Mean, Median, and Mode.

//Mean, Median, and Mode.

Mean, Median, and Mode.

  As people who have been keeping up with these daily updates know. I would like to become a teacher. However there are some things I think require some dramatic overhauling, mainly because we are at an unusual cross roads where I feel misinformation is causing a dramatic (and ill directed) change in the education system.

  For further discussion later I will note that I am strongly opposed to most (if not all) of the premises behind the No-child-left-behind act. The logic behind taking money away from worse off schools and shoveling that money into better off schools until the worse off schools implode is terrible at best. In Seattle quite a few schools closed down because of the policy and the funding cuts and it makes me wonder when you shove 5 school’s worth of children into less than 5 schools how you are not leaving children behind.

  However it’s painfully obvious that if you wrote a bill that legalized the beating of babies and called it “The don’t beat babies bill.” people would vote for it for fear of voting otherwise leaving them labeled baby beater.

  We’ll leave that there for now and I’ll come back to it on another day. For today I’d like to discuss why all exams should be looked over with the mean, median, and mode or shouldn’t be looked over at all.

  In most of my psychology courses the professor would explain how they feel the exam tested well because the average grade was a low B. This sounded cute and fulfilling till you looked at all the possible situations where the mean or average gives you no idea of the ‘average’ performance in a course.

  Say you have an exam for simplicities sake that has 10 questions. We’ll assume each class has the same amount of students.

7 – 7 – 7 – 7 – 7
Class Average: 7 (Or 70%)

  When we thing of the average grade in a class we think of it like this (or perhaps a more direct 9 – 8 – 7 – 6 – 5 setup). If the average was 7 then everyone got roughly a C and passed the class. That’s fine and dandy in theory however the average really tells us nothing about how the class overall really performed.

10 – 10 – 5 – 5 – 5
Class Average: 7 (Or 70%)

  As you can see in this example over half the class failed the exam. If you walked into a course and the professor told you that over half of you would fail would you stay? What if they told you the class average was 70%? It is likely that the latter would trick you into assuming the course was doable when for all intensive purposes the people passing appear (when looking at the performance of the whole) to have the knowledge necessary to pass regardless of how the professor or the book educates them.

  I’ve noticed in courses with much larger numbers of students you’ll have a small group that performs exceptionally well and a vast majority that perform at just below par or quite far below par. This offsets to some ‘theoretically’ comfortable average and when seen by other faculty or staff the average alone gives the allusion of proper examination and instructional procedures.

  Likewise I feel that the next source of information is by itself relatively worthless. The median essentially tells us nothing about the performance of a class. I’ll again provide two examples that return the same result but are dramatically different. The median for those curious is the middle number when all numbers are placed in numerical order (IE. from least to greatest or vice versa). In the case of an even number you generally would take whatever is in between the two. If you meet at the middle with 7 and 6 the median would be 6.5 else if you met with 6 and 6 you’d end with 6.

  10 10 7 1 1
  Class Median: 7 (or 70%)

This seems appropriate for what the median does by itself. Indeed when you lined up the grades of all the students in your class the median grade was 70%. This is a pretty respectable performance, however you still have just barely under 50% of your course failing, 40% to be exact. This again is offset by the fact that generally speaking the people who do exceptionally well on an exam that the majority of students do on par or sub par would have performed at such high rankings regardless of the professor or the book.

8 – 8 – 7 – 7 – 3
  Class Median: 7 (or 70%)

  So the reason this bothers me is that as you can see the performance of these two classes is dramatically different. In the bottom set 80% of the class has passed, likewise the upper performance is not perfect which may hint towards a more accurate examination to teaching style. I’ll come back to why the above example is better than the first (of the median examples) in a short while.

  We finally move onto the mode, a mode is the most common digit in a set of digits. For example if you have 3 numbers and two of them are the same then the mode would be whatever that number is. However much like its cousins (or brothers/sisters what have you) the mode is utterly meaningless by itself.

10 – 7 – 7 – 2 – 1
  Class Mode: 7 (or 70%)

  A 70% is essentially the bare minimum you can receive in a course before you pseudo-pass it. When you pseudo-pass something you receive a ‘passing’ grade however you are strongly requested to retake the course. It’s essentially failing without failing and I know in the one case where it happened to me it was treated as worse than failing (which I found odd).

  Modes can get far more hokey when you get into larger groups of people. On an exam with 50 possible outcomes you could end up with only one outcome being performed more than once leaving it as the ‘mode’ where really the only thing it has on others is that its merely one larger. The mode is a support function when looking at grading and really means nothing by itself.

7 – 7 – 7 – 7 – 7
  Class Mode: 7 (or 70%)

  This case is dramatically different from the first case, your entire class passed the test which is a good thing however nobody performed better or worse than anyone else. This tends to show failure on the part of the examiner for either providing misleading study suggestions, poorly worded questions, or some other mistake that is all too common. The main reason for pulling this out again (as it matches up with the very first example) is that it leads into my main point (took a while to get here…perhaps unnecessarily so…but I’m rarely as succinct as I want to be).

  The only case in which statistics for a course are acceptable without the full print out is when the mean, median, and mode all are fairly close to one another. If any of them is dramatically different than the other an investigation should be taken. Not necessarily by the FBI but someone should look into the teaching or testing style of the professor/teacher. This is usually a good sign that something is wrong and generally when an entire classroom is effected its not all the students (people aren’t quite that homogenous yet).

  A few examples are as follows.

8 – 8 – 8 – 8 – 7
  Class Mean: 7.8 (or 78%)
  Class Median: 8 (or 80%)
  Class Mode: 8 (or 80%)

  So in the above case you have a class where everyone passed, we don’t see a case of a ceiling or floor affect (everyone neither got a 10 nor did everyone get a 1), and roughly speaking all three M’s are very close to one another. It is difficult to create a situation where all 3 are the same and you don’t have an accurate idea (without seeing the individual performance of all the students on an excel printout) and I don’t have one handy but I’ll give it a shot. (Update: In retrospect I feel I failed. Feel free to comment if you have a working example).

9 – 7 – 7 – 6 – 3
  Class Mean: 6.4 (or 64%)
  Class Median: 7 (or 70%)
  Class Mode: 7 (or 70%)

  I believe in this case we do see a relatively broad range of grades (as high as 90% and as low as 30%) however the overall performance is so poor that it doesn’t matter. These kinds of cases should always spark curiosity in the institutions that they unfold in. Maybe I’m being close minded and there is a grand example of a mean, median, and mode all showing great performance yet most of the class failing but I’m not convinced that is entirely possible (I will not say it isn’t though).

  The perfect case is obviously when everyone in the course gets the exact same grade, that’s the point of popping out the lucky 7’s scenario. Ideally I would hope everyone would get ‘lucky’ 8’s or 9’s but it seems odd that we immediately assume that all people are ‘above average’…I would think that if everyone is above average then they are not above average they are average.

  These three functions are almost meaningless by themselves, each can in certain (and numerous) situations provide dramatically misleading information supporting all sorts of flimsy or hokey ideals. The use of any of these three functions by themselves when fashioning policies or judging the performance of an entity is likely to end in misinformation and failure. I propose that either professors and teachers produce all three pieces of information or provide none because it has become all too apparent at the very least at Western Washington University, that Mean is being abused worse than the proverbial red headed step child.

  The psychology department routinely fails large portions of their students (or D’s them) and yet the average make it appear that people are performing at or slightly above average performance. This is unacceptable and an additional reason I’m not the least bit troubled that the college has lost 35 (possibly more) million dollars.

  If you can’t see the flimsy nature of the mean function than you probably cannot see the danger of living dramatically far beyond your means (pun intended and in at least one sense its not even a pun).

For Next time: I’ll likely discuss the idea of Indeterminism.

By | 2009-02-11T17:52:27+00:00 February 11th, 2009|Journal|Comments Off on Mean, Median, and Mode.