So recently, the amount of threads that are around asking about how Study Scores and ATARs are calculated have been numerous, so I hope this will be a be all and end all guide to those questions. This thread assumes that you have a good understanding of statistics and probability which is covered in Mathematical Methods - even if you don't, it will still make a lot of sense, as long as you skip over the mathematical parts.
Study Score Your study score is a normally (Gaussian) distributed number which represents your performance in comparison to everybody else who did your subject that year. It is a probability density function with a mean of 30 and a standard deviation of 7. Essentially, the cut off for the Study Score is 50, which means that even if you are the best student in the state by a massive margin, you will still be awarded a 50.
A study score of 30 means that you are the median student in the state, approximately 50% of students are worse than you and 50% of students are better than you. Other commonly asked about numbers are 35 (top 22% of the state), 40 (top 9% of the state) and 45 (top 2% of the state) - note that these are only for studies which have a high number of enrollments, studies with fewer students may not be perfectly normally distributed.
Scaling Scaling refers to a constant amount added or subtracted to the mean study score (30) - this means that its effects will always be greatest around 30 and it will be diluted as you move up or down, away from 30. Essentially scaling is introduced in order to reflect the amount of competition in the different studies. Certain studies, such as Specialist Maths (for example) have a much higher level of competition than a study such as Further Maths - because the Specialist Maths students are better at their subject than the Further Maths students usually are. This is why a student that is in the top 2% of Specialist Maths students is deemed to be better than a student in the top 2% of Further Maths students.
This is why Specialist Maths is then scaled up, whereas Further Maths is then scaled down.
How are Study Scores calculated Study Scores are calculated by looking at the distribution of each of your three Graded Assessments (GAs). Every subject will have three GAs, each weighted differently depending on the subject. If the subject has one exam, the GAs will be such that - GA1 is Unit 3 SACs, GA2 is Unit 4 SACs and GA3 is the exam. Subjects with two exams will have one GA which represents combined SACs - which means that for these subjects, there are no Unit 3 SACs or Unit 4 SACs, they are all summed up as one mark.
So what happens is that you will get an arbitrary number for each GA which will represent your mark for the section. Usually for the exams, it will be your Exam mark, in SACs, it will be your scaled SAC mark. Now VCAA will plot your all the marks for a particular GA on a normal distribution and standardise it so that you get a Z-score - this is done by finding the mean and the standard deviations of the exam mark and using the formula to standardise (Z = (X-m)/o).
VCAA will then do the same thing with all of your three GAs, and will end up with three Z-scores. It will then do a weighted sum of the three scores and come up with the Z-score for your Study Score - which they will then "un-standardise" by introducing the mean of 30 and standard deviation of 7.
You might ask why they decided to use a mean of 30 and a standard deviation of 7. Well I think this is purely because 50 will then be (around) 3 standard deviations of the mean and 50 is a convenient number to use. (99.7% of values are within 3 SDs of the mean).
How are SAC GAs calculated SAC GAs are often where most students get lost - and it is probably because most are not familiar with it.
First of all, what is important here is your overall SAC rank. This means that if you are ranked first, it doesn't matter if you are ranked first by a big margin or by a small margin. If you are ranked first, you are ranked first.
So now, say you are ranked
n th. What VCAA will do is they will give you the
n th exam mark in your cohort as your SAC mark.
So if there are three students, Thushan, Dan and Paul in a cohort and they each get an average SAC mark of 100, 70, 60 respectively, this means that Thushan will be ranked 1, Dan will be ranked 2 and Paul will be ranked 3. Say they all sit the exam, and on the exam day, Dan's beard has grown so much that he can't see his exam paper anymore, thus, their exam marks are 100, 20, 70.
This means that for the SAC GA - Thushan will get 100, Dan will get 70 and Paul will get 20.
Shadows came across this video created by 'jpbreheny13878' on YouTube, which might be worth watching if you're still confused. Particularly on the point of what role do raw sac marks have to play in the statistical moderation process, do also take a read of the discussion later in this thread, especially the posts
Re: Guide to how Study Scores and ATARs are calculated! onwards as they provide a bit more detail in addition to this post and the video.
VIDEO
That video was posted by shadows in this thread
SAC SCALING EXPLANATION But isn't this unfair Yes, in some way or another this is unfair, especially if you have a small cohort, however, they need a way of
standardising SACs so that students who go to good schools aren't disadvantaged. For example at MHS, it is apparently very difficult to get 100% on Specialist Maths SACs, but if I go to some not-so-well-off school, it might be a breeze. So if it was all raw marks, I would just go to that not-so-well-off school - not to mention schools would then purposely set easy SACs so that their students would dominate.
It becomes a complicated system if we just consider the raw marks.
How is the ATAR calculated The ATAR is calculated by taking your English (or EL/Lit/ESL) score and adding it to your top three scaled scores, then adding 10% of the highest two of your remaining scores and this becomes the aggregate. The aggregate is then plotted as a cumulative probability density function and you are given the number that corresponds to your aggregate.
This means that if you have gotten an ATAR of 98.50, your aggregate is better than 98.50% of the state.
Questions? Post below! 
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