Class Averages... Mean or Median?
The purpose of putting a 'class average' on a report card is so that a student and his or her parents can get some indication of how the student is doing, in comparison to other students in the class.
For example, if your Social Studies mark was 69%, not in itself a very good mark, it might be useful to know that the class average was only 52%. In that case, you would be almost 20% above the average mark!
Alternately, if your mark of 69% in Social Studies is reported with a class average of 79%, you are well under the average for the class.
In either case, the class average gives a little more information about how you did, in comparison to the rest of the class.
But a problem occurs when the size of the class is small ... say, 10 students or less. In that case, how you work out the class average will make a big difference! There are in fact many ways to do an average.

First, let's look at three of the ways you can actually calculate a class average.

Mean
This is what people normally do when they calculate an average; add all the numbers and divide by how many there were.
Median
The median is the middle mark, when the marks are listed in order.
Mode
Mode is the mark or mark range there is the most of.
For large sets of data (a class of 20-30 students, for example), these three methods are usually equivalent. In other words, the mean, median, and mode will usually be about the same. So there is no reason not to work out the class average by just doing an ordinary 'mean'.

What happens when the class size is very small? If the marks are well spread out, not much.

For example, suppose we look at a class of only 5 students. Their Social Studies marks on the last report card looked like this:



These marks are spread out fairly widely. If we were to calculate a class average, using the mean and median methods, here is what we would find:


MEAN: 64%    MEDIAN: 64%


Notice that the mean and median are the same. Even if they were slightly different (by a few percent), it wouldn't matter much, because the marks are symmetrically distributed around the middle value.

Unfortunately, that is seldom the case with small sets of marks. Often there are one or two marks that are very high or low, compared to the others. In a large class, that wouldn't matter much. But with a small class, this will have a disastrous effect on the mean.

For example, consider these marks:


This often happens with small classes. Most of the marks are in one range, with one mark far higher than the others.

How would you describe this class's marks? Clearly the best description would be something like this, in words: "Most students got a mark in the 60's. Only one scored in the 90's".

The problem comes when a teacher tries to report this to parents using the report card. He's supposed to enter something called a 'class average'.

If he calculates the MEAN, the answer will be 70%! Try it and see!

This means that four of the five students will have report card marks that are below the class average!

Clearly this is not an accurate description of the class results, since we've already described them as being 'all but one in the 60's'. What is happening is that, in a small set of data, the mean mark is being influenced by a small number of unusually high values (in this case, one), and is not describing the data very well.

Now let's look at the MEDIAN. Here's the set of marks again:


The mean was 70%. If you list the marks in order and find the middle one, you get a median of 65%. Clearly, 65% is a much better way to describe this set of data.

You could also describe the data using the MODE, which would be 'marks in the 60's'.
Either the median or the mode would fairly describe this class's report card marks, which were all but one in the 60's.

What is happening can be summarized as follows:

For a small set of data numbers, that are not evenly distributed, the mean may be influenced by one or more high marks. The median and mode probably won't be.

A similar effect will occur if there are a few very low marks in a small class ... most of the students will show a mark well above the class average, completely distorting the true situation.

This is a well-known phenomenon in statistics, and it should mean that a teacher would want to use the median instead of the mean, for a more accurate description of the class results, when the class size is small and the marks aren't symmetrically distributed.
In fact, what the teacher should do is work out both the mean and median, and use the median if the mean is more than a percent or two different.

Unfortunately, very few report cards do this. Which means that you will always be unsure if the class average really means what it purports to mean. If in doubt, ask your teacher!

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