Below is the autograder. When entering your answers, make sure there is a comma between each question. If you skipped the question, it would look like ",,".

Small example: "A,B,CD,,E,E,E"

The score is based off of two variables: time and accuracy. The time makes up 20% of the overall score, with accuracy the other 80%. The formula can be written as follows:

$S = 0.20 \cdot f(t) + 0.80 \cdot P$

Where $S$ is score, $f(t)$ is points from time, and $P$ is points from the exam, all in percent.

The score in time is a piecewise function, meant to allot full points to those who use half the max time or less. Afterwards, it linearly decreases until it hits zero points at the max time. The formula is shown as follows:

$f(x) =
\left\{
\begin{array}{ll}
1 & \quad t \leq \dfrac{t_{MAX}}{2} \\
1 + \dfrac{t - 0.5 \cdot t_{MAX}}{0.5 \cdot t_{MAX} - t_{MAX}} & \quad t \geq \dfrac{t_{MAX}}{2}
\end{array}
\right.$

Where $t_{MAX}$ is the maximum time alloted for the exam. This ensures that "rushing" the exam will award no extra points compared to spending half the alloted time.

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:

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

For the inputs: $k$ is total number of options, $c_a$ is number of correctly chosen answers, $w_a$ is number of wrongly chosen answers, and $c$ 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 $P_c$ is the number of points awarded, $P_w$ is number of points subtracted, and $P_{tot}$ 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 $P_w$ is:

$P_c(k, w_a) =
\left\{
\begin{array}{ll}
0 & \quad \textrm{if } w_a = 0 \\ \\
\dfrac{w}{k - w}\ln\left(\dfrac{k}{w}\right) & \quad \textrm{if } w_a \ge 0
\end{array}
\right.$

The equation for $P_c$ is:

$P_c(k, c_a, c) =
\left\{
\begin{array}{ll}
0 & \quad \textrm{if } c_a = 0 \\ \\
\ln\left(\dfrac{k}{c}\right) & \quad \textrm{if } c_a \ge 0 \textrm{ and } c = 1 \\ \\
\ln\left(\dfrac{k}{3 - c_a}\right) & \quad \textrm{if } c_a \ge 0 \textrm{ and } c = 2
\end{array}
\right.$

Similarly, the equation for $P_{tot}$ is:

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

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 $c + 1 - c_a$ and can be extended to have more than two keys. However, it may not be reliable for more than two keys.

There's not a lot of cases to cover for this, but I will give a couple examples. Assume that the exam is 60 minutes long, meaning you get full credit if you turn it in within 30 minutes.

- Someone submitted the exam after a minute (somehow): 100%
- Someone submitted the exam in 30 minutes: 100%
- Someone submitted the exam in 31 minutes: 96.67%
- Someone submitted the exam in 59 minutes: 3.33%

Accuracy is a lot more interesting. 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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