More about measurement scales

One of the key vocabulary/concepts in an introductory statistics’ course is that of how to identify and use quantitative and categorical data. For some reason I woke up this morning thinking about it and the thought that quantitative data can be converted into categorical data and that categorical data can be converted into quantitative data.
Letter Grades

That idea seemed, very early this morning, possibly confusing to beginning stats students. But in the same early morning dream state, I thought of a perfect, to me, example to use: grades and grade point averages (GPA) – every student’s worry.

By the time students get to college, they are intimately familiar with grades and grade point averages. They all know that the most common grading scheme is A, B, C, D, and F. And they know that most schools use a numerical scale that ranges from 0 to 100 which instructors use to grade an assignment or quiz.

And usually, any score on an assignment below 60 is an “F” and a grade 90 or above is an “A,” though their instructor may not indicate that when they return the graded assignment. But they know course instructors add up their scores on homework, quizzes, exams, papers, etc., and divide by the number of graded items to get an average, say 89.1%, on all the work during a term. They know that the instructor of a course will then use some logic to decide whether or not to decide to award that 89.1 average a “B” or to somehow round-up to an “A.”

In other words, the numerical, or quantitative, averages are thrown into bins or categories of letter grades. So they know an actual process where a quantitative value, the 89.1 average score, is converted into a categorical value, A or B, for the course. (A more statistically precise way to describe this process is that the average scores were “recoded” into grades.)
And they also know that as they accumulate individual letter grades on classes, they are building a grade point average. Someone or something at the school, let’s call it the “system,” keeps track of all their course grades and converts them into a GPA. Typically, A’s are valued at 4 points, B’s at 3 points, C’s at 2 points, D’s at 1 point, and F’s get 0 points.

A question then arises: have the course letter grades been converted from categorical back into quantitative? it is tempting to say “yes” because we are talking about ‘numbers’ again. But the reality is that these ‘numbers’ are, in one sense, still aliases for the categorical letter grades.

Still, the “system” adds up all the points a student earns on their courses and divides by the number of courses to get an average grade – the GPA – which is a quantitative value, say 3.48. So the categorical letter grades have indeed now been converted back into a quantitative data point. (I should note that some statisticians would not agree with this ‘conversion’ of categorical into quantitative – the whole ordinal versus interval discussion we will save for a later date when we talk about measurement scales.)

Quantitative converted to categorical converted back to quantitative.

Cat to Quant Cycle

And the process can continue. Some schools categorize graduates into bins again – “graduate” and “graduate with distinction,” or “college-ready” or not, based on their GPA’s.

Then my mind wandered into another area where my students keep making mistakes – distinguishing discrete from continuous numbers.

But I’ll save that for another post.



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