[Math] Formula to generate a score from 1 to 100 based on 2 percentages

algorithmsanalysisprobabilitystatistics

I am trying to come up with a formula that will result in a score of 1 to 100 (never anything lower or higher). I have two numbers that I can use to come up with this score, a specific percent and an average percent (from all of the others in the set).

The score would essentially not be able to reach the upper or lower bounds, just get infinitely closer. The idea would be to assign a score of 50 if the specific percent matched the average percent, and scale from there. I know I've done something like this in the past, but I've been racking my brain off and on all day and haven't been able to come up with anything. Any ideas?

Edit: I forgot to mention the most important part. The percentages will for the most part be between 0.01% and 2.0%. So the average could be something like 0.42%, etc.

Best Answer

Do you need to come up with the function before you have the data, or can you look at the whole data set before choosing the function? If you have all the data, percentiles sound like they do what you want.

If you can't examine the data in advance, it would still help to have some idea of the scale. For example the specific scores might always stay within +-10% of the average, or they might range by a factor of 1000. If you choose a function that keeps the ones a factor of 1000 within bounds, the ones within 10% will be very compressed. One example would be to start with the logistic function $f(t)=\frac{1}{1+\exp(-t)}$, which takes input in $(-\infty,\infty)$ and returns values in $(0,1)$. If values in (0,100) are acceptable, you could set t=(specific percent-average percent)/scale, then score$=\frac{1}{1+\exp(-t)}$, choosing scale to reflect the range of interest.

Added: with your edit, maybe setting average percent to 1% and scale to 0.5% or 1% will meet your needs. You could try some values in a spreadsheet and see what you think.

Added2: that is why I talked about the scale factor, as your values are so close arithmetically. If scale$=1, t=0.003-0.0042=-0.0012$ and score$=0.4997$. But maybe you should use a scale of $0.001$. Then $t=-1.2$ and score$=23\%$. An idea how this works using average$=0.0042=0.42\%$, scale$=0.1\%$ is (all amounts in percent-the percent signs ruin the formatting) $$\begin {array}{c c} raw & score \\ 0.00 & 1.48 \\ 0.10 & 3.92 \\ 0.20 & 9.98 \\ 0.30 & 23.15 \\ 0.40 & 45.02 \\ 0.50 & 69.00\\ 0.60 & 85.81\\ 0.70 & 94.27\\ 0.80 & 97.81\\ 0.90 & 99.18\\ 1.00 & 99.70 \end{array}$$

Related Solutions

[Math] Dropping a Lowest Score

Let $t_i$ and $s_i$ be the possible points and actual points scored, respectively, for each individual assignment, and let $T$ and $S$ be the respective sums over all assignments. Assuming that the goal is to produce the highest percentage for the student, you’re trying to maximize ${S-s_i\over T-t_i}$. With a bit of algebraic manipulation, this can be rewritten as $\frac S T+{St_i-s_iT\over T(T-t_i)}$, so drop the assignment that maximizes ${St_i-s_iT\over T(T-t_i)}$.

For the small example in your comment to Travis’ answer, these values are $0.129$, $-0.017$, $-0.026$ and $-0.060$, so the first score should be dropped. For the slightly larger data set in those comments, this method selects the 15/35 score ($0.036$), just as you had originally computed. For the example in your question, “Hydrate” is the winner with a value of $0.045$, so it should be dropped, which also agrees with the method you’d been using.

Of course, if you’re doing this in a spreadsheet anyway, you can compute ${S-s_i\over T-t_i}$ for each assignment and have the spreadsheet find the highest value for you, as Henry points out in his comment to your question.

Algorithm For Ranking Dates Based On Deviation From Specific Date

I've read through your descriptions and examples and it is not at all clear what you are trying to say, but I was able to glean a few things.

I think you probably want a normal distribution:

$$y = e^{-a(x - d)^2}\times 100\%$$

In programming parlance

y = 100 * exp(-a*(x - d)^2)

Where $x$ is the current date, $d$ is the target date (Christmas, in your example), $a$ is an adjustable parameter, and $y$ is the output value.

$y$ will be at $100\%$ when $x = d$, but the greater $x$ differs from $d$ in either direction, the lower $y$ will be. The parameter $a$ controls how fast $y$ decays as $x$ moves away from $d$. The bigger $a$ is, the faster $y$ will fall.

Some important characteristics of this function:

It is symmetric about $d$. 3 days before will have exactly the same value of $y$ as 3 days after.
$y$ never drops all the way to $0\%$. This is unlike your attempted formulas, which will drop below $0$. But the normal distribution just tapers off. 10,000 years in the future would still give a ridiculous small, but non-zero value (in exact mathematics - in computing, at some point it becomes too small for the computer to handle, so after that it will just return $0$).

There are a multitude of functions that you could use. If these characteristics are not right for your purpose, then there are other choices.

Best Answer

Related Solutions

[Math] Dropping a Lowest Score

Algorithm For Ranking Dates Based On Deviation From Specific Date

Related Question