Z-SCORES (STANDARD SCORES)

We can use the SD (s) to classify people on any measured variable.

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

X

z

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

1

)( 22

NNX

Xs

Scores for current managers 10, 25, 32, 35, 39, 40, 41, 45, 48,

55, 70 N=11 Need the mean

Need the standard deviation

N

XX

Let’s Do ItX 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

4011

440

N

XX

58.1511111)440(

20030

1

)(

2

22

NNX

Xs

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 SD’s 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 we’ve 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 doesn’t change (Think of it as re-labeling) 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 z-SCORE TRANSFORMATION here’s an example of this in your book (on pg.

161). I’m 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

sketch the two distributions and locate his score in each distribution

Standardize the distributions by converting every score into a z-score

OR 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 test’s parameters Take the z-score and turn it into a “raw”

score using the second test’s parameters. Standard Score = mnew + zsnew

See “Learning Checks” in text, these are a general idea of what might be on the exam