What is the Meaning of "Average"?

Date: 03/28/2007 at 16:24:13
From: Danny
Subject: What is the meaning of average

Can you please give a detailed description of average and its meaning?
I'm not looking for a definition like "average is a certain # divided
by a certain total #."  I don't quite understand the real meaning of
average.  

Date: 03/28/2007 at 23:07:43
From: Doctor Peterson
Subject: Re: What is the meaning of average

Hi, Danny.

There are several different meanings of "average".  The most general
is a "measure of central tendency", meaning any statistic that in some
sense represents a typical value from a data set.  The mean, median,
and mode are often identified as "averages" in this sense.

The word "average" is also used (especially at elementary levels) to
refer specifically to the mean, which is the kind of average you
mentioned: add the numbers and divide by how many there are.  This
kind of average has a specific meaning: it is the number you could use
in place of each of the values, and still have the same sum.  

For example, I'll illustrate the idea by making several piles of, say,
beans.  Suppose I make 5 piles, containing 4, 10, 9, 6, and 11 
respectively.  If I wanted to redistribute them into five piles each
of which had the same number, I would gather them all together, count
them (4+10+9+6+11 = 40), and then divide them evenly into 5 piles of 8
(40 / 5 = 8).  Thus, the average is the number I get when I distribute
a sum evenly; it smooths out the variations in the numbers.

Here is an explanation of this kind of average, using a different example:

  What Does Average Mean?
    http://mathforum.org/library/drmath/view/52809.html 

The idea of a mean can be applied to other situations where addition
is not the relevant operation.  The mean we just talked about is the
"arithmetic mean", meaning that it is based on addition.  There is
also a "geometric mean", based on multiplication, which is the number
you could replace everything with and keep the same product, rather
than the same sum.  For example, the geometric mean of 12 and 27 is
the square root of 324, which is 18.  This is because the product of
12 and 27 is 324, and the product of 18 and 18 is also 324; we could
replace both numbers with 18 and the product would be the same.  There
are other kinds of mean based on other ways of combining them, which
are used on the basis of how the numbers involved "want" to be
combined.  See these pages:

  Arithmetic vs. Geometric Mean
    http://mathforum.org/library/drmath/view/52804.html 

  Applications of Arithmetic, Geometric, Harmonic, and Quadratic Means
    http://mathforum.org/library/drmath/view/69480.html 

See also:

  Average
    http://mathforum.org/library/drmath/view/57613.html 

If you have any further questions, feel free to write back.

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

Date: 04/11/2007 at 12:00:23
From: Danny
Subject: What is the meaning of average

Hello Doctor,
 
Thanks for the helpful response.  In your letter, you mentioned that 
average refers to central tendency.  Let me give my interpretation of 
what central tendency means.  Please correct me if I am wrong.  For 
example, if our data shows that it rains 10 times over 100 days, then 
it means that the sky "tends" to rain 10 times per 100 days.  10 
divided 100 gives a frequency value of 0.1, which means that it rains 
0.1 time per day on average.  This average refers to how frequently it 
rains.  For example, if it rained 11 times (more frequent than 10), 
then you would get 11/100, which is a bigger value than 10/100.  
Thus, 11/100 is more frequent than 10/100.  Is this interpretation of 
central tendency correct?

I also think central tendency is the average value that tends to be
close to MOST of the various values in the data.  For instance, if my
data set is (4,6,1,3,0,5,3,4) the central tendency is 3.25, which is a
value that tends towards 3 and 4.  There are two 3 values and two 4
values in the data, which make up most of the data set.
  
So are both of those ways of thinking about central tendency?

Date: 04/11/2007 at 22:43:28
From: Doctor Peterson
Subject: Re: What is the meaning of average

Hi, Danny.

What you are saying in both cases is a reasonable example of the mean,
and fits with my description of average rainfall, though I used the
inches of rain per day rather than the number of rainfalls.

But central tendency is intended to be a much broader term.  It's 
meant to be vague, because it covers not only means but also the
median, the midrange, and even the mode.  Its meaning is "any 
statistic that tends to fall in the middle of a set of numbers";
anything that gives a sense of what the "usual" or "typical" value is,
in some sense, can be called a measure of central tendency.

The *median* is, literally, the number in the middle--put the numbers 
in order, and take the middle number in the list, or the average of 
the two middle numbers if necessary.  So that's clearly a "central 
tendency".  The *midrange* is the exact middle of the range--the
average, in fact, of the highest and lowest numbers.  So that, too, 
has to lie in the middle, though it doesn't take into account how the 
rest of the numbers are distributed.  The *mode* is the most common 
value, if there is one; it really doesn't have to be "in the middle", 
or even to exist, but it certainly fits the idea of "typical".  The 
(arithmetic) *mean*, like all the others, has to lie within the range 
of the numbers, and it represents the "center of gravity" of all the 
numbers.  So each of these fits the meaning of "measure of central 
tendency", each in a different way.

Taking your set of numbers as an example, here are the values of the
various measures of central tendency.  Your numbers are

  4,6,1,3,0,5,3,4

which when sorted in increasing order are

  0,1,3,3,4,4,5,6

  midrange: (0+6)/2 = 3
  median: (3+4)/2 = 3.5
  mode: both 3 and 4
  mean: (0+1+3+3+4+4+5+6)/8 = 3.25

All of these are "middle" numbers, and for many real data sets they
will be close together.  The geometric mean in this case is 0; it
doesn't work well when zero is allowed!

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

Date: 04/12/2007 at 17:35:33
From: Danny
Subject: What is the meaning of average

Hello doctor, thanks for your insights, I now have a better idea of 
average.  Here is one more question about probability.  Let's say 
that I was sick 40 times out of 1000 days.  So based on this 
information, the probability of me getting sick on a random day is 
40/1000.

40/1000 says that, in 1000 days I was sick 40 times, and that is how
frequently I was sick.  Simplifying this 40 to 1000 ratio, we get 1 to
25 ratio.  That is, on average, I was sick 1 time per 25 days (or 0.04
times per day).  So the probability of me getting sick on a random day
is 0.04 based on how frequently I was sick.  

This leads me to conclude that the probability of anything is based on
the past data, and we can make good predictions of future events 
because of the law of continuity, meaning that things in the universe
always follow a pattern.  If we lived in a universe without
continuity, then the knowledge of probability is useless.

So if I was sick 40 times out of 1000 days in the past, then the 
probability of me getting sick on a random day is the average value of 
40/1000 = 0.04.  I like to point out that the average 0.04 doesn't 
have a real physical meaning, because it says that I was sick on 
average 0.04 times per day (0.04 times? that makes no sense).  I think 
0.04 is just a number that corresponds to or represents 40/1000 (40 
times per 1000 days is meaningful).

Please verify what I have written and correct my errors if there are 
any.  Thank you so much.

Date: 04/12/2007 at 23:12:39
From: Doctor Peterson
Subject: Re: What is the meaning of average

Hi, Danny.

>the probability of me getting sick on a random day is 40/1000.

This is really a whole different question, at least on the surface; 
but I can see the connection between averages and probability, and
perhaps you really had probability in mind from the start.

What you're talking about here is called empirical probability: just a
description of what actually happened, which can't say anything about
why, or what could happen another time.  It's simply a ratio: how does
the number of occurrences of sickness compare to the number of days
under consideration?  Out of those 1000 days, 40 of them were sick 
days; so "on the average" 40 out of 1000, or 4 out of 100, or 1 out of
25 were sick days.  If they were evenly distributed--the same idea as 
a mean--then every 25th day would have been a sick day.

>This leads me to conclude that the probability of anything is based
>on the past data, and we can make good predictions of future events
>because of the law of continuity, meaning that things in the universe
>always follow a pattern...probability is established on the basis of
>continuity in the universe.

Now you've made some big jumps!  Not ALL of probability is just about
past data; that's just empirical probability.  And we can't always
extrapolate from past events to the future.  Sometimes that works,
sometimes it doesn't.  In part, it's the job of statistics to look at
the data you've got and determine how valid it is to expect the same
probabilities to continue--how good a sample you have.  But even
beyond that, whether we can assume that patterns will continue depends
on other knowledge entirely, such as science.  If we find a mechanism
that explains a pattern, we have much better grounds for expecting it
to continue than if we don't.

To make a broad statement that "things in the universe ALWAYS follow a
pattern" is to indulge in philosophy, not math.  In probability, we go
the other way: we make an ASSUMPTION that things will continue as they
are, in order to be able to apply probability to predicting anything;
we leave it up to scientists (or sometimes philosophers) to decide
whether that is a valid assumption.  The scientist will most likely do
some experiments to see if the predictions based on his theory work
out, and if so he has some evidence that it is valid, and he can
continue to make predictions.  If not, then he tries another theory!
He certainly would not say that probability forces him to believe that
things work a certain way.

And perhaps that's what you mean to say: probability applies to a
situation beyond the data we have only if there is consistency in the
causes underlying the phenomena.

>So if I was sick 40 times out of 1000 days in the past, then the 
>probability of me getting sick on a random day is the average 
>value: 40/1000 = 0.04.

Again, the empirical probability in itself says nothing about whether
you will continue being sick at the same rate.  It only says that IF
you continue at the same rate, then you can expect to be sick 1 day
out of every 25, on the average over a long period of time.

This, in part, is an expression of what is called the Law of Large
Numbers: that IF there is an underlying pattern such that on each day
(in your example) there is a 4% chance of being sick, then OVER A
SUFFICIENTLY LONG period of time, you can EXPECT to be sick on 4% of
all days.  So you're right that the probability says nothing about any
particular day, and to express it as if it meant you would get sick
1/25 of a day each day is silly.  You should say that you get sick
1/25 of ALL days, IF in fact you do!

The difference between this and the general idea of averages is that
an average can apply to any collection of numbers, not just to the
frequency of an occurrence.  We can talk about the average speed of a
car; regardless of how its speed has varied along a route, we can use
the total distance traveled and the total time it took to determine
the average speed, which is the speed it might have been going
throughout the entire trip, in order to get the same total distance in
the same total time.  There is nothing probabilistic about this; but
like probability, we are taking something that may vary "randomly" and
condensing all its variations into a single number.  The average speed
does not mean that at every moment the car was going that fast, and
the probability does not mean that out of every 25 days you are sick
on one of them, or, worse, that on every day you are sick for 1/25 of
the time.  Averages and probability both ignore unevenness and look
only at the big picture.

And that makes your question a very good one.  I've been noticing the
connections between probability and averages in several areas lately,
and it's good to have a chance to think more about it.

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

Date: 04/13/2007 at 14:37:16
From: Danny
Subject: What is the meaning of average

Hello doctor,
 
Thanks for you help and patience.  I have one last question.  It seems
like sometimes averages have no meaning.  For instance, in a class of
10 students, 2 got 100 on a test, 8 got 0.  The test average is 
200/10 = 20.  So on average every person got a 20 on the test.  If I 
am correct in thinking that an average value is an estimate of the
various values in the same data set (like you said, an average is like
a center of gravity in the data set, so all the numbers in the data
set should lean towards the average), then the average 20 is closer to
the REAL scores of the 8 students who got 0 than to the REAL score of
the 2 people who got 100.  

This average gives a vague idea of how badly most people did, but it
has "hidden" the two perfect scores.  The average may tell us that
most of the people must have done badly so that the average comes out
to be so low.  However, we can't know that some people did perfectly
just by looking at the average.
  
This leads me to believe that the average taken in this case shows 
that MOST people did badly.  On the contrary, it does not give an 
overall picture of how EVERYBODY did.

Let me give another example.  In a class of 10 students, 3 students 
got 70 on a test, 5 students got 80, and 2 got 60.  If we take the 
average here, it comes out to be 73 points per student.  Now this 
average is a good estimate of how EVERYBODY did.  Because it is close 
to the scores of 70, 80, and 60.  In the previous example, the average 
20 is just too far away from 100 to tell us anything about the 
students who got 100.  With this case, the average 73 gives a better 
idea of how EVERYBODY did.

So this shows that average values sometimes do give a overall or 
general picture of how EVERYBODY did.  But some other times, it only
shows how MOST people did.  Without looking at the actual data, you
can't be sure what means what.  So averages are vague in meaning...I
think.  Is what I said correct?

Date: 04/13/2007 at 20:51:49
From: Doctor Peterson
Subject: Re: What is the meaning of average

Hi, Danny.

Several of the pages on our site that discuss mean, median, and mode
talk about why you would choose one rather than another.  Each has its
uses, and what you're saying is that for some purposes the mean is not
the appropriate "measure of central tendency".  That doesn't mean that
it is meaningless, or that it is never a valid concept; only that it
doesn't tell you what you'd like to know in this situation.

The mean is the "center of gravity"; and there are many objects
(speaking physically, now) whose center of gravity is not within the
object.  The center of gravity does NOT mean "where most of the atoms
in the object are".  That doesn't mean the center of gravity is
meaningless; it's what determines how the object will balance.  But
sometimes balance isn't what you're interested in!

In the case of scores on a test, the median is usually considered the
most reasonable measure; in your example, the median would be zero,
showing that over half (in fact, more than that) scored zero.  So if
you choose your statistic carefully, it will tell you what you want to
know.

Another classic example of this is median income.  If in your town 999
people earned $1000 a year, and one man earned $9,000,000 a year, the
average (mean) income would be 10,000 a year, even though NOBODY made
that amount.  The median income gives a much better picture, if you
want to know how the "average" person is doing; but that entirely
misses the fact that there is one person who is rich.  No matter what
"average" you use, you'll be leaving someone out.

Another example is the rainfall I like to use to illustrate the idea
of the mean.  If the average rainfall is 1 inch a day, say, it might
actually have been dry as a bone for 99 days, and then there was a 100
inch flood on the last day.  The average accurately reflects the TOTAL
amount of rain over the 100 days, but that isn't all it takes to
decide what plants can survive there.

Again, the whole idea of an average is to try to boil down a lot of
information into one number.  That necessarily means that you have to
lose some information.  (That's why people don't want to be treated as
mere numbers; they are more complex than that.  Even a set of numbers
doesn't like to be replaced by a single number!)

I think I've said all along that averages are meant to be "vague" in
the sense that they deliberately ignore all the details.  As I showed
above, you're actually being too generous in saying that the average
shows how "most" people did; it may show how NONE of them did.  Your
last few statements are exactly right: the average is not enough to
tell what's really happening.

In trying to use any kind of average to say how EVERYBODY did, you are
misusing the whole concept.  Unless the numbers are all close 
together, there's no way for any number to tell you how they all did.
It's ridiculous to expect that!

But there are other statistics that can come to your aid--not 
averages, but "measures of dispersion", that tell you how FAR APART
the numbers are.  The most famous of these is the standard deviation,
which is actually the square root of the mean squared deviation from
the mean (not that I expect you to make any sense of that).  This
number tells you how accurately you can expect the mean to tell you
anything about most of the population. 

So if you had not only the mean but also the standard deviation, you
would have enough information to decide not only what the "middle" is,
but also how far "most" of them are from that middle.  Even then,
however, you would be missing a lot of details, because you've boiled
a whole class down to two numbers, which isn't a lot better than one.
If you really want to get a sense of how the class (as a group of
individuals) is doing, you'll make a graph of some kind.  When I give
a test, I do exactly that: my spreadsheet calculates the mean, the
median, and the standard deviation, and I also make a graph of the
distribution of grades so I can see where the "outliers" are--the
individuals who are entirely missed by the simple statistics, and
whose existence has to be recognized.

Incidentally, I've sometimes noticed in teaching, as a result of these
statistics, that I can't "teach to the middle" of the class, because
there is no middle.  Sometimes I find a bimodal distribution, which
means that I have a lot of F's and a lot of B's, and no one in between
where the median and the mean both lie.  (The last word there is an
interesting, and very appropriate, pun!) So I have to ignore the
statistics and teach the students.

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

Date: 04/16/2007 at 20:24:40
From: Danny
Subject: What is the meaning of average

Hi doctor,

I have a better idea now, and have learned a lot.  Thanks!