From marks obtained by $n$ students in $2$ subjects, how to decide the subject in which students have higher knowledge level?

Question

Given the marks obtained by $n$ students in $2$ subjects (say A & B), which measure of central tendency - Mean or Median - should be taken into account to decide the subject in which the students have a higher level of knowledge ?

For example, consider the marks obtained by $20$ students in subjects A & B -

$20$ students in $2$ subjects A & B">

Mean of marks in A $= 44$

Mean of marks in B $= 45.5$

i.e., Mean marks in A < Mean marks in B.

Also,

Median of marks in A $= 45.5$

Median of marks in B $= 40.5$

i.e., Median of marks in A > Median of marks in B.

So, which subject is the knowledge level of students said to be higher in? In other words, does this decision depend on the Mean or the Median of marks ?

According to me, both the Mean & the Median should be considered for making such a decision, since both values measure the average performance of the students, but I'm unable to say conclusively if this is indeed true. I need to understand why one value estimates the knowledge level better than the other value (if this is the case).

Any help is highly appreciable.

Answer 1

This is, to a large extent, a philosophical question, because it's not all that clear what it means for the students, as a group, to have higher knowledge level (as your example shows).

It makes sense, however, to discuss the properties of the mean and median, and give the typical reasons why you might prefer one or the other. Here are the most relevant points:

1. (reason to use mean) An important property of the mean is The Law of Large Numbers. As per Wikipedia, "According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed." For example, if you get to make a lot of deals with random rewards, choosing the deal with best expected mean would maximize your long-term winnings.
2. (reason to use mean) The mean is easy to compute and work with. You don't have a lot of numbers, so it doesn't matter that much to you, but in general, it's much easier to compute a mean than a median. It has nice mathematical properties such as linearity, which can be useful in many applications.
3. (reason to use median) On the other hand, the median is more robust - that is, less sensitive to outliers. If a student got a 0 or 100, that would affect the mean much more than it would affect the median. Robustness is typically good, because if a student performed unusually, it could as well be because that student was ill (or cheated). There are, of course, situations where you might not want robustness.

In this particular case, if I'm interested how well a lot of students to work on a large decentralized project (where students mostly work individually, not all together), I'd probably look at the mean. On the other hand, if I'm interested in how well an average student would do on his/her own, median would be the better choice.