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Definitions: Average, Mean, Mode


Date: 5/20/96 at 22:6:20
From: Anonymous
Subject: Definitions of average, mean, and mode

I see that the dictionary says the average is the arithmetical mean 
and that the geometrical mean is different, but I would like to find a 
simple definition comparing the meaning of the 

Mean
Average
Mode

Everything I have found is either too complex and over my head, or not 
complete.

Thanks very much.

Margaret
San Francisco


Date: 5/29/96 at 0:5:22
From: Doctor Pete
Subject: Re: Definitions of average, mean, and mode

"Mean" is a general term, but is most commonly used as an abbreviation 
for "arithmetical mean."  Now, say you have a set of numbers, say,

  { 5, 6, 1, 1, 1, 7, 4, 3 } .

In this case, there are 8 numbers, but we can have as many as we want.  
Then the *arithmetic mean*, or *average* is

  (5+6+1+1+1+7+4+3)/8 = 28/8 = 3.5 ;

that is, we add all the elements and then divide by the number of 
elements.

The *geometric mean* is

  (5*6*1*1*1*7*4*3)^(1/8) = 2.6618 ;

so we multiply by all the elements and then take the 8th root.  The
*harmonic mean* is

  8/(1/5+1/6+1/1+1/1+1/1+1/7+1/4+1/3) = 1.9546 ;

which is like the arithmetic mean except everything is "flipped 
around."

The *mode* is

  1 ;

this is simply the number which appears the most often in your set.

So as you can see, "mean" can have more than one meaning (no pun 
intended). Often we are taught that the mean is, in a sense, a 
"representative" value for a set of data; that is, if you took a test 
5 times, the arithmetic mean of those 5 scores would be an indicator 
of your average performance.  But the arithmetic mean is not always 
the best representative value to choose - sometimes it's better to 
take the mode.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
Middle School Definitions
Middle School Statistics

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