Introduction
How should we think about the mean? Let me count the ways:
1. It is the sum of the measurements divided by the number of measurements.
2. It is the amount that would be allotted to each observation if the measurements were re-distributed equally.
3. It is the fulcrum (the point at which the measurements would balance).
4. It is the point for which the sum of the deviations around it is equal to zero.
5. It is the point for which the sum of the squared deviations around it is a minimum.
6. It need not be one of the actual measurements.
7. It is not necessarily in or near the center of a frequency distribution.
8. It is easy to calculate (often easier than the median, even for computers).
9. It is the first moment around the origin.
10. It requires a unit of measurement; i.e., you have to be able to say the mean
"what".
I would like to take as a point of departure the first and the last of these matters and proceed from there.
Definition
Everybody knows what a mean is. You've been calculating them all of your lives.
What do you do? You add up all of the measurements and divide by the number of measurements. You probably called that "the average", but if you've taken a statistics course you discovered that there are different kinds of averages. There are even different kinds of means (arithmetic, geometric, harmonic), but it is only the arithmetic mean that will be of concern in this chapter, since it is so often referred to as "the mean".
inches, the mean is in inches; if the measurements are in pounds, the mean is in pounds; if the measurements are in dollars, the mean is in dollars; etc.
Therefore, the mean is "meaningful" for interval-level and ratio-level variables, but it is "meaningless" for ordinal variables, as Marcus-Roberts and Roberts (1987) so carefully pointed out. Consider the typical Likert-type scale for measuring attitudes. It usually consists of five categories: strongly disagree, disagree, no opinion, agree, and strongly agree (or similar verbal equivalents).
Those five categories are most frequently assigned the numbers 1,2,3,4,and 5, respectively. But you can't say 1 what, 2 what, 3 what, 4 what, or 5 what.
The other eight "meanings of the mean" all flow from its definition and the requirement of a unit of measurement. Let me take them in turn.
Re-distribution
This property is what Watier, Lamontagne, and Chartier (2011) call (humorously but accurately) "The Socialist Conceptualization". The simplest context is financial. If the mean income of all of the employees of a particular company is equal to x dollars, x is the salary each would receive if the total amount of money paid out in salaries were distributed equally to the employees. (That is unlikely to ever happen.) A mean height of x inches is more difficult to conceptualize, because we rarely think about a total number of inches that could be
re-distributed, but x would be the height of everybody in the group, be it sample or population, if they were all of the same height. A mean weight of x pounds is easier to think of than a mean height of x inches, since pounds accumulate faster than inches do (as anyone on a diet will attest).
Fulcrum (or center of gravity)
Watier, et al. (2011) call this property, naturally enough, "The Fulcrum
Conceptualization". Think of a see-saw on a playground. (I used to call them teeter-totters.) If children of various weights were to sit on one side or the other of the see-saw board, the mean weight would be the weight where the see-saw would balance (the board would be parallel to the ground).
The sum of the positive and negative deviations is equal to zero
This is actually an alternative conceptualization to the previous one. If you subtract the mean weight from the weight of each child and add up those differences ("deviations") you get zero, again an indication of a balancing point.
The sum of the squared deviations is a minimum
This is a non-intuitive (to most of us) property of the mean, but it's correct. If you take any measurement in a set of measurements other than the mean and
calculate the sum of the squared deviations from it, you always get a larger number. (Watier, et al., 2011, call this "The Least Squares Conceptualization".) Try it sometime, with a small set of numbers such as 1,2,3, and 4.
It doesn't have to be one of the actual measurements
This is obvious for the case of a seriously bimodal frequency distribution, where only two different measurements have been obtained, say a and b. If there is the same numbers of a's as b's then the mean is equal to (a+b)/2. But even if there is not the same number of a's as b's the mean is not equal to either of them.
It doesn't have to be near the center of the distribution
This property follows from the previous one, or vice versa. The mean is often called an indicator of "the central tendency" of a frequency distribution, but that is often a misnomer. The median, by definition, must be in the center, but the mean need only be greater than the smallest measurement and less than the largest measurement.
It is easy to calculate
Compare what it is that you need to do in order to get a mean with what you need to do in order to get a median. If you have very few measurements the amount of labor involved is approximately the same: Add (n-1 times) and divide (once); or sort and pick out. But if you have many measurements it is a pain in the neck to calculate a median, even for a computer (do they have necks?).
Think about it. Suppose you had to write a computer program that would
calculate a median. The measurements are stored somewhere and have to be compared with one another in order to put them in order of magnitude. And there's that annoying matter of an odd number vs. an even number of measurements.
To get a mean you accumulate everything and carry out one division. Nice.
The first moment
Karl Pearson, the famous British statistician, developed a very useful taxonomy of properties of a frequency distribution. They are as follows:
The first moment (around the origin). This is what you get when you add up all of the measurements and divide by the number of them. It is the (arithmetic) mean.
The term "moment" comes from physics and has to do with a force around a
The first moment around the mean. This is what you get when you subtract the mean from each of the measurements, add up those "deviations", and divide by the number of them. It is always equal to zero, as explained above.
The second moment around the mean. This is what you get when you take those deviations, square them, add up the squared deviations, and divide by the number of them. It is called the variance, and it is an indicator of the "spread" of the measurements around their mean, in squared units. Its square root is the standard deviation, which is in the original units.
The third moment around the mean. This is what you get when you take the deviations, cube them (i.e., raise them to the third power), add them up, divide by the number of deviations, and divide that by the cube of the standard deviation.
It provides an indicator of the degree of symmetry or asymmetry ("skewness") of a distribution.
The fourth moment around the mean. This is what you get when you take the deviations, raise them to the fourth power, add them up, divide by the number of them, and divide that by the fourth power of the standard deviation. It provides an indicator of the extent of the kurtosis ("peakedness") of a distribution.
What about nominal variables in general and dichotomies in particular?
I hope you are now convinced that the mean is OK for interval variables and ratio variables, but not OK for ordinal variables. In 1946 the psychologist S.S.
Stevens claimed that there were four kinds of variables, not three. The fourth kind is nominal, i.e., a variable that is amenable to categorization but not very much else. Surely if the mean is inappropriate for ordinal variables it must be inappropriate for nominal variables? Well, yes and no.
Let's take the "yes" part first. If you are concerned with a variable such as blood type, there is no defensible unit of measurement like an inch, a pound, or a dollar. There are eight different blood types (A+, A-, B+, B-, AB+, AB-, O+, and O-). No matter how many of each you have, you can't determine the mean blood type. Likewise for a variable such as religious affiliation. There are lots of
categories (Catholic, Protestant, Jewish, Islamic,...,None), but it wouldn't make any sense to assign the numbers 1,2,3,4,..., k to the various categories, calculate the mean, and report it as something like 2.97.
Now for the "no" part. For a dichotomous nominal variable such as sex (male, female) or treatment (experimental, control), it is perfectly appropriate (alas) to CALCULATE a mean, but you have to be careful about how you INTERPRET it.
The key is the concept of a "dummy" variable. Consider, for example, the sex variable. You can call all of the males "1" (they are male) and all of the females
"0" (they are not). Suppose you have a small study in which there are five males and ten females. The "mean sex" (sounds strange, doesn't it?) is equal to the
sum of all of the measurements (5) divided by the number of measurements (15), or .333. That's not .333 "anythings", so there is still no unit of measurement, but the .333 can be interpreted as the PROPORTION of participants who are male (the 1's). It can be converted into a percentage by multiplying by 100 and affixing a % sign, but that wouldn't provide a unit of measurement either.
There is an old saying that "there is an exception to every rule". This is one of them.
References
Marcus-Roberts, H.M., & Roberts, F.S. (1987). Meaningless statistics. Journal of Educational Statistics, 12, 383-394.
Stevens, S.S. (1946). On the theory of scales of measurement. Science, 103, 677-680.
Watier, N.N., Lamontagne, C., & Chartier, S. (2011). What does the mean mean? Journal of Statistics Education, 19 (2), 1-20.
CHAPTER 17: THE MEDIAN SHOULD BE THE MESSAGE