Interpret the z-score (like it's your job)
Most of our students will have no trouble calculating a z-score. But interpreting the z-score is a much more important concept for their future success. Here is one approach to get them there:
Start with the why:
Rather than shoving the formula in front of students, try to put them in a situation where they discover the need for the concept of a z-score. Students are given this scenario and asked to discuss in pairs:
The Chapter 1 AP Statistics test had an average of 70 and a standard deviation of 10 while the Chapter 1 AP Psychology test had an average of 70 and a standard deviation of 5. Luke scored a 90 on the AP Statistics test and an 85 on the AP Psychology test. Which of Luke’s test scores is more impressive?
AP Stats test. Mean = 70 SD = 10 Luke test score = 90
AP Psych test. Mean = 70 SD = 5 Luke test score = 85
Of course at first glance, the AP Stats test looks more impressive as 90 > 85. But some students will start to realize that the AP Psych test is actually more impressive, as the score is 3 full standard deviations above the mean while the AP Stats score is only 2 standard deviations above the mean.
When students share out their thoughts on this short activity, ask them to explain the (simple) mathematics they used to arrive at 3 and 2, and we get this:
At this point, don’t worry at all about using Greek letter notation, as we are building a concept.
Develop the concept of the z-score.
Next, we calculate the z-score for several students for the AP Stats test (mean = 70, SD = 10).
Luke test score: 90 z-score = 2
Biff test score: 60 z-score = -1
Marty test score: 88 z-score = 1.8
In pairs, have students discuss the meaning of the z-score. What is a z-score of 1.8 telling us? And we arrive at the goal of the lesson:
Student: “Marty’s Chapter 1 test score is 1.8 standard deviations above the mean”
Make it someone's job.
Because the z-score interpretation is so important, I hire a student to be in charge of this interpretation. This is a full year commitment and is a paid position ($1 paid in full at the end of the year…but it is a great resume builder).
Now every time in class that we calculate a z-score (and later a t-score), I call on that student to interpret: “Student, what is this z-score telling us?”
I often follow up with the question “Is this far from the mean?”
Why does this matter?
Many stats teachers will claim that the reason we need students to calculate and understand z-scores is so that they can use a Table to find area under normal distributions. But now technology exists (applets, normalcdf) that can do this for us.
So why do we still bother with z-scores?
OK, so you might be in your first year and this is not humanly possible. But next year you will have this test written before teaching the unit. You must be familiar with your own test and the rubric used to grade it. This will inform your instruction. You will know better which information to stress and you will choose your activities to help support your students’ learning and ability to be successful on the test. If you have a well-written test, then everything that happens in class should be moving your students towards success on that test. (yes I just said teach to the test).
Take me to the P-value
The concept and interpretation of the z-score later becomes the concept and interpretation of the test statistic when we get to significance testing. We will then use the test statistic to calculate a P-value and ultimately make a decision.
We all know that the P-value is the holy grail of introductory statistics. But we want students to understand the interpretation of the P-value, not just the calculation. How likely is it that we would get this result? Is this result unusual? To answer this question, we must know how far our results are from what was expected (z-score!!). And this is the reason why the follow up question is so important:
Start: “Student, what is this z-score telling us?”
Follow up: “Is this far from the mean?” ‘
And later when doing normal distribution calculations: “What kind of an area are you expecting to get for an answer here”
We want students to start to think about how likely it is to get our result. If the result is fairly close to the mean (low z-score), then it is not surprising. If the result is far from the mean (high z-score), then it is unusual.