Grading Scheme

The score is based off of two variables: time and accuracy. How time is calculated depends on the year:

  • 2021 to 2023 allocates 20% to speed, with more points awarded the faster you finish
  • 2024 and forward uses a points multiplier to your score to discourage guessing under 30 min.

Accuracy

The accuracy function is based on a Danish paper, with modifications made to allow for multiple key (correct options) questions. Overall, it results in a value between -1 and 4 being awarded for each question. To get a score, you simply add all the points together and divide by the number of questions times four. The function is as follows:

P(k,ca,wa,c)={0if (ca+wa)=k or (ca+wa)=0Pc−PwPtototherwiseP(k, c_a, w_a, c) = \left\{ \begin{array}{ll} 0 & \quad \textrm{if } (c_a + w_a) = k \textrm{ or } (c_a + w_a) = 0 \\ \\ \dfrac{P_c - P_w}{P_{tot}} & \quad \textrm{otherwise} \end{array} \right.

For the inputs: kk is total number of options, cac_a is number of correctly chosen answers, waw_a is number of wrongly chosen answers, and cc is number of keys (correct options). This function awards zero points if all or none of the options are chosen, as they are equivalent. If only some options are chosen, the second equation is used, where PcP_c is the number of points awarded, PwP_w is number of points subtracted, and PtotP_{tot} is used to scale the points. This will always be -1 if only wrong options are chosen, and +4 if only right options are chosen. Examples will be covered in a later section.

The equation for PwP_w is:

Pc(k,wa)={0if wa=0wk−wln⁡(kw)if wa≥0P_c(k, w_a) = \left\{ \begin{array}{ll} 0 & \quad \textrm{if } w_a = 0 \\ \\ \dfrac{w}{k - w}\ln\left(\dfrac{k}{w}\right) & \quad \textrm{if } w_a \ge 0 \end{array} \right.

The equation for PcP_c is:

Pc(k,ca,c)={0if ca=0ln⁡(kc)if ca≥0 and c=1ln⁡(k3−ca)if ca≥0 and c=2P_c(k, c_a, c) = \left\{ \begin{array}{ll} 0 & \quad \textrm{if } c_a = 0 \\ \\ \ln\left(\dfrac{k}{c}\right) & \quad \textrm{if } c_a \ge 0 \textrm{ and } c = 1 \\ \\ \ln\left(\dfrac{k}{3 - c_a}\right) & \quad \textrm{if } c_a \ge 0 \textrm{ and } c = 2 \end{array} \right.

Similarly, the equation for PtotP_{tot} is:

Pc(k,a,c)={0if ca=0ca+wak−(ca+wa)ln⁡(ca+wak−(ca+wa))if ca≥0 and c=13−ca+wak−(3−ca+wa)ln⁡(3−ca+wak−(3−ca+wa))if ca≥0 and c=2P_c(k, a, c) = \left\{ \begin{array}{ll} 0 & \quad \textrm{if } c_a = 0 \\ \\ \dfrac{c_a + w_a}{k - (c_a + wa)}\ln\left(\dfrac{c_a + w_a}{k - (c_a + wa)}\right) & \quad \textrm{if } c_a \ge 0 \textrm{ and } c = 1 \\ \\ \dfrac{3 - c_a + w_a}{k - (3 - c_a + wa)}\ln\left(\dfrac{3 - c_a + w_a}{k - (3 - c_a + wa)}\right) & \quad \textrm{if } c_a \ge 0 \textrm{ and } c = 2 \end{array} \right.

The zero removes undefined behavior of the natural log and fraction. The second function defines points when there is a single key. It was provided by the Danish paper on grading multiple choice questions. The third function, if present, flips the funciton by the horizontal axis. It follows the rule c+1−cac + 1 - c_a and can be extended to have more than two keys. However, it may not be reliable for more than two keys.

Examples

For all examples, assume that there are five options.

First, we will cover the basic cases: none equivalent, all wrong, and all correct. The points awarded is capped at -1 for all wrong answers and +4 for all correct answers.

  • No options were chosen: 0 points
  • All options were chosen: 0 points
  • 1 wrong option was chosen: -1 point
  • 2 wrong answers were chosen: -1 point
  • 1 correct options were chosen: +4 points
  • 2 correct options were chosen: +4 points

If both right and wrong answers are chosen, less points are awarded. The idea is that the student got it right, but was not as sure as an all-correct student. This tapers off from the maximum +4 down to 0. For these examples, assume that there is only one correct option:

  • 1 correct and 1 wrong: +2 points
  • 1 correct and 2 wrong: +1.3 points
  • 1 correct and 3 wrong: +0.9 points

This gets more complex if there are two correct options. Picking only one of the two options will award less points, then choosing both options. This is compounded by choosing wrong options. For these examples, assume there are two correct options:

  • 1 correct only: +1.5 points
  • 1 correct and 1 wrong: +0.7

If both correct options are chosen and wrong options are additionally picked, it simply follows the "one correct option" examples.

For an interactive view, check out the Desmos.

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