Correctly averaging grades when tests are over different points

Question

Suppose in a school the grading system is from $60,$ and you've done $3$ tests, $1$ over $60,$ and the other two were smaller tests over $40$ and $20$ points respectively. I am trying to learn what the right way of adding up and averaging the grades should be. Admitteldly, right way is a vague statement, so allow me to present two most straightforward ways that come to mind. For illustration, we will assume the following results of a student: $\displaystyle\frac{35}{60},$ $\displaystyle\frac{15}{40}$ and $\displaystyle\frac{20}{20}.$

(A) Adding up the results of the 2nd and 3rd tests and averaging with the 1st

Since the 2nd and 3rd tests were over $40$ and $20$ they add up to $60$, so we can treat them together as one complete test over $60$. In this approach the student's average is: $$ a = \frac{35+(15+20)}{2}=35. $$

(B) Covert the 2nd and 3rd grades to count over 60 then average of 3 tests

Plan: We will simply use a cross-multiplication (rule of three) to convert the grades to their equivalent over $60$. Then we have 3 grades to average over.

Denoting the equivalent grades of the 2nd and 3rd tests over $60$ points with $x_2$ and $x_3$ respectively, we have: \begin{align} \frac{15}{40} &= \frac{x_2}{60} \Leftrightarrow x_2=\frac{15\cdot 60}{40}=22.5 \\ \frac{20}{20} &= \frac{x_3}{60} \Leftrightarrow x_3=\frac{20\cdot 60}{20}=60. \end{align}

Therefore, in this approach the average is now over 3 tests which is: $$ b = \frac{35+22.5+60}{3} = 39.1\overline{6} \approx 39. $$

Observations and questions

• The most noticeable aspect is the $4$ point difference between the two averages of the students' grades. Approach (B), at least from this example, seems to be favourable for the student. But I don't know if this is true both for those having low grades and as well as those scoring always high.
• My intuition for the latter is that, the conversion bears the effect of making good scores in small quiz count as much as good scores in full tests over 60. This does seem a bit unfair as it is generally harder to score well in longer tests, so intuitively, a good grade over $60$ ought to be given a higher merit than the collection of smaller tests.
• An advantage of (B) is that the tests do not have to carry over total points that necessarily add up to 60. This makes the grade distribution per test easier.

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1. Are these two approaches expected to result in very different averages in general?
2. To me approach (B) makes the most sense, because the averaging is at least performed over the actual number of tests done. But is there a (simple) statistical argument that allows to discern between the two? For instance, is there an unjustified assumption that is being made in either method?
3. Are there other approaches (e.g., averaging differently) one could consider, that would maybe be "fairer"? For instance in view of the fact that either approach seems to favour mainly one specific group of students (very low grades, high scores, or in between cases).

Answer 1

I would rather use relative scores (as opposed to absolute scores, as you are trying to do). So for your example this would be the mean of $35/60 \approx 0.5833333$, $15/40 = 0.375$, $20/20 = 1$, the mean being $\approx 0.6527778$.

Alternatively, if you want to take into account the effect of the tests, i.e. tests with more points are harder and this should be considered when evaulating this mean, then I would use a weighted average, with the total possible scores as the weights. In this case the weighted mean would be $\approx 0.5833333$.