Statistics Grading Question

[Crimson419]

Hi all... sorry if this isn't posted in the right forum, but I wasn't sure where to post this. 

 

I am a law student at a very prestigious university, and we just had a grading scandal in one of your classes. I am not really grasping the formula the professor used (though, I don't think he has any idea what he was doing, either).

 

There were 3 sections on the exam: A, B, C, and each section had a raw score. The professor used an excel sheet to figure out the standard deviation of each section. The formula he used to calculate the final grade was this:

 

(ARawScore / Standard Deviation of A) + 3((BRawScore + CRawScore))/Standard Deviation of BRawScore + CRawScore)

 

Here is what I don't get. Is there any mathematical reason to ever divide a raw score by the standard deviation for the set of all raw scores for that section? It really makes no sense to me. What the professor purported to do was weight section A 25%, Section B 25%, and Section C 50%, but, I don't see how that formula gets that result. I think the correct formula should be:

 

(ARawScore x 0.25)+(BRawScore x 0.25)+(CRawScore x 0.50), which is quite easy... so, either I'm totally lost, or the professor has no idea what he is doing.

[TakeruK]

This sounds like "z-score" or "standard-score", where the "z-score" (http://en.wikipedia.org/wiki/Standard_score) is usually calculated by:

 

(Raw score - average score) / standard deviation

 

This sounds like what your prof did (even though you didn't specify the subtracting off the mean, it's hard to tell without knowing how everything is calculated). This is a useful statistic because it tells you how well you did compared to the rest of the class. For example, if your "z-score is +1 then it means you are 1 standard deviation above the class, which is basically the top 68th percentile, if you assume the scores are normally distributed. 

 

So it sounds like what your prof did is not only weighting "A" less than "B" and "C", but also giving lower weights to parts where the class did well (i.e. easier) compared to harder parts. That is, if you got 90% in the section where the average was 60%, that would be worth more to your grade than getting 90% in a section where the average was 80%.

 

The formula you give does not scale grades at all -- it simply makes some section worth more than others and your score is unaffected by the performance of others. 

 

Overall, scaling is a controversial practice and there is no "right" or "wrong" way to scale marks -- profs can use different schemes depending on what they want to achieve. I think the real question is whether or not grades should scaled at all, and whether or not the assumptions that go into the scaling algorithms are reasonable!

[cyberwulf]

The logic behind dividing by the standard deviation of the scores on a test (or the sections thereof) is that getting a high grade is less "meaningful"/"impressive" if there is a lot of variability in the scores than if there is little variability.  For example, consider the following sections of a test:

 

In Section 1, Student X scores a 90%. On this section, half the students scored 90%, and half the students got 50%.

In Section 2, Student X scores a 90%. On this section, all the students (except Student X) got 70%.

 

In both cases, the average section score is ~70%, but it could be argued that Student X's performance on Section 2 is more "remarkable" and should therefore be rewarded more than their performance on Section 1. Dividing scores by the standard deviations (larger in Section 1, smaller in Section 2) is one way of accomplishing this.

[Crimson419]

This sounds like "z-score" or "standard-score", where the "z-score" (http://en.wikipedia.org/wiki/Standard_score) is usually calculated by:

 

(Raw score - average score) / standard deviation

 

This sounds like what your prof did (even though you didn't specify the subtracting off the mean, it's hard to tell without knowing how everything is calculated). This is a useful statistic because it tells you how well you did compared to the rest of the class. For example, if your "z-score is +1 then it means you are 1 standard deviation above the class, which is basically the top 68th percentile, if you assume the scores are normally distributed. 

 

So it sounds like what your prof did is not only weighting "A" less than "B" and "C", but also giving lower weights to parts where the class did well (i.e. easier) compared to harder parts. That is, if you got 90% in the section where the average was 60%, that would be worth more to your grade than getting 90% in a section where the average was 80%.

 

The formula you give does not scale grades at all -- it simply makes some section worth more than others and your score is unaffected by the performance of others. 

 

Overall, scaling is a controversial practice and there is no "right" or "wrong" way to scale marks -- profs can use different schemes depending on what they want to achieve. I think the real question is whether or not grades should scaled at all, and whether or not the assumptions that go into the scaling algorithms are reasonable!

 

Maybe I misstated the question... The formula the professor used didn't scale the grades, it just produced a "score" that was then used to scale the grades. The formula I gave was a formula that I thought would represent the proper weights given to the sections and produce a score that would then scale the grades.

 

I understand the z-score, and they may have been what the professor was trying to do, but he didn't subtact the raw score from the mean score before dividing by the standard deviation. My contention is that the professor literally had no clue what he was doing and used a formula that has no real meaning, and it seems that your response confirms that in that there is no reasonable mathematical reason to divide a raw score by the standard deviation... That is, dividing by any arbitrary number would yield a result as meaningful as using the standard deviation.

[33andathirdRPM]

The professor may not know how to properly apply the technique. Following up with them on this could be a bit touchy though.

[cyberwulf]

The only thing subtracting the mean of each section does is shift everyone's scores by a constant factor, so computing z-scores it isn't really essential other than to obtain the "nice" interpretation of standard deviations above or below the overall mean. 

 

You said in your initial post that the professor wanted to achieve a 0.25/0.25/0.5 weighting for the three sections. The formula he used will only do that under very specific circumstances relating the standard deviations and number of points in each section (you can work out some example conditions -- fun exercise!). So, if that was indeed the goal, then perhaps your professor is clueless. But there isn't anything inherently "incorrect" about the scaling used.