Notes for Introducing Z-Scores

My statistics students are moving into z-scores.  Z-scores are the sort of thing that sound terrifying, but they aren’t really that bad if you understand what the number actually means.  While trying to figure out how to introduce the topic, I asked my fiance for advice.  

Shaun suggested that I have students find z-scores based on the definition of a z-score before introducing them to the formula.  Of course, he also informed me that they were pronounced as “zed-scores”.

That idea led to this handout:

I read the definition of z-score to them and set them loose to work through the two problems in their groups.  I did instruct them that values above the mean would have a positive z-score and values below the mean would have a negative z-score.

I expected them to struggle with finding the z-scores that ended up being decimals, but soon I heard groups throwing out decimals and fractions on their own.

After going through each answer, I decided to pose a few scenarios to see how they were understanding z-scores.

Imagine your teacher gave you back your test and instead of writing your score, they wrote your z-score.  What type of z-score would you want to see?  

What would a positive z-score mean?  What would a negative z-score mean?  What would a z-score of zero mean?

What if we measured everyone in class and assigned everyone a z-score based on their height?  What type of z-score would you prefer?  What would the worst possible z-score be?  What would that look like?

I was amazed by the discussions that happened as a result of posing these scenarios.  Only after I made sure they had a conceptual understanding did I pass out a formula sheet for my students to glue in their notebooks.

Next, I told them we were going to find out their z-scores for various things.  We took a vote on three pieces of data we could collect.  I quickly vetoed the idea of collecting data on how many people they had kissed.  Teenagers…

We ended up with these three questions:

1) How many letters are in your first name?

2) How many sports do you play?

3) How many years have you attended school in Drumright?

The students wrote their answers on the SMART Board, and they calculated the mean and standard deviation for the class using their TI-84.  Then, they found their z-score and wrote a sentence describing what their z-score meant.

I think I need to do this more often.  I’m usually so quick to jumping to give my kids the formula that they just learn how to use the formula and never really grasp what the number we find means.