Grade Calculations

GB2 Grade Calculations

The calculations for the new gradebook have two main modes:

• points-based calculations : in which the ratio of an item's max points versus the total points possible for all items scored is used to establish the item's 'weight' within the gradebook
• weight-based calculations : in which the product of two assigned weights at the category and item levels is used to establish the item's 'weight' within the gradebook

In combination with these modes, there are a couple of additional parameters that influence how grades are calculated:

• #Project Extra Credit toggles whether extra credit calculations will be 'projected' or scaled against the overall gradebook or if they will be relative to only the items that have been already graded
• #Weight By Points allows an instructor to use points-based calculations to determine the relative weight of items within a weighted category, rather than requiring additional 'weights' to be set

Points Based Calculations

There are two cases for points based calculations, but the distinctions between the two are very slight. Whenever the 'organization' of the gradebook is set to "No Categories" or "Categories", points-based calculations will be used, independent of whether the "grade type" is set to "points" or "percentages". In setting up the gradebook, an instructor must provide a maximum number of points per item, or if one is not entered, a default value of 100 will be used.

So, in "No Categories" mode, we might create the following gradebook:

• My Default Gradebook
  • Item 1 (25 points)
  • Item 2 (25 points)
  • Item 3 (25 points)
  • Item 4 (25 points)
  • Extra Credit Item (10 points)

This gradebook would have a total possible non-extra-credit point value of 100, with 10 extra credit points possible.

Each student's course grade is then calculated according to the point value (or in percentages mode, the percent of max points) assigned by the instructor, as per the following equation:

S = sum of all non-extra-credit points earned
E = sum of all extra-credit points earned
P = sum of all non-extra-credit points possible for scored items

% score = (S+E)/P

Of course when we include the ability of instructors to excuse an individual student from an individual item, we discover that P can vary on a student-by-student basis, as shown below:

Fig. 1

Note that in this case "Joe" gets more of a boost from his 10 point extra credit item than Melody does, simply by virtue of having been excused from one item . . . this is logical, since 10/75 (0.133) is larger than 10/100 (0.100), but it may be unintuitive at first glance to some instructors.

In "Categories" mode, calculations are still point-based, and work along the same fundamental principle, though when it comes to extra credit there is a further complication. By organizing the points into categories, we end up with the idea of an item that contributes extra credit to that category specifically, where it doesn't make sense to determine the category contribution to the overall score unless at least one non-extra-credit item has been graded for that category.

• My Default Gradebook
  • Category 1
    • Item 1 (25 points)
    • Item 2 (25 points)
    • Item 3 (25 points)
    • Item 4 (25 points)
    • Extra Credit Item (10 points)

So under this case, we have the same basic calculation as above, but if we add a student who only completed the extra credit item, we get another non-intuitive case:

Fig. 2

As you can see, it is not possible to calculate a 'category score', or in the case of this gradebook, a course grade, for Francis... s/he has received 10 extra points, but since P is zero, we get a division by zero exception.

Weight Based Calculations

• My Default Gradebook
  • Category 1 (60% of course grade)
    • Item 1.1 (25 points) (25% of category, 15% of gradebook)
    • Item 1.2 (25 points) (25% of category, 15% of gradebook)
    • Item 1.3 (25 points) (25% of category, 15% of gradebook)
    • Item 1.4 (25 points) (25% of category, 15% of gradebook)
    • Extra Credit Item (10 points) (10% of category, 6% of gradebook)
  • Category 2 (40% of course grade)
    • Item 2.1 (15 points) (40% of category, 16% of gradebook)
    • Item 2.2 (15 points) (30% of category, 12% of gradebook)
    • Item 2.3 (15 points) (30% of category, 12% of gradebook)

Weight based calculations are used whenever a gradebook is in "Weighted Categories" mode. In this
case the instructor must lay out the desired weighting of each category and each item within those categories.

These weights are effectively a best case scenario, in which a given student has scores assigned for every item. If a score has not been assigned for an item or if the student has been excused from that item or if that item has been dropped as a lowest score within a category then that item will not be included in the calculation for that student and the weights will be proportionally modified based on the items that are scored.

So in the table below, the hyphen indicates unscored items, and otherwise a weight is provided to indicate the course grade weight that item will contribute:

Fig. 4

That is, Joe has a score for all items, and so receives the default weights for those scores. Melody has one item in Category 1 ungraded, and so each of the other three items in that category increase their weight proportionally to 33.33% of the category or 20% of the course grade; she also is missing a score for Item 2.1, so Item 2.2 and 2.3 each increase to 50% of the category and 20% of the course grade. In Francis' case, since she has only one item in each category graded, they each make up the full category weight, 60% and 40% respectively. Roderick, finally, has only one item graded and so that item's score is 100% of his course grade.

This is an essential strategy in order to cleanly project grades throughout the quarter/semester/trimester, so the instructor can see how well his or her students are doing based on the items that have been graded, even before all the scores are in.

Calculating Weighted Category Grade w/ Drop Lowest

When calculating a category score with the number of drop lowest items > 0, it is necessary to first calculate the re-weighted scores for all items for that student based on which items are graded and not excused, then to remove the lowest values and recalculate the weights again.

This procedure is illustrative of the process in general, since when drop lowest = 0, the secondary recalculation is no longer necessary.

A category can only drop the lowest X items if all items under that category are equally weighted. Otherwise the drop lowest value is assumed to be 0.

w _i = item weight within a category
W = sum of all included item weights within a category excluding extra credit items (the instructor's "best case" weights)
r = 1 / W
p _i = percentage score for a given item (points earned / points possible)
x _i = p * w (weighted score for a given item: x ₀, x ₁ . . . x _N)
y _i = w * r (scaled weighted score for a given item: y ₀, y ₁ . . . y _N), though with extra credit items r is fixed at 1.00
Z = sum of all the student's scaled scores for a given r

1. Sum the student's scaled weighted scores (y ₀, y ₁ . . . y _N)
2. To determine the drop lowest order, sort the scaled scores
3. If a drop lowest number is defined, drop that many items from the calculation
4. Recalculate W based on the items that are still included
5. Recalculate r based on the new W
6. Recalculate Z based on the new r

So, if we were to calculate the "Category 1" grade for Joe, we first need to examine the desired category weights for each item, as in the table below:

Fig. 5

W = 1
r = 1

We can then look at Joe's actual percentage scores for each item and calculate W, r, w0..wN, x0..XN, and Z:

In this case, of course, x _i = y _i, since r = 1. This means that Joe has been graded for all items in this category.

Therefore,

Z = 0.2375 + 0.2000 + 0.2175 + 0.1975 + 0.1000
Z = 0.9525

That is, his base grade for this category without including extra credit it 85.25% with an extra credit of 10%, so his category grade overall is 95.25%.

We can check this calculation easily, since in this particular case all items are equally weighted, so we can simply take the average of the four non-extra-credit items

average = (0.95 + 0.80 + 0.87 + 0.79) / 4 = 0.8525 = 85.25%

We can then repeat this calculation for some arbitrary "Category 2" scores for Joe:

W = 1
r = 1

So again, we can calculate W, r, w0..wN, x0..XN, and Z:

Therefore,

Z ₂ = 0.3400 + 0.2640 + 0.2610
Z ₂ = 0.8650

Calculating the projected course grade

Once the category scores have been calculated, a similar procedure is followed to determine if the the category weights must be redetermined (for example, if a given category has no graded items below it).

g _j = category grade for each included category, as calculated from algorithm directly above
c _j = category weight for each included category
h _j = g _j * c _j
C = sum of all included category weights within a gradebook (the instructor's "best case" weights) where g _j for that c _j is not null
r = 1 / C
v _j = h _j * r
% score = sum of all h's above (h ₀, h ₁ . . . h _N)

So, if we were to calculate the course grade for Joe, assuming the following:

C = 1
r = 1

% score = 0.5715 + 0.3460 = 0.9175 = 91.75%

Melody's Case

So, if you remember from the table above and copied below, Melody had two items unscored or excused, one in each category. The result was that the overall course grade percentages for her graded items increased in value, as shown in this table. *Note that these percentages are not category weights, but course grade percentages, that is, the product of an item weight and a category weight:

If we assign some arbitrary scores for these items, we can perform the same calculations we did above for Joe, but taking into account the excused/missing items.

Melody's "Category 1" Item Weights:

W = 0.75
r = 1.3333333 (repeating)

So again, we can calculate W, r, w0..wN, x0..XN, and Z:

Therefore,

Z ₁ = 0.2967 + 0.3100 + 0.3267 + 0.1000
Z ₁ = 1.0334

but since a category grade cannot contribute more than the desired weight for that category, we truncate this down to 1

Z ₁ = 1.0

Melody's "Category 2" Item Weights:

W = 0.6
r = 1.666666 (repeating)

So again, we can calculate W, r, w0..wN, x0..XN, and Z:

Therefore,

Z ₂ = 0.435 + 0.455
Z ₂ = 0.89

Melody's projected course grade

C = 1
r = 1

% score = 0.6000 + 0.3560 = 0.956 = 95.60%

Summary of example scores

Note that, taking each student above individually, the hyphens above could be the result of any of the following:

• Dropping the lowest score in a category
• Excusing that student from that one item in that category
• Choosing not to enter a grade for that student for that item

Project Extra Credit

Basically the idea is that in "categories" mode, there are two options:

(1) Project Extra Credit = TRUE
(2) Project Extra Credit = FALSE

With (1) the instructor is saying that she wants to see extra credit contribute to the score ONLY as much as it will once all grades are entered. So if the total non-extra-credit points in a gradebook are 100, and you enter 5 points of extra credit, the percentage contribution of those 5 points will be +5/100 = 0.05

With (2) the instructor is saying that he wants to see extra credit contribute EXACTLY the number of points that are entered. So if only 1 non-extra-credit item (with 10 points possible) of the above gradebook is graded, then the percentage contribution of the 5 points extra credit will be +5/10 = 0.50

In "weighted categories" mode, the impact of "Project Extra Credit" is similar, but it has a substantially different impact on scores if you have an Extra Credit category with more than one item in it. That is, when "Project Extra Credit" is TRUE in "weighted categories", it means, assume that the extra credit contribution can be for the whole category weight even if only one item is graded. If "Project Extra Credit" is false, then the extra credit contribution will be bounded to the % grade for that item.

Weight By Points

Weight by points is only applicable when the mode of the gradebook is set to "Weighted Categories". The grade type can be any valid grade type: points, percentages, letter grades. In "Weighted Categories" mode a checkbox labeled "Weight By Points" will be visible for any category when editing.

Checking this box will recalculate the % category values for the items under this category to match the relative values of the underlying points and scaled as though the total points for that category represent 100% of the category weight. So if you have 10 items with 10 points each, of course they will each have 10%, but equally, if you have 10 items with 40 points each, they will have 10% each.