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.
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:
Positive: above the mean; Negative: below the mean
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.