Zero and Statistical Integrity

Part III of the reasons to avoid zeros in grading…

For folks who don’t or won’t buy into Doug Reeves’s arguments about the futility and just plain wrongness of putting zeros in a grade book and dumping them into a mean average, here are some words of wisdom from a statistics textbook author who apparently also had issues with at least one of his teachers.

“One of the most useful ideas in statistics is the representation of a collection of measures by a measure of central tendency, that is, a single value chosen in such a fashion as to by typical of the collection…

…Just what constitutes a typical (or average) measure will depend to a large extent on the level of measurement and the manner in which the measures are dispersed throughout the collection.” (Fundamental Research Statistics, John T. Roscoe, Holt, Rinehart, Winston, Inc., New York, 1969.)

The above paragraph tells us that a grade, as in report card grade, needs to be an alpha or numeric indicator of a collection of scores (marks or grades) that is truly representative of the scores.

“[The mean] is the only one of our three [measures] that is dependent upon the value of every measure in the collection. This latter characteristic is usually a valuable one, but not always, as the mean is much affected by extreme scores… (Yes, that would be a zero. — Hugh)

…Thus, the mean would not be very typical of a student’s work if the teacher makes a practice of awarding zeros when the student is absent or other wise unproductive.” (Italics mine.) (Fundamental Research Statistics, John T. Roscoe, Holt, Rinehart, Winston, Inc., New York, 1969.)

So what’s a poor teacher to do? For starters, face up to the fact that zero is just wrong for a lot of reasons. (See The Responsibility Paradox.) Zero’s not a score, or a mark, or a grade. It can’t legitimately be included in a mean average.

There’s more to this story, but for now, consider abandoning zero altogether, and give some thought to going to median averages for that part of the grade that you feel the need to calculate.

Tip: If your gradebook mean and median averages are close, you’re in good shape. If the mean is significantly off the median, you’ve probably included the forbidden “outliers” or extreme scores in the mean, which render it invalid.


12 thoughts on “Zero and Statistical Integrity

  1. I NEVER give a zero for a grade when a student is absent for a test. Either the student gets to take the test at another time, or I just don’t figure that particular test into his or her average.

    Glad to hear that, CTG. — Hugh

  2. Repairman, your posts on the zero integrity of the zero grade have transformed my teaching in so many ways. Are you, perhaps, planning on coming to the East Coast to present a workshop?:)

    I’m glad we made that connection, Miss Profe. You more than make up for the rock chuckers. 🙂

    I’m doing some sessions later this week here in Portland, but I haven’t branched out to the East Coast yet. I’ve been thinking, though… — Hugh

  3. We neither give late exams nor accept late projects. A zero is an excellent representation for the fact that the student was too lazy or irresponsible to do the work. And indicating no work *is* an accurate representation of work.

    Don’t turn it in, get a zero.

    Professor, we seem to be coming from different philosophical perspectives. In my K-12 public school world, our fundamental purpose is to help all children achieve the learning goals mandated by the state. To do this, we strive to perfect our craft, and seek never to let our grading practices produce inaccurate profiles of student achievement, or let those same grading practices deter students from achieving at a high level.

    About five percent of my undergrad and graduate profs practiced their craft in the same way, and they were revered by their students. The others “profs” who practiced “real world” grading and lectured to sleeping students were merely tolerated as a means to an end…the sheepskin.

    Your insistence that a zero is valid in a percentile frame of reference is to say that you can add an orange to a collection of apples, and still arrive at an apple that is a pure indicator of the collection of apples. (“Orange” being a zero given for late or missing work.) How do you do that? 😉 –Hugh

  4. Hey! loved this post. Son #1 has d’s in most classes as a result of 0’s. Otherwise, he would have a low A, high B.

    In my presentations today, I talked about my son going through that same crapola. I could hear Harvard’s door slam clear out here in Oregon. Getting on this trail never helped him much (as I originally hoped it would), but he roots for me every time I talk to a group about it.

    Don’t worry…your son will be fine. Another part of my educational philosophy is that no matter what we do to them at school, in terms of lousy teaching, they still wind up alright! 🙂 (I just wish the journey could be a little smoother, and that’s why I’m still in the conversation about standards-based grading seven years after my first exposure to the concepts.) –Hugh

  5. I is puzzled!
    So, if I want to get an average grade for the homework that a student does over the term, then the fact that they refuse to do it should be ignored in the mean score I report on at the end of term. Similarly, if they actually do score a zero on a test (not because they are absent) but because their answers are not correct, or they don’t provide any answers at all, that should not be represented in the mean score over the term?

    Why do you want to calculate an average grade for homework? Homework is formative assessment, and unless you’ve given a take-home test, you need only record whether or not it was done, with your evaluation, or with an INC or a zero, or a picture of a rhinoceros.

    Currently, formative assessment — and homework is only one aspect of this — is regarded as the most powerful tool we have to facilitate greater student achievement, but we have to leave it out of the grade calculation, or we compromise it’s effectiveness. (See Black & Wiliam, 1998.)

    With regard to your hypothetical case of the student actually scoring zero…you cannot escape the fact that the mean is ruined by extreme scores. You might consider using the median average, which is less affected by outliers. –Hugh

  6. I think we disagree on our understanding of what a cumulative grade represents. IMO, it is not simply a measure of the quality of a student’s work (in which case I would agree with you); it should represent the quality of a student’s work in relation to the standards. Unless each assessment covers ALL the standards, a student who has turned in all summative assessments and done well on them SHOULD have a better grade than a student who has turned in half of the assessments and done very well on them. Otherwise you are simply taking on faith the student’s ability to do well on the uncompleted (and therefore unproven) work, and that’s not fair either.

    On tests, do you excuse answers that are not completed? How far do you think this should be taken?

    Clix, at the risk of sounding like the daily horoscope, zero is wrong. Wanting it to be right won’t make it right. Find other alternatives. — Hugh

    PS: After reading Clix’s posts for today on her blog, I realize that I didn’t do justice to the question above. My apologies, Clix. The discussion that could have happened here is over on Epic Adventures Are Often Uncomfortable. — H.

  7. Grades are not as important to me as they are to the student, so I always provide students with the opportunity to make up work. I also allow them the opportunity to improve grade by re-doing assignments and re-taking tests. Doing these things are no skin off my nose, and I find that both students and parents tend to appreciate this courtesy greatly.

    That was my policy, too. 🙂 — Hugh

  8. Hi, I just stumbled on this discussion by accident and this is something that we in my Intensive English Program are worrying about.

    Many of us have no choice in the type of assignments we give our students. Because of certain “standards” and accreditation issues, we have to tow the company line on how we assess students. However, could this example lessen the problem with zeros?

    ——->Let’s say a student does not do a test or a project. Instead of , for example, a zero out of 100 points, could I enter, 50? It would not make the impact that a zero would.

    Thanks for this discussion.

    You’re correct. A 50 would have less impact and make the percentile scale function more like an A-F scale that features just one point between grades.

    If you have no other alternatives for a summative assessment, a 50, instead of a zero, would be the way to go. For a brief discussion of this particular subject, here’s a short post with a link to a likewise short paper on the subject by Douglas Reeves, Ph.D., published in Phi Beta Kappan.

    If you are required to include formative assessments in report card grades, the system needs someone — on the inside or outside — to call attention to the direction mainstream K-12 education is heading with regard to assessment for learning. Including formative assessment in report card grades essentially neutralizes our strongest teaching tool (see Black & Wiliam, 1998).

    My alternative for a delinquent summative assessment would be to enter an incomplete in the grade book and explore with the student the ways to accomplish the learning and get the assignment completed. Then, evaluate the assignment strictly on its academic merits.

    Current support ideas include mandatory assignment completion “opportunities” before or after school, or during lunch time, assisting the student with time management and organizational challenges, parent phone calls, and other methods that are suitable in your environment.

    Glad you stopped by, and thanks for engaging in the conversation. — Hugh

  9. OK, months later, but better late than never.
    Homework can be both formative AND/OR summative assessment, depending on what you set. And the reason I want to work out the average is because it is required by my school for the purpose of reporting.

    Yes, homework can be either formative or summative assessment, depending on the purpose of the assignment. The real problem with zeros, besides wrecking the mathematics of central tendency measurement, is that a teacher’s focus can be riveted to task completion rather than the learning that is supposed to take place. I always tried measure the learning, not whether the student managed to jump through a particular hoop. I have a feeling that you wouldn’t disagree.

    BTW, the most recent issue (Dec/Jan 2008) of Educational Leadership has a great little article about formative/summative assessment by Steve and Jan Chappuis of the ETS Assessment Training Institute here in Portland, OR — Hugh

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