ZScores

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Z-Scores

• Location of scores and standardized distributions

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Figure 5-1 (p. 139)Two distributions of exam
scores. For both distributions, µ 70, but for
one distribution, s 12. The position of X 76
is very different for these two distributions.
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Z-Scores

• Z-scores are a way of standardizing a score with
  respect to the other scores in the group.
  
• This is done by taking account of the mean and
  standard deviation (SD) of the group.
  
• A Z-score expressed a particular score in terms
  of how many standard deviations it is away from
  the mean.
  
• By converting a raw score to a Z-score, we are
  expressing that score on a z-score scale, which
  always has a mean of 0 and a standard deviation
  of 1.
• In short, we are re-defining each raw score in
  terms of how far away it is from the group mean.

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Figure 5-2 (p. 141)The relationship between
z-score values and locations in a population
distribution.
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Figure 5-3 (p. 145) An entire population of
scores is transformed into z-scores. The
transformation does not change the shape of the
population but the mean is transformed into a
value of 0 and the standard deviation is
transformed to a value of 1.
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Figure 5-4 (p. 146) Following a z-score
transformation, the X-axis is relabled in z-score
units. The distance that is equivalent to 1
standard deviation on the X-axis (s 10 points
in this example) corresponds to 1 point on the
z-score scale.
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Calculating Z-scores

• First, we find the difference between the raw
  score and the mean score (this tells us how far
  away the raw score is from the average score)
  
• Second, we divide by the standard deviation (this
  tells us how many standard deviations the raw
  score is away from the average score)
  

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What happens to the distribution when raw scores
are converted to Z-scores?

• Nothing happens to the distribution.
• It is only that the scores become standardized

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Advantages of using Z-scores

• Clarity The relationship between a raw score and
  the distribution of scores is much more clear.
  It is possible to get an idea of how good or bad
  a score is relative to the entire group.
• Comparison You can compare scores measured on
  different scales.

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Example 1

• An IQ test has a mean of 100 and a standard
  deviation of 15
  
• 263 MSU Students were tested
• A psychology student scores 130.
• What is the Z-score for this student and what
  does that score convey about that students
  standing relative to the entire distribution of
  IQ scores?

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Example 2

• A math student scores a 60 on the IQ test.
• What is this students Z-score and what does his
  score convey about his standing relative to the
  entire distribution?
  

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Other key points

• Given the formula to calculate a Z-score, you
  should be able to solve for the mean, standard
  deviation, and the raw score.
  
• Give Z (X µ)/s
• X µ Zs
• µ X Zs
• s (X µ)/Z
• What is the sum of Z-scores? (Sz?)
• Sz0
• Sz2SSN
• Why?

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Other standardized distribution

• It is also common to standardize a distribution
  by transforming Z-scores to a distribution with a
  predetermined mean and standard deviation that
  are whole round numbers.
• In doing so, a new standardize distribution can
  be created with simple values for the mean and
  standard deviation.
  
• Ex for IQ scores, the mean is always 100
  (instead of 0) and the standard deviation is
  always 15 (instead of 1).

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Figure 5-6 (p. 151)The distribution of exam
scores from Example 5.6 The original distribution
was standardized to produce a new distribution
with µ 50 and s 10. Note that each individual
is identified by an original score, a z-score,
and a new, standardized score. For example, Joe
has an original score of 43, a z-score of 1.00,
and a standardized score of 40.
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To compute new standardized score

• First transform each of the original raw scores
  into Z-scores, given the original mean the
  original standard deviation.
  
• Next, Change the Z-scores to the new,
  standardized scores, given the new mean and the
  new standard deviation.
  
• New Standard score µnew Zsnew
• Ex µ57, s14, µnew 50, snew 10
• Given X64, what is the new standard score?

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Figure 5-7 (p. 153) Using z-scores to determine
whether a specific score or sample is near to the
population mean (representative) or very
different from the population mean
(nonrepresentative).
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Figure 5-8 (p. 153)A diagram of a research
study. The goal of the study is to evaluate the
effective-ness of a treatment. One individual is
selected from the population and the treatment is
administered to that individual. If, after
treatment, the individual is noticeably different
from the original population, then we have
evidence that the treatment does have an effect.