I decided to do the math, and see what is the probability of each possible grade. I assumed a bell curve, which means that most people will get middle range grades and fewer get the lower-tail and upper-tail grades.
Unfortunately, I don't know what's the equation for a bell curve. I probably need to use e somewhere, but I've no idea how to tweak that into a bell curve so I used the best equation I had on hand: a sine curve. I used 700sin((pi x)/100) and integrated that with respect to the mark ranges, and found the area of each section as a percentage of the whole area under the curve. 700, because I approximated my year to have 700 students, and I multipled x by pi/100 because I needed to widen the period so that it peaks at 50 and reaches the axis at 100.
And these are my calculations!
Click click
In descending probability:
U
A
B
D
E
C
S
Assumptions made:
- That no one can get 0 or 100. That's because I don't know how to tweak the equation to JUST include 0 and 100.
- That the bell curve peaks at the mid-range marks. I should take into account that NJ is a better school, so maybe the bell curve peaks slightly more to the right? Or we have harder papers, so the curve peaks slightly more to the left?
Therefore, you really shouldn't feel bad if you got a U, but neither should you be overjoyed if you got an A, because it's just probability working in your favour.
6 comments:
Sad fact of JC life...
Btw, don't you think that resembles a curve about how we look in a lifetime? We'd all peak at one point & our looks will slowly decline from then on.
Another sad fact of life!
- Cheryl
cheryl: The bell curve relates a lot of things, actually. Wealth and generosity, looks and amount of make-up, etc.
The equation for a normal distribution (bell curve) is e^(-x^2/2)/(sqrt(2pi)). Unfortunately it doesn't integrate nicely, but there are plenty of tables out there which will tell you the area under the curve between -infinity and whatever point you're looking for. You can modify adjust this curve to suit whatever model you want by adjusting the offset and variance parameters, so in this image you can see three different functions.
The red is your curve, scaled down to have an area of 1 between 0 and 100. The green curve is a normal distribution for what you were trying to represent. Due to the nature of the function, not all of the area can ever be contained within finite bounds, so I opted for the endpoints to be 3 standard deviations from the midpoint at 50, though 2.5 or 2 might be more appropriate*. This gets the majority (99.74%) of the area between 0 and 100.
Example:
In my experience, the majority of grading scales are designed to have the peak around 70% (my guess is that it's more likely to be around 78%, but 70 makes for nice math) on a A-F scale (A=exceptional, B=Above Average, C=Average, D=Below Average, F=Failing).
I'll compute probabilities according to the scale I'm used to using:
Math
I'll leave it as an exercise to the reader to find the proper equation and percentages for your own situation. ;)
* I calculated this by taking (100-50)/3 ~= 16 and in the second case (100-70)/3 = 10.
You could possibly try modelling it with a log-normal distribution of sorts as well....
Don't you all have MF11 which will have all the values for a normal curve?
eryanv: Lolz. You actually did the working? Got lots of time huh.
shen ting: I could, but I don't quite know how.
hejin: It's summer and there's not a whole lot to do around here. So, might as well help people on the other side of the world with their mathematical problems, right?
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