# Who’s the ‘Better’ Scholarship Candidate? (z-scores)

The following problem is what I use each year I introduce the notion of z-scores and converting position values on a Normal distribution with mean μ and standard deviation σ [ N(μ, σ) ] to scores on the standard Normal distribution with mean 0 and standard deviation 1 [ N(0, 1) ].

We use YMS The Practice of Statistics, 3rd edition, in the AP Stats class I teach. Standardized scores makes its first appearance in Chapter 2. We cover this content in early September, a time where many college bound students are busy filling out college applications, preparing resumes, and requesting letters of reference. Since our school is in the Midwest, virtually all students are familiar with the ACT. Few know about the SAT; in particular, few know the maximum possible score on sections of the SAT. This activity leads to a nice thought experiment also, where the students must put themselves in the shoes of scholarship committee members making decisions that affect students' lives.

Here's what the example above looks like worked out on the Promethean board:

This decision is pretty simple to make. The student with an ACT math score of 33 has a relative performance far more impressive than the student with the SAT math score of 705. I start with this example because the values are fairly clean and the decision is easy to make.

But what happens when the computations reveal values that do not yield an 'easy' decision? Here's the example I use to immediately follow the ACT vs SAT issue.

I like this example because it requires the students to reflect on the choice they will make about units of length. Should we convert the feet & inches measurements to decimal feet? Or to inches? Many students choose to use inches. I show students the problem and put three minutes on a countdown timer.

Then I circulate the room as students work through the problem. I listen carefully for the discussion, for the argument of which student would be the better scholarship candidate. I randomly select a student to go to the front of the room to show the work they did and to explain their thinking. An example of some student work is below.

Listening to the students argue about this decision is fascinating. Some will insist that because the female has a z-score that is a whole unit higher (5.11 versus 4.109), the female deserves the scholarship.

Others will argue that because the normalcdf command on the TI-84 yields the same value to four decimal places, it does not matter which candidate we choose; both are equally good. (This four decimal claim is because the rounding convention we use as a default is to the nearest ten-thousandth when not specified).

Another school of thought amongst the students is that because the procedure does not yield a clear result, further analysis is needed, such as academic performance, financial need, or an assessment of each athlete's moral character. These factors of consideration are student-centric.

I challenge students to think from the perspective of the athletic team or the institution. Perhaps the conference is loaded with strong female athletes, so we need a strong female athlete to be competitive. Perhaps we need to choose the athlete whose family could provide more financial support in the event we have to split the scholarship value later. Many of these institution-centric considerations do not occur to the students naturally.

Using high quality problems like this one provides another hidden instructional benefit. I always have a conceptual hook on which to hang the process of standardizing scores. If I ask "Do you remember the process for standardizing scores on the Normal curve?" and get little to no positive responses, I can always quickly follow with, "Think back to the scholarship problem, where you had to compare two different candidates to see which candidate was better." This cuts down on the time I have to spend reteaching and allows us to be more efficient during class time.

As I look to summer when I have additional time to better my practice, this is one of the first problems I will look to film when I test the waters of 'flipping' the classroom.

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