# Z-SCORES (STANDARD SCORES) - Michigan State University Z-SCORES (STANDARD SCORES) We can use the SD (s) to classify people on any measured variable. X z Why might you ever use this in real life? Diagnosis of a mental disorder Selecting the best person for the job Figuring out which children may need special assistance in school

EXAMPLE FROM I/O Extraversion predicts managerial performance. The more extraverted you are, the better a manager you will be (with everything else held constant, of course). AN EXTRAVERSION TEST TO EMPLOYEES Scores for current managers 10, 25, 32, 35, 39, 40, 41, 45, 48, 55, 70

N=11 Need the mean X X N 2 ( X ) X 2 N s N1 Need the standard deviation

X X2 10 100 25 625 32 1024 35 1225 39 1521 40

1600 41 1681 45 2025 48 2304 55 3025 70 4900 440

20030 Lets Do It X X 440 40 N 11 ( X ) 2 X N s N1 2 (440) 2 20030 11 15.58 11 1 SOMEBODY APPLIES FOR A JOB AS A MANAGER

Obtains a score of 42. Should I hire him? Somebody else comes in and has a score of 44? What about her? What if the mean were still 40, but the s = 2? HARDER EXAMPLE: Two people applying to graduate school

Bob, GPA = 3.2 at Northwestern Michigan Mary, GPA = 3.2 at Southern Michigan Whom do we accept? What else do we need to know to determine who gets in? SCHOOL PARAMETERS NWMU mean GPA = 3.0; SD = .1 SMU mean GPA = 3.6; SD = .2

THE MORAL OF THE STORY: We can compare people across ANY two tests just by saying how many SDs they are from the mean. ONLY ONE TEST it might make sense to rescore everyone on that test in terms of how many standard deviations each person is from the mean. The curve z-SCORES & LOCATION IN A DISTRIBUTION

Standardization or Putting scores on a test into a form that you can use to compare across tests. These scores become known as standardized scores. The purpose of z-scores, or standard scores, is to identify and describe the exact location of every score in a distribution z-score is the number of standard deviations a particular score is from the mean. (This is exactly what weve been doing for the last however many minutes!) z-SCORES The sign tells whether the score is located above (+) or below (-) the mean The number (magnitude) tells the distance between the score and the mean in terms of number of

standard deviations WHAT ELSE CAN WE DO WITH z-SCORES? Converting z-scores to X values Go backwards. Aaron says he had a z-score of 2.2 on the Math SAT. Math SAT has a m = 500 and s = 100 What was his SAT score? USING Z-SCORES TO STANDARDIZE A DISTRIBUTION Shape doesnt change (Think of it as relabeling)

Mean is always 0 SD is always 1 Why is the fact that the mean is 0 and the SD is 1 useful? standardized distribution is composed of scores that have been transformed to create predetermined values for m and s Standardized distributions are used to make dissimilar distributions comparable DEMONSTRATION OF A zSCORE TRANSFORMATION heres an example of this in your book (on pg. 161). Im not going to ask you to do this on an exam, but I do want you to look at this example. I

think it helps to re-emphasize the important characteristics of z-scores. The two distributions have exactly the same shape After the transformation to z-scores, the mean of the distribution becomes 0 After the transformation, the SD becomes 1 For a z-score distribution, Sz = 0 For a z-score distribution, Sz2 = SS = N (I will not emphasize this point) FINAL CHALLENGE Using z-scores to make comparisons (Example from pg. 112) Bob has a raw score of 60 on his psych exam and a raw score of 56 on his biology exam.

In order to compare, need the mean & the SD of each distribution Psych: m = 50 and s=10 Bio: m = 48 and s=4 FINAL CHALLENGE II You could OR

sketch the two distributions and locate his score in each distribution Standardize the distributions by converting every score into a z-score Transform the two scores of interest into z-scores PSYCH SCORE = (60-50)/10 = 10/10 = +1 BIO SCORE = (56-48)/4 = 8/4 = +2 *Important element of this is INTERPRETATION* OTHER LINEAR TRANSFORMATIONS Steps for converting scores to another test

Take the original score and make it a z-score using the first tests parameters Take the z-score and turn it into a raw score using the second tests parameters. Standard Score = mnew + zsnew See Learning Checks in text, these are a general idea of what might be on the exam