 5, 15, and 45, the arithmetic mean is
65/3 = 21 2/3, and the geometric mean is (5*15*45)^1/3 = 15.

As far as deciding which mean to use when, it's pretty valuable to see what
any of them are, and then you'll know that much more about your data.  For
instance, the two values for mean above tell us that the data values
aren't grouped that close together, since the two means are pretty well
separated.

This stuff shows up a bunch in statistics, and statisticians have some
pretty well-developed techniques for analyzing means.

-Ken "Dr." Math