Lecture notes, Chapters 5.

(Note, in the material below M = myuu for populations, x-bar for samples, SD = sigma for populations, s for samples)

Chapter 5. Z-scores: Relating scores to distributions

In the first 4 classes we focused on understanding what a distribution is, and how to display and summarize the information contained in a distribution-in tables, graphs, words, numbers. The focus has been on describing the whole distribution. Now we move to locating a single score within a distribution: Z-scores.

Z-score: A standardized score that indicates where a score is in a distribution

Z-scores help us know the relative value of a score--how far it is from the mean. We need M and SD for the formulas.

Formulas:

To find a Z when you know the X: Z = (X-M)/SD

To find an X when you know the Z: X=(Z)(SD)+M

More on Z-scores:

• Can be positive or negative

Positive: above the mean; Negative: below the mean

• Mean is always 0, Variance is always 1, Standard deviation is always 1.

Skills you need: transform X scores into Z scores and back again. Memorize the formulas:

or memorize the meaning and the process of translation using a number line.

Exercise: Z-scores are for the birds. Taking duck as the perfectly prototypical bird, with z-scores of 0 on all dimensions (weight, size, wingspan, flying speed, football prowess, etc. ). For the following birds (cardinal, chicken, eagle, barn owl), identify (1) a feature that would have a positive z-score (2) a feature that would have a negative z-score.

What are Z-scores good for?

Allow us to quickly see the relative position of a score within a distribution. Consider a doctor looking at a long printout of all kinds of tests on a patient who is ill. What is more useful-raw scores or z-scores?

Z-scores also allow us to compare things measured in different units (speed, weight, wingspan)

Allow us to look for statistical relationships among things measured in different units.

Example: Are wingspan and maximum flight speed positively correlated? Hard to translate inches into miles per hour. Converting both into z-scores (standardizing the variables) allows us to calculate correlation, with both measured in the same standard units.

Z-scores are also very important for locating sample means in the normal distribution...more about this in later chapters.