How to calculate Probability of "Ranking"?
I took a course with other 10 students. After the final exam, the professor reported statistical result of scores to us: 1. The range of score: 46~98; 2. The average score: 73.3; 3 The S.D of score: 14.3; I got 81, without any further information, how do I know the probabilities for me to be ranked in the 2nd place or the last 2nd place?
If you happen to find the similar question to this on other sites with a reasonable answer, I'd like to know. Or if you can incorporate your own code in whatever language, I'll appreciate your answer very much!
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$\begingroup$My way to approach this problem is to assume the distribution of the score is like a normal distribution (may not be the case in reality). This is a reasonable assumption especially when the total number of samples is large, like 1000 as you said.
Then suppose your score is $x$ and the mean and standard deviation of the sample are $\mu$ and $\sigma$. Then you can calculate the standard score $z = \frac{x-\mu}{\sigma}$ and use the normal distribution calculate the probability P(Z < z). Then this gives you the percentage of students who scored less than x.
Use your case as an example, $z = \frac{81-73.3}{14.3 }=0.538$, P(Z < 0.538) = 0.70471 which means about 70% of students score less than 81 and hence you are likely to be in the top 30%.
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