The case of the geometric mean in Education

When students get marks for different components of assessment in courses at school or university, then the overall mark for their course is usually some kind of weighted arithmetic mean. In this short post I want to make the case that use of the weighted geometric mean would in many cases be much more useful.

Some Equations

Lets first talk some equations, only two however, that tell us exactly what the calculational differences are between the weighted geometric and arithmetic means.

The weighted arithmetic mean

The weighted arithmetic mean of n component-marks attained for n assessments in the course, where each component is assigned a weight, and where all the weights add up to 1, is given as follows:

This is the usual way in which in most educational programmes and courses marks for different components are aggregated into a final mark. The main advantage of this way of averaging is its calculational simplicity. This calculational simplicity has however lost its competitive edge with the advent of calculators and computers ever since the 1970’s. Remarkably however it is still the main method for collating together marks from different components almost everywhere in almost every situation.

The weighted arithmetic mean however has some serious didactical downsides. Let’s just consider a simple case of a course with a mark coming from a presentation with a weight of 0.2 and a mark coming from an research report with a weight of 0.8. The student could utterly fail (i.e. get a mark of 0) in any of these components and still end up passing overall, or at least getting a non-zero mark overall. If the marks are given on a scale of 1-10 and the pass-mark is 5, then getting a 7 in the report and a 0 in the presentation would yield a mark above the pass-mark. Even getting a 0 in the report and a 100 in the presentation would do the trick.

We can formulate this characteristic as follows: in the arithmetic mean the trade-off between marks for the different components is constant and independent of the marks achieved. In the example above every 4 marks lost on the presentation can be compensated can be compensated by a single added mark on the report. It doesn’t matter what the two marks are, their trade-off ratio is fixed. This constant trade-off ratio has a consequence for the effort that the student might be willing to invest in the different components.

As a result, depending on how much the student values the effort expended on other activities (opportunity cost of learning), some assessment components will simply not be ‘worth’ investing effort in at all. For the weighted geometric mean this will not be the case and hence its use completely changes the nature of the incentives students have to learn and study.

The weighted geometric mean

The weighted geometric mean is calculated as follows

In this calculation the weight enters as the power of the mark achieved rather than as a pre-factor. The aggregation of marks is multiplicative rather than additive as with the arithmetic mean. What is immediately clear from this equation is that getting a mark of 0 in anything is not an option for the student as that will default the overall mark to 0, irrespective of the other marks obtained. This highlights an important aspect of the weighted geometric mean: the student will be required to spend some effort on all components of assessment.

The downside of using this mean is that it feels less intuitive and students will find it harder to estimate which mark they need to get in their next assignment in order to achieve a certain mean. This is especially true when your course is an aggregate of more than two marks. Calculating these means is however not difficult and basic software such as Excel will do so quickly and without much of a problem.

The great advantage of the weighted geometric mean is that is makes the trade-off between marks from different components non-constant. In fact what happens is that the different between a mark of 0 or 1 (in our example) would be very large whereas the difference between a mark of 9 or 10 would be small. The weighted geometric mean gives relatively high rewards for making at least a minimal effort while it (also relatively) discourages spending a lot of effort on achieving perfection. As a result it is an ideal aggregating tool for a series assessment components where you as a teacher think each of them is important to be taken by the student at least to a certain minimum level.

A comparison

The graph below shows you what the ‘mean’ marks would look for a course that has two assessment components (a presentation and a project report) with weighted of 0.2 and 0.8 and where marks are given between 0 and 100.

The straight lines here are the combinations of marks that yield a weighted arithmetic mean of either 40 (lower line) or 70 (upper line). The straight lines clearly show how ‘expendable’ the presentation mark is. The curved lines are the combinations of marks that yield a weighted geometric mean of 40 (lowest curve), 50, 60 and 70 (highest curve). As you see the straight lines are tangent to the curved lines in the point where both assessment components achieve the same, mean mark. The way in which the curved lines curve up from the straight lines into the higher mark regions shows how, in order to achieve a certain overall mark, making a minimal effort is important and equally how relatively mild the additional reward is for achieving only one extremely high mark.

Interpretation as workload indicators

Finally there is one more interesting observation about such a weighted geometric mean that I would like to report here. You might argue that a possible downside of it is that the weights are so ‘difficult to interpret’ for a student. Well, fortunately quite the opposite is true. If we assume, for simplicity, that

  • the mark for each of the assessment components is proportional to effort expended on that component;
  • that the overall effort is constrained to a certain maximum;

then a straightforward calculation shows that the weight of an assessment component is the optimal fraction of effort to be expended on the corresponding assessment item. The total mark achieved under that assumption would then simply be proportional to the total effort spent and the geometric mean of the efficiencies with which a student translates effort into achievement.

Conclusion

It seems to me there are good reasons to try out the use of weighted geometric means in education as a way to aggregated assessment outcomes across courses and curricula. This is not to say they are always better then arithmetic means, they are not. But the interpretational and incentivisation aspects of the geometric mean certainly has justifiable application in many contexts.

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