z-scores and Transforming Data
Chapter 2 - Day 2 - Lesson 2.1
Find and interpret the standardized score (z-score) of an individual value in a distribution of data.
Describe the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and variability of a distribution of data.
Activity: How Did I Do?
First of all, let’s take care of the movie reference. If you don’t know, you’re not thinking fourth dimensionally. It’s the most perfect blockbuster ever made.
Now, about the activity. Students should work in groups (or pairs) on page 1. By the end of the page, they will write a formula for a z-score. For now, we prefer to use words rather than symbols:
z-score = (value – mean) / SD
We don’t want students to have to wrestle with mu vs xbar and sigma vs s. We will save that discussion for Chapter 7. For now, the most important idea in this lesson is that students understand the interpretation of the z-score:
A z-score tells us the number of standard deviations above or below the mean.
In fact, this idea is so important that you want to choose a student and make it their job (we pay them in full at the end of the school year….$1 but it is a great resume builder). Now any time throughout the year when you calculate a z-score as a class, have this student interpret the z-score. Also, here is the answer to the bonus question.
On page 2 we are killing two birds with one stone:
To understand the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and variability of a distribution of data.
Practice with z-scores and understanding that standardizing a distribution maintains the shape of the distribution, but changes the mean to 0 and the standard deviation to 1.
Preparing for Inference
Why do we place so much emphasis on interpreting the z-score? We are planting seeds that will later grow into inferential thinking and reasoning. Later, when we perform a significance test, we will want to know how likely a certain result would be under a certain condition (P-value!). To understand “how likely is this result”, we must first think about “how far is this result from what was expected”.
Take any set of data, and transform all of the values into z-scores (standardize the distribution) by subtracting the mean and dividing by the standard deviation. Subtracting the mean will shift the mean to 0. Dividing by the standard deviation will dilate the variability such that the standard deviation will be 1. We will need this idea to understand why we can use Table A to do normal distribution calculations in Lesson 2.2. In that lesson, we learn that you can take any normal distribution and standardize it into a standard normal distribution (with mean 0 and standard deviation of 1), allowing us to then use Table A to find area.