ZSCORES STANDARD SCORES

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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

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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).

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AN EXTRAVERSION TEST TO EMPLOYEES

• Scores for current managers
• 10, 25, 32, 35, 39, 40, 41, 45, 48, 55, 70
• N11
• Need the mean
• Need the standard deviation

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Lets Do It
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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?

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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?

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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.

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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

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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!)

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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

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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?

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USING Z-SCORES TO STANDARDIZE A DISTRIBUTION

• Shape doesnt 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

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DEMONSTRATION OF A z-SCORE 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)

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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 s10
• Bio m 48 and s4

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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

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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