How can I think about, and understand, "means"?

[Amwil]

Hi all!

So far I know that the mean is a number that when multiplied by the number of numbers in the set will equal the sum of the numbers in the set.
With only two numbers I can see that calculating the mean gives you the number midway between them -- but I don't really understand how to think about the mean of more than two numbers.

How can I think about a midway point between say three numbers? Does anyone have any tips?

Thanks.

[Dr.Peterson]

So far I know that the mean is a number that when multiplied by the number of numbers in the set will equal the sum of the numbers in the set.

With only two numbers I can see that calculating the mean gives you the number midway between them -- but I don't really understand how to think about the mean of more than two numbers.

How can I think about a midway point between say three numbers? Does anyone have any tips?

One way to think of a mean is as the "center of gravity" of the numbers. That is, if you put equal weights on a number line at the points indicated by the numbers, they will balance at the mean. Clearly that number is "in the middle" of the collection of numbers.

When the numbers are integers, your first sentence provides a good image. Suppose n people each have various numbers of objects, and they put them all into one pile, then each person takes the same amount from the pile. The amount each then has is the mean. The mean relates to sharing the total equally.

[Veru]

Just to review the process:

So you have 3 numbers: 1, 2, and 3. To get the mean, first, you add 1+2+3 to give you 6. And you had 3 numbers to begin with, so to find the mean, you divide the sum of the numbers by 3. So, you divide 6/3 to get 2. So, 2 is the mean of 1, 2 and 3.

That is such a basic example, though, it can be hard to see what the mean actually does. To help explain this, you can call the mean by a more functional name: the average. Why the mean is called "the average" becomes more noticeable when you are given different numbers, though.

For example:

You have 2, 10 and 12.
Add the three numbers together, 2+10+12 = 24
Divide 24 by the # of numbers, 24/3 = 8

So 8 is the mean. But how is it the average?

Let's say you have 2 sticks, 10 sticks, and 12 sticks in separate piles. You put them all together into 1 pile. Then, you divide them up again into 3 equal piles. There will be 8 sticks in each pile. So, the average is 8. This is your mean as well.

[mmm4444bot]

… you can call the mean by a more functional name: the average …

For beginning students, the words 'mean' and 'average' are synonyms. They both refer to the same thing. They may be used interchangeably.

Later on, you might learn about different kinds of means (eg: harmonic mean, geometric mean -- what you're doing right now is the arithmetic mean). Examples of other types of averages are the median and the mode.

But, for right now, 'mean' and 'average' are the same.

What do you have in mind, when you say that 'average' is a more functional name?

[JeffM]

Hi all!

So far I know that the mean is a number that when multiplied by the number of numbers in the set will equal the sum of the numbers in the set.
With only two numbers I can see that calculating the mean gives you the number midway between them -- but I don't really understand how to think about the mean of more than two numbers.

How can I think about a midway point between say three numbers? Does anyone have any tips?

Thanks.

The mean is one kind of descriptive statistic that summarizes multiple numbers. It is a simplification to permit understanding. There are several means; the arithmetic mean that you are discussing is just one of them. Means measure the "central tendency" of a collection of related numbers, but there are other measures of central tendency such as the median and the mode. Which measure is best is studied in statistics. But the arithmetic mean that you are asking about is far and away the most frequently used measure of central tendency, in part because it is amazingly easy to calculate: you add all the numbers up and divide that sum by the number of summands. A third grader can do it.

Descriptive statistics are not very valuable if you are summarizing a small number of numbers. But suppose you have to deal with ten thousand numbers. They will become a big blur. Summarizing is necessary if the numbers are to be useful to the human mind.

Now you are technically correct that the arithmetic mean of a collection of related numbers multiplied by the number of numbers in the collection equals the sum of all the numbers in the collection. That is a true statement. But it gives no clue as to why the arithmetic mean is somehow "typical" of the numbers in the collection. Suppose you have a collection of n numbers and another collection of n numbers each of which is equal to the arithmetic mean of the numbers in the first collection. The sums of the numbers in each collection will be identical: the two collections have the exact same sum.

Collection A consists of 1, 2, 6, 11. Collection B consists of 5, 5, 5, 5. Both collections have the same number of elements, namely 4. And both collections have the same sum, namely 20. So it is reasonable (at least for some purposes) to say that we can think of 5 being the typical number of Collection A even though 5 is not even in Collection A.

As I say, the process does not seem very useful when dealing with a collection of just four numbers, but think about a collection of forty numbers, or four hundred numbers, or four million numbers. You will want to summarize them. And you will KNOW that a collection of the same size consisting solely of the arithmetic mean repeated over and over again will have the same total value.