It seems common for teachers to ask students which “measure of central tendency” (average) is best. Sometimes they ask this in a particular context (say, to report the “average” salary at a company); other times there is no context (other than perhaps some unidentified data). In the latter case, we generally consider the question to be inappropriate; in the former, it is merely subjective. Let’s look at several ways we have answered such questions.
Background
First, if you are not quite sure what these terms mean, here is an introduction to all of them, which hints at the basic answer to our question:
Range, Mean, Median, and Mode I have some questions that you may want to answer for me: 1. Why do we have to study range, mean, median, and mode? 2. Could you help me understand them more? 3. How is it going to help me later in life?
Doctor Stacey explains what each is, and why it might be used; a key comment near the end is, “Now, mean, median and mode are all good types of averages, and each works best in different types of situations.” None of them is universally “best”; that’s why they all exist! What’s best depends on your needs in a particular situation (context). That will serve as a useful foundation.
A question with context
Here is a typical question, from 2001:
Which is the Best Description? I am asked to answer the question; "Mode, mean, range, median: which best describes the number of times "Popular Song" was played on the radio per day?" given the following information: Sunday: 9 Monday: 13 Tuesday: 13 Wednesday: 12 Thursday: 14 Friday: 3 Saturday: 10 I calculated that the range is 11, the median is 12, the mode is 13, and the mean is 10.5. How should this question be answered? Is there an answer? I would guess the answer to be mode, since 13 occurs twice in the list. But I don't know how to answer the question.
Here, there is a specific context, so at least Doctor Stacey’s ideas can be applied. But is there really a correct answer? Doctor TWE starts off with his own example:
Let's say we were comparing two basketball players, Anne and Rich. In five games they've made the following points:
G# Anne Rich
-- ---- ----
1 12 10
2 0 13
3 13 9
4 25 8
5 0 10
When comparing means, Anne and Rich seem to be equivalent, since each averages 10 points per game. If we look at their median scores, Anne seems to be the better player; her median is 12 as compared to Rich's median of 10. If we consider their modes, the "most likely outcome" is that Anne will score 0 points, but Rich will score 10. Rich's mode is better than Anne's.
Let's look at these numbers all together:
Anne Rich
---- ----
Mean 10 10
Median 12 10
Mode 0 10
Who's better? It depends on what you're looking for.
If, for example, you were to have an "office pool" on how many points Anne or Rich will score in the next game (and you had to get the score exactly to win the pool), you'd want to use their modes. If you had a pool and the winner was the one who came closest to the player's score, you'd want to use their means. If you wanted to know what score you'd have to get to have a better-than-even chance of outscoring Anne or Rich, you'd want to use the median (or more precisely, the median + 1). The numbers don't indicate a clear "winner," but they do paint a good picture of the types of players Anne and Rich are.
The point is that which is “best” depends not only on the data and their meaning (both of which we are given), but on how you will be using the result (which we are not). He continues,
My assumption is that the problem was not asked with a right or wrong answer in mind, but rather to see if the student can justify his or her choice. Does it ask the student why he or she gave their answer? This type of question can be used to test the student's understanding of the underlying concepts rather than just their ability to "number crunch."
This should be posed as an essay question, in which the reasoning is more important than the specific answer. Unfortunately, Sandy replied that it was given as a multiple-choice question. Doctor TWE responded:
This is why I dislike closed form (i.e. multiple choice, true/false, etc.) interpretation questions. There are valid arguments that can be made for any of the first three choices, depending on why the information is being sought. But in lieu of more information, I would say that the mode *probably* best describes the number of times the song was played. As a teacher, I avoid using multiple choice except for "factual" recall questions. (E.g. "which of the following is the median of the data set?") When it comes to interpreting data and understanding concepts, I want to see what the student is thinking, and the only way to do that is with open-ended questions. (E.g. "which measure of central tendency best describes the number of times the song was played on the radio per day? Defend your choice.")
I imagine that all of the Math Doctors, whether we are teachers or not, would agree with him on the last point; but we may well give different answers to the question itself! (I personally think the mean is a better answer here.)
A question without context
Here is a question from 2009, with less context to the question, and more thought from the student:
Choosing between Median and [Mean] as the Best Representation I recently did a quiz relating to measures of central tendency (mean, median, mode, range). On it was this question: The set of numbers are: 1, 3, 9, 10, 13, 15, 25, 39, 58, 63. Which measure of central tendency best represents the data? I thought it was the mean but my teacher thought it was the median. The mean of the numbers is 23.6 The median of the numbers is 14 There is no mode I am unsure as to what measure of central tendency best represents the set of numbers. I believe it is the mean because it factors in all of the numbers, including the very low and very high numbers, therefore it is not distorted by any very high or very low numbers. My teacher disagrees and thinks is the median. My teacher believes that the mean is distorted by the bigger numbers, however, I believe that the median is distorted by the smaller numbers.
Here we are given the values, but are not told what they represent, or what our goal is. But Henry has given thought to the reason for his choice (bravo!), and so has the teacher. Can we declare a winner? I didn’t:
I don't think there is a correct answer to the question. It depends on your point of view and the purpose of your measurement, as you stated nicely in your last paragraph. The question is, what do you mean by "represent"? For example, suppose you listed the annual pay for all the employees of a company. The mean would represent how much each would be paid if the total payroll were divided evenly among the employees. That might be the natural way for the manager to summarize the pay, since the total payroll is important to him. It well represents the effect of all the salaries on the bottom line. But if a few have very high salaries while most are not paid much, the mean would make it look as if everyone earned a lot of money, "on the average". The mean would be too greatly influenced by those "outliers". The employees, on the other hand, might consider the median to better represent their average pay, since it would show how much an "average" (typical) employee made (focusing on the individual rather than the bottom line). So which is most useful or important depends on your point of view; and each contains different information. The employees and the manager are both right; they just have different interests, and different ideas of what it means for an "average" to represent the data. (It just happens that the mean also favors the manager in making the company look generous, while the median favors the employees by making them look underpaid. But I don't think that's why they would tend to make the choices they would!)
(Sometimes people answer this question based on a somewhat cynical view that each would make the choice that favors his own position, and that is probably true in many situations. But I think it’s important to recognize that these are also logical choices, and both are objectively correct based on their interests.)
I applied this to Henry’s problem:
In your case, your teacher seems to be more like the employees, focusing on the individuals, while you are more like the manager, thinking of the whole.
One way to clarify this is to diagram the whole distribution to get a better sense of how the numbers relate to one another. If there were more data I'd use a histogram, but I'll just use a "dot plot":
o o oo oo o o o o
+-------+-------+-------+-------+-------+-------+-------+
0 10 20 30 40 50 60 70
^ ^
median mean
Both measures are very much in the middle; the median is more "in the middle" of the individuals, while the mean is more "in the middle" of the whole.
Distorted by outliers?
I pointed out earlier that the mean is highly influenced by outliers: that is, just changing the highest or lowest salary will change the mean, while it will have no effect on the median. Henry seems to be saying the opposite: “the mean … is not distorted by any very high or very low numbers” because it “factors in all of the numbers, including the very low and very high numbers.” I suppose he is thinking of the fact that every number contributes to the mean; but I wouldn’t say that eliminates “distortion”.
The difference lies in what each considers to be a “distortion”: “My teacher believes that the mean is distorted by the bigger numbers, however, I believe that the median is distorted by the smaller numbers.” Technically, this data set doesn’t have any outliers (isolated values far from the main body); but the relatively few high numbers do “pull the mean up”. On the other hand, the median is affected (directly) only by the middle numbers, so it really doesn’t make sense to say that small numbers distort it. Rather, I think he is saying that the concentration of many low numbers causes the median to be low. But using the word “distortion” assumes that there is some right answer that is being messed up, which is circular reasoning. Why shouldn’t the “average” be low, when many of the numbers are low? And why shouldn’t high numbers have an influence? All we really have here are two different ideas of average, not a right one and a wrong one!
Let’s see what I had to say about this:
My inclination is to agree with the teacher, if I had to take sides; the median also seems closer to what the mode would be if you were to group the data, since it is densest around 9-15. But I also notice that this is not quite the kind of situation I described for employees, where a very few numbers are far above all the others. It's hard to say that the many numbers clustered toward the low end "distort" the median (which is where most of the numbers are, anyway), or that the numbers smeared out toward the high end "distort" the mean (which is not so far away from the median).
The fact is that the median is closer to more of the data, and in that sense it represents the data better. But here is an interesting grammatical point: The word “data” (taken straight from Latin) is technically a plural, and if you take it that way (as I did just now in saying “more of the data”), then we are focusing on the individuals, and the median is best. But today many people don’t know Latin, and take data as a singular, referring to the whole collection. And from that perspective, the mean may be better!
Nevertheless, as I closed, I agreed with Doctor TWE:
So I still don't really know what it means to "best represent the data" without a context! This sort of question works better as an essay topic than a multiple choice.
Outliers, symmetry, and ease of use
Here is one more example, from 2003:
Mean or Median? In working on finding textbook readability for a math project, we have to find the mean number of words per sentence. One question asks: Why does the formula use the mean number of words per sentence instead of the median number of words per sentence? When I found both the mean and the median, they had the same value, 13. They are both measuring the center, so what's the difference? I think it's supposed to have something to do with outliers.
This turned into a long discussion with Doctor Ian about how each is affected by outliers, and under what conditions one is larger then the other. I will not quote at length, but he starts with a salary example like mine (emphasizing that the mean might be used just to give a good impression, but both are valid). Then he corrects a misunderstanding of the phrase “more resistant to outliers”, and points out some practical reasons why the mean might be preferred (namely, easier calculation). Then he gets to Jill’s question about why the mean and median are about the same for her data (it’s often a result of symmetry), and shows how either could be greater (depending on how the data are skewed). Finally, Jill shows her actual data set, which turns out to have an outlier at each end, and otherwise fairly symmetrical.
We never quite get to a direct answer to the project’s question: “Why does the formula use the mean number of words per sentence instead of the median number of words per sentence?” I have no idea what the answer might be; you’d have to ask the author! I probably would have wanted to ask for the entire statement in the project, to make sure I knew what “the formula” is, and where it came from. If it is referring to something like the Flesch readability index, I suspect Doctor Ian’s suggestion is right: The mean is just easier to calculate for a large piece of text, because you only have to count words and sentences. In my experience, that seems to be the usual reason for using the mean!