Almost everybody knows that measures of central tendency are an important part of statistical analysis, and that mean, median, and mode are the most common approaches to finding a representative central tendency. However, what is often found lacking is an insight into which of these will serve our purpose in a given scenario, and an understanding of the drastic effect that our choice will have on the result we conclude on. Here are some examples of some usages of mean, median and mode that I have encountered and found to be inappropriate.
Let’s start with how the mean (arithmetic mean, to be precise), is often misused. A good example is how per-capita income is sometimes interpreted as the income of an average person. To understand how misleading this can be, we must consider a sample set that is at least qualitatively realistic. So, let’s say there are 100 persons in a country. If 20 of them earn $1, another 20 earn $2, another 20 earn $3, another 20 earn $4, next 10 earn $10 and the next 10 earn $100, the per-capita income of this population will be $13. Note that more than 80% of the population earns less than one third of this figure. This figure might be useful in comparing the earnings of two populations that differ in size (by bringing them to a common base). What it is not, is an indicator of how much an average person in that population earns. To get an idea of the average person’s earning, the median is a much better choice.The median is also often used in situations where the results can be misleading. There are online courses (offered by Coursera, Udacity, etc.) for which tens of thousands of students enroll, making it impossible for the instructor to personally evaluate their tests, assignments and projects. So usually they are reviewed and assessed by five of your peers and you, in turn, evaluate the work of 5 others. So far, I have no objections. Now there are some courses in which students are asked to grade a work on a scale of 1 to 5 (say very poor, poor, average, good and excellent), and your net grade is the median of the 5 individual grades you get. This, I think is not a very good way to combine the individual grades, because most reviewers when they are not sure will choose the safe option of assigning a 2. Thus, you would have people getting [1,1,3,3,3] in individual assignments and those getting [3,3,3,5,5]. While the first set of people were decisively rated poor by two reviewers and the latter decisively rated excellent by two reviewers, they both end up being rated as average. In a course taken by 30000 students worldwide including many not-so-serious ones, the individual rating of 3 in many cases just means that the reviewer is not sure or cannot make up his mind. If we use the arithmetic mean to combine these individual grades, the first set will get 2.2 and the second set will get 3.8. We may even round these to 2 and 4 and this, I believe, would be a more correct depiction of their performance.In the grading example, some might think it common sense to assign the grade that most people think is deserved. This is the approach known as mode and, if the university had chosen to use it (which they did not), the result would have been same as when using median. The mode is useful in a variety of cases, including when dealing with non-numeric data. In an election process where everybody casts their vote for a candidate of their preference and the candidate with the most number of votes are chosen as a “representative” of the population, we are actually using the mode!
While these three are very useful statistical tools, there are many situations in which it makes no sense to interpret a result statistically. One of the best examples I have heard is the story of an engineer, physicist and a statistician who go hunting. When they see a deer, the physicist takes out his gun, positions it at an angle based on the velocity of the bullet and the distance to the target, and fires. Since he does not take the air resistance into account, the bullet falls a metre short of the target. Next, the engineer shoots, and this time he makes all calculations carefully and taking air resistance also into account. However, since he is used to making calculations with a slightly higher “factor of safety”, his bullet goes beyond the target by a metre. The statistician who was watching them both cries out “We got him”! Of course, this is only a story and no statistician would use statistics so inappropriately. But it gave me a good laugh when I heard it, and I think it serves to remind us that we have to be careful about when and how we use statistics to interpret data.