## A problem with averages

This xkcd comic demonstrates a big problem with averages.

By the way, this same problem occurs when you average grades as well.

• It’s worse with grades based on percents. If a student receives a 91 percent on a first test and then receives a zero because of whatever reason, the student will need to achieve nine 100 percents to lift the grade back to an “A”.

• David Wees wrote:

Yeah, I hate averaging zeroes into a mark to "teach students lessons" or "because they didn’t do the work." It’s a horrible practice that needs to end. Find some other way to deal with the problem of students not turning in assignments!

• Most subjects other than Math or Science do not use a 0-100 scale for their grading – they rely on letter grades only… thus the lowest possible score is whatever corresponds to an “F” – usually either a 60 or a 70. Nothing prevents Math teachers from doing the same.

However, with our hopefully more confident numerical skills, nothing prevents us from using a totally different numerical scale, say 80-100, then mapping it to 60-100 at grade reporting time. In that way, a “0” would map to an 80, and would have a much smaller impact on a student’s overall grade.

What do we want for a mean grade? Say we decide on 85 for a mean.
What do we wish the standard deviation to work out to be? Suppose we expect grades to be normally distributed, and thus decide on (100-85)/3 = 5.
Based on this approach, the minimum numerical grade we assign should be not much lower than 85-15 = 70

All food for thought.

• I don’t think “the same problem occurs when you average test grades”. The point of this cartoon isn’t that one bad review unfairly drops a rating; the point is that only one of the four reviews speaks to the fundamental question “Does the product do what it was designed to do?”. All evidence is not created equal, and therefore, should be not be weighted equally.

As far as grades go, I understand that receiving a 0 on a 100-point scale is excessively punitive, however most teachers I know look to help students succeed, not to punish them. Teachers constantly create opportunities for students to meet standards and earn credit in a reasonable way. But even reasonable teachers have to have a limit: if a student doesn’t do the work, hand in the assignment, make any attempt on a test or a quiz, I don’t think giving them an 80 is a reasonable response.

• David Wees wrote:

The way I see it, averaging grades hides problems in student’s understanding, just like averaging ratings hides problems with the product. If I’m a parent, or next year’s teacher, and a student comes to me with an 80%, I have no way of knowing how they got there from the 80% average itself. I don’t know in what they are strong or weak. The grade itself doesn’t provide me with enough information to ensure that I am either able to help my child (as a parent) or target instruction to help my student (as a teacher).

• It seems like you are expecting a single number (or letter) to convey an unreasonable amount of information. A course grade is typically an average, and averages, by design, sacrifice information for simplicity. If you want more detailed information about a student’s performance, where are you going to look? Probably at more targeted averages (How does this student do on tests? Homework? Projects? Algebra tests? Geometry test? etc). Or you’ll look at narratives, or portfolios, or interviews. Or whatever combination works for you and your students.

Is the broader argument that “grades” are useless? Maybe there’s something to that, but in the end, isn’t there always going to be some kind of summary assessment when a unit of learning is complete? Does it make a big difference whether the labels are “Exceeds Expectations / Meets Expectations / Almost Meets Expectations”, “90 / 80 / 70”, or “A / B / C”?

• David Wees wrote:

"It seems like you are expecting a single number (or letter) to convey an unreasonable amount of information."

Aaah, but that’s the rub isn’t it? I’m not the one who expects this number to convey this information, but our society absolutely is. We use these grades to determine a grade point average, which is factored into decisions about the student’s life, like "does this student get a scholarship to go to college?" or "am a good student?"

• Factored is a key word here, right? It’s one piece of data considered by colleges, companies, etc.

Look, I’m certainly not going to defend colleges, corporations, and Congress for making poor decisions based primarily on poorly constructed metrics that, at best, tell only part of the story. But I think changing society’s perception of “grade” is bit beyond my sphere of influence. 🙂

Plus, we’re math folks, David. We’ve got to have some kind of belief in quantitative reasoning, don’t we!

• David Wees wrote:

As an alternative to a single number approach, I really like the International Baccalaureate Middle Years Program approach. It’s not perfect, but it does share much more information and is relevatively easy to implement. See http://stutzfamily.com/mrstutz/math7/MYPCriteriaedited.html for a sample implementation.

• I guess my primary point is: how do we decide on an appropriate value for a “lowest possible grade”. If an English teacher assigns an F (=60), why should a Math teacher assign a 0?

In fact, why do Math teachers have to use numerical grades and/or a 0-100 scale for students at all? Why not stick with the A-F scheme, as most other subjects do, and translate them into numbers only for the purpose of averaging?

Alternatively, why not have a 0-20, or 30, or 40 scale for grading all student work? It can be translated into a letter grade when grades are due to the administration.

I believe students react primarily to the intent of the grade. They know if they got the worst grade possible. The averaging scheme for grades will be important mostly to the grade-grubbers, and they are not likely to get zeroes to begin with. The students who are struggling with math need encouragement, support, and engagement. If they did not put in an effort, did not do assigned work, etc. – then they clearly are either not engaged, or have other issues that need resolving in their life. It is our challenge as teachers to find ways of engaging them. Giving them a zero might shock them out of complacency, but it not likely (by itself) to engage them.

In fact, giving a zero, which on a 0-100 scale is a very punitive grade if 60 or 70 is considered passing, can risk dis-engaging students – at which point they can be even more challenging to re-engage.

Perhaps this all is just an argument in favor of using rubrics to assess student work instead of percent of possible points earned.

• I agree that the 0 – 100 scale is kind of silly. It’s easy to see the appeal (we all like and understand percentages), but generally grades are not uniformly (or even normally) distributed from 0 to 100. NYC use the percentage system (not letter grades) for all subjects, and you can’t even really enter a grade less than 50 for a student. As with most things in schools, responsibility ultimately falls to the teacher to work out the details, to give “reasonable” grades, and to be accountable for the grading system.

I hear your point about dis-engaging students, but if a student is already dis-engaged, is receiving a 60/100 on an assignment they didn’t do going to help re-engage them? And if an engaged student is getting 0s on assignments, the issue isn’t that the 0s are going to unfairly fail the student; the issue is that the student is obviously not doing what’s necessary to be successful. That’s what the student, and teacher, need to address.

• Anonymous wrote:

a 0 mark will only be given when the assignment is not attempted at all. Illness and other forms of misadventure have the opertunity to be re-evaluated, other opertunities given to garner the marks. How can you justify giving a student >0 if they have done 0 work?

• David Wees wrote:

At my school we have separated out behaviours from performance. Students get graded on a rubric for their approaches to learning, and they get a summative grade (which includes only assignments they have handed in). Our comments that we give to parents would talk about assignments missed by a student, and we would probably contact parents to discuss these kinds of issues. When a student doesn’t complete an assignment, we don’t include it in their grade at all. If a student does not complete enough assignments for us to be able to determine their performance (in the various criterion we use) then we would probably need to give them an incomplete, although this doesn’t happen very often for us because our summative assignments are generally done during class time.

This allows us to report on the learning habits of students (such as completing assignments) separately from their performance on those learning activities.

• Another comic about issues with averaging long-term data in order to describe “current state” of growth/understanding: http://bit.ly/q9VATi Cool to see xkcd tackling a similar issue!