What Does Average Mean?

Date: 12/05/2001 at 21:50:18
From: John Profeta
Subject: Average

Can someone explain what average is? For example what is the average 
rainfall in New York? I understand the formula needed to find the 
average, but what is the meaning of average?

Thanks, 
John

Date: 12/05/2001 at 23:24:11
From: Doctor Peterson
Subject: Re: Average

Hi, John.

Sometimes "average" just means "normal." Other times it refers to some 
number "in the middle"; there are several different ways to define an 
average. But the usual average, which you are familiar with, called 
the "arithmetic mean," is calculated by summing all the values of 
something and then dividing by the number of values you used. 

What this really means is that you are finding a single number that 
you could use in place of all the different numbers and still get 
the same total. For example, suppose that the rainfall one day is 
2 inches, the next day 0 inches, and the next day 1 inch. The total 
rainfall over the three days is 3 inches, so it is just as if there 
had been 1 inch each day. So the average daily rainfall over that 
period is 1 inch.

Do you see that an average "smooths out" the numbers? It's as if there 
were two inches of dirt on one block of sidewalk, none on the next, 
and one on the third. If I spread all the dirt out evenly, so it was 
all the same level, there would be one inch over the whole sidewalk. 
In the same way, if we spread the rain out evenly over all three days, 
there would be one inch every day. Of course, everything isn't "just 
as if it rained an inch every day"; then you could never have a picnic 
in Central Park! But just from the perspective of the total amount of 
rain, it's the same. And that's why there are different kinds of 
average: if something else is important other than the total, then you 
can get a different "average."

It's also important to ask what data are represented by the average 
you are talking about. Is the "average rainfall in New York" the 
average per day over some particular number of days? Is it the average 
over a whole year? Is it the average per year over a number of years? 
Or the average amount on December 5 over several years? The actual 
amount will vary with the season as well as from year to year, and you 
can average the data to eliminate either variation, or both, depending 
on what your goal is.

That may not fully answer your question, so write back if you have 
more to ask.

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