0
$\begingroup$

If I have a collection of student scores, what is a good way to create a 'single vs. group' statistic? What I would like to know is how the single score compares to the group of scores against which it is juxtaposed.

So, for example: lets say that the test permits values from 1 to 9 (inclusive), and we are looking at a score of 3. We could say that 3 is around 33% - but this doesn't factor in the other scores: what if 3 was the lowest score of the group? What if it was the highest?

Even just a one-word answer pointing me in the right direction of some research I could do would be appreciated. Obviously even better if you can help me understand :-).

$\endgroup$

1 Answer 1

1
$\begingroup$

You can use the empirical cumulative distribution function $ecdf$, which tells you what fraction of all scores are equal to or below some score.

Say your scores are $\{1, 1, 1, 1, 3, 3, 3, 4, 5, 5, 5, 6, 6, 6, 8, 8, 9, 9, 9, 9\}$. Then $ecdf(3)=0.35$, meaning that student is in the top 65% of the class, there are 35% equal or below.

In R you can do it like this:

a <- c(8, 6, 9, 8, 3, 5, 5, 9, 1, 6, 3, 1, 9, 5, 3, 1, 1, 4, 9, 6) # sample data
f <- ecdf(a) # compute the ecdf
f(3) # how many at or below 3?
[1] 0.35 # 35%
f(a) # check all students
[1] 0.80 0.70 1.00 0.80 0.35 0.55 0.55 1.00 0.20 0.70 0.35 0.20 1.00 0.55 0.35 0.20 0.20 0.40 1.00 0.70
$\endgroup$
2
  • $\begingroup$ Thanks for your answer Julián, I will read further on it. An additional question though - if we were looking at ecd f(3) as you proposed, I can see that 35% of the sample obtained 3 or below. But could you say that a score of 3 is in the 'top' 65%? Intuition tells me that a score of 3 is in fact the top 80% (16 scores of 3+ out of 20 = 0.8). Is my logic flawed here? Just trying to understand the relationship between the 'top' and 'bottom' percentages. Thanks $\endgroup$
    – Katstevens
    Jun 24, 2013 at 1:47
  • $\begingroup$ Yes, it all depends on how you deal with ties. As ecdf gives you "less than or equal" it is always safe to say "bottom X%". $\endgroup$ Jun 24, 2013 at 1:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.