# Comparing Grades Across Grading Systems

## Statistics can help guide our understanding of grades. Here's how.

Posted Feb 21, 2021

My son Andrew is currently in the midst of the college application process. My daughter Megan is a junior in college. And in my job as a college professor, I constantly find myself in conversations surrounding grades. In my world, it's hard to avoid thinking about grades at times.

A pretty typical issue pertains to the fact that different schools often use different grading systems. And these systems may differ from one another in a variety of ways, such as:

- Some schools have GPA (grade point average) on a scale of 0-100 while others have it on a scale of 0-4.0.
- Even for schools that use the same scale, one school may have a higher mean (i.e., average) GPA than another school (the average GPA at one school might be 75 while at another school it might be 92, for instance).
- At some schools, some additional points may be added to correct for taking relatively difficult classes (such as AP classes) while this same correction may not be implemented at other schools (or a different mathematical correction might be implemented at other schools).

The other day, we were sitting around discussing a specific situation. One kid goes to a school that allows GPA, based on the adding of extra points, to get not only over 4.0 but, actually over 5.0. That kid's GPA was in the 5.3 range. We were like: *What does that even mean?* *How can that grade be compared to someone whose GPA is at a school that is on a genuine 0-100 scale, with 100 being the absolute highest?*

As millions of high school seniors and their parents scramble to get a grip on this stuff as they try to negotiate with schools regarding financial packages, etc., I thought it'd be useful to provide a clear statistical framework for understanding this broader issue.

**Meet Tito and Teresa**

So here is a simple example: Tito and Teresa are high school seniors at two different high schools. Tito's GPA is 94. Teresa's GPA is 3.73. The grading systems are on different scales. You are on the undergraduate admissions committee at a university and are trying to figure out which of these two students, who are basically equal on all other parameters, should be ranked higher in your list of applicants.

Luckily for you, you have taken a basic course in statistics years ago. You recall your professor saying, "This stuff will be relevant in life at some point. I promise!" You realize at this very moment that, in fact, your professor was right about this premonition.

**The Beauty of Standardized Scores**

While I have served on graduate application committees on several occasions, I have never formally worked in an admissions office. This said, whenever faced with a situation like Tito and Teresa's, I automatically think about something that statisticians refer to as *standardized scores* (see Geher & Hall, 2014). A standardized score provides a way to think about a score regardless of the scale that the score came from.

Perhaps the most basic form of a standardized score is a z-score. A z-score is essentially a comment on a standardized score. It tells us exactly *how many standard deviations above or below the mean (average) that a particular score is* on the scale that it came from.

Note that *standard **deviation* is an index of how much the scores from that original sample (or population) vary from one another. One standard deviation roughly corresponds to the average amount that scores in a sample (or population) deviate from the mean. So if the mean of a sample is 85 and the standard deviation is 8, then roughly, scores in that sample vary from the mean on average by about eight points.

The z-score formula is very intuitive. It is this: *Z = (X-M)/SD*, meaning that someone's standardized score is a function of how much that person's raw score deviates from the mean (or average) score (*X-M*) relative to (i.e., *divided by*) the standard deviation. So if Tito's GPA is 94 and the mean for the population of scores is 85 and the standard deviation is 8, then Tito's z-score is, based on the z-score formula, 1.125. That simply means that his raw score of 94 is 1.125 standard deviations above the mean based on where it is relative to all the scores for the seniors at his school.

Now imagine that at Teresa's school, the mean GPA is 2.8 and the standard deviation is .6. We can now convert her GPA to a z-score to see where her score is relative to the other seniors at her school. We'll use the same simple formula, *Z = (X-M)/SD*. Here, we get a z-score of 1.55. Thus, compared with all the seniors at her school, Teresa's GPA of 3.73 is 1.55 standard deviations higher than average.

Now we can compare Tito and Teresa's scores with each other because we have put them on the same scale as one another. Tito's z-score is 1.125, while Teresa's is 1.55. Both students should be applauded for their academic efforts. Both have grades that are more than one standard deviation above the means in their classes. However, Teresa's GPA is a bit higher relative to her grade in comparison with Tito's GPA. All things equal, Teresa is looking slightly stronger than Tito academically when we use the standardized scoring method to compare their GPAs directly with one another.

**Bottom Line**

As students and others around them think about grades and the college admissions process, it might be useful to think about the simple elegance of standardized scores which provide a way to directly compare grades from across two different grading systems with one another. Which score is better: a 94 or a 3.73? Only standardized scoring can really address this kind of question.

I hope people find these ideas useful in thinking about the broader issue of comparing scores of the same construct (e.g., high school grades) from across different scales. And I hope this post gets the reader to truly see that statistical processes are applicable to everyday life. It's true!

*For a broader survey of statistics written in a straightforward manner, check out my book (co-authored with Sara Hall), *Straightforward Statistics: Understanding the Tools of Research.