How to work out an average grade
How do we work out the average of the grades we got at school? First let’s look at the simple method, where all grades have the same weighting. Imagine that you get these grades in five tests (on the 9–1 scale, where 9 is the best):
That last test really spoiled the average for us, didn’t it? To work out the average, we add all the grades together:
Now we divide this result by the number of grades, that is, we divide by five:
And that’s all. The average grade is exactly 7. It can happen that we don’t get a whole grade. For example, if we had revised harder for the last test and got a 5, the total would be
and the average would be
But on the school report we would probably still get a 7.
Weighted grades for end-of-term tests
Sometimes some tests are more important than others. For example, an end-of-term test may count twice as much. How would we work out the average grade if our ordinary tests gave us grades 7, 5, 3, 5 and from the end-of-term test we got a lovely 9? If a grade has double weighting, we work out the average grade in the usual way, but we count the grade with double weighting twice. So we work out the sum of the grades
and divide by the number of grades, but the end-of-term test counts as two grades. We have four ordinary grades and “two” grades for the end-of-term test. So we divide the sum by six:
We get an average grade of 6.333, which looks like a solid 6 on the school report. If the end-of-term test had three times the weighting, we would count the 9 three times:
and when dividing we would also count it as three grades, so we would work out
After rounding, the average grade would be 6.71.
What about a 7−?
What if you get a 7− in a test? Well, it is hard to say how to count it, because “minus” is not exactly a precise mathematical term in this case. A “7−” usually means “a weaker 7”, which we could turn into a number as if we had got a 6, halfway between 7 and 5. So if we got the grades 9−, 7−, 7, 5, 9, we work out the mean as
and divide by five as normal, because we are working with five grades:
The average grade is 7. If as well as minuses you also get exclamation marks, pluses or I don’t know what else, you will have to ask at school for the conversion rate 🙃.
Even more weighted grades
What if the weightings of the grades were even more complicated? We could, for example, have these grades: 3, 5, 5, 1, 3, where the first grade has half the weighting, the second a third, the next the ordinary weighting, the next double and the last two and a half times the weighting. Let’s write it in a table to make it clearer:
| weighting: | ½ | ⅓ | 1 | 2 | 2.5 |
| grade: | 3 | 5 | 5 | 1 | 3 |
In the first row we have the weightings, in the second row the grades. To work out the average grade, we multiply each grade by its weighting. We store the result of each product in the third row of the table:
| weighting: | ½ | ⅓ | 1 | 2 | 2.5 |
| grade: | 3 | 5 | 5 | 1 | 3 |
| product: | 1.5 | 5/3 | 5 | 2 | 7.5 |
Now we work out the sum of all the products, that is, we add up the third row:
Now we work out the sum of all the weightings, that is, we add up the first row of the table:
Now we take the sum of the weighted grades and divide it by the sum of the weightings:
The average grade is approximately 2.8 after rounding.
See also
Chapters: Mean (average), Median, Mode, How to work out an average grade, Geometric mean, Harmonic mean
