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Chapter 5
Z-Scores
Review
► We
have finished the basic elements of
descriptive statistics.
► Now we will begin to develop the concepts
and skills that form the foundation for
inferential statistics.
Review (cont.)
► Chapter
5: Present a method for describing
the exact location of an individual score
relative to the other scores in a distribution.
► Chapter 6: Determine probability values
associated with different locations in a
distribution of scores
► Chapter 7: Apply skills from Ch. 5 & 6 to
sample means instead of individual scores.
Z-Scores
► Purpose
of z-scores, or standard scores, is
to identify and describe the exact location of
every score in a distribution.
► To do this, we use the mean and standard
deviation.
Z-Scores
►A
statistical technique that uses the mean
and the standard deviation to transform
each score (X value) into a z-score or a
standard score.
► Purpose of a z-score or a standard score is
to identify the exact location of every score
in a distribution.
Why are z-scores useful?
► If
you got a 76 on a test, how did you do?
► You would need more information.
► You need to know the other scores in the
distribution.
► What is the mean?
Why are z-scores useful? (cont.)
► Knowing
the mean is not enough.
► You also need to know the standard
deviation.
► The relative location within the distribution
depends on the mean and the statistical
deviation as well as your score.
Figure 5.1
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Two distributions of exam scores
Score is in the extreme right
hand tail – one of the highest
in the distribution
Score is only slightly
above average.
Purpose for z-scores
►A
score by itself does not necessarily
provide much information about its position
within a distribution
 These are called raw scores
► To
make raw scores more meaningful, they
are often transformed into new values that
contain more information.
► This transformation is one purpose for zscores.
Purpose for z-scores (cont.)
► We
can transform scores into z-scores to
find out exactly where the original scores
are located.
► A second purpose is to standardize an entire
distribution.
 IQ scores
►All
are standardized with a mean of 100 and s.d. of
15
►An IQ score of 95 is slightly below average and an IQ
score of 145 is extremely high no matter what IQ
test
To describe the exact location of the
score within a distribution
►A
z-score transforms an X score into a
signed +/- number
 + above the mean
 - below the mean
 The number tells the distance between the
score and the mean in terms of the number of
standard deviations.
Example
► In
a distribution of standardized IQ scores
with m = 100 and s = 15 and a score of
X=130
► The score of X=130 could be transformed
into z= +2.00
► z value indicates + (above the mean)
► by a distance of 2 standard deviations (30
points)
Definition
►A
z-score
 Specifies the precise location of each X value
with a distribution
 The sign of +/- signifies whether the score is
above or below the mean
 The numerical value of the z-score specifies the
distance from the mean by counting the number
of standard deviations between the X and m.
Z-scores
► Consist
of two parts
 +/ Magnitude
► Both
parts are necessary to describe
completely where a raw score is located
within a distribution.
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Figure 5.2
The relationship between z-scores and locations in a distribution
a z-score of z = +1.00 corresponds to a
position exactly 1 standard deviation above
the mean.
► a z-score of z = +2.00 corresponds to a
position exactly 2 standard deviations above
the mean.
► The numerical value tells you the number of
standard deviations from the mean.
►
Figure 5.1
Two distributions of exam scores
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Now use a z-score to describe the position of X=76.
Z= +2.00
Z= +0.50
The score is located above
the mean by exactly 2 s.d.
The score is located above
the mean by 1/2 s.d.
Learning Check
►A
negative z-score always indicates a
location below the mean.
► What z-score value identifies each of the
following locations in a distribution?




Above the mean by 2 s.d.
Below the mean by ½ s.d.
Above the mean by ¼ s.d.
Below the mean by 3 s.d.
Learning Check
a population with m = 50 and s = 10, find
the z-score for each of the following scores:
► For
 X = 55
 X = 40
 X = 30
Z= +0.50
Z= - 1.00
Z = - 2.00
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Figure 5.4
A z-score transformation
10
30
40
= 50
Learning Check
a population with m = 50 and s = 10,
find the X value corresponding to each of
the following z-scores:
► For
 z = + 1.00
 z = - 0.50
 z = + 2.00
X= 60
X = 45
X = 70
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Figure 5.4
A z-score transformation
10
30
40
= 50
60
70
Formula for transforming z-scores
►z
=X–m
s
Example
distribution of scores has a mean of m =
100 and a standard deviation of s = 10.
What z-score corresponds to a score of X=120
in this distribution?
Z = X – m = 120-100 = 20 = 2.00
s
10
10
►A
Transforming z-scores into X values
►X
= m + zs
= 60 + (-2.00)(5)
= 60 + (-10.00)
= 50
m= 60
s= 5
Z = -2.00
Using z-scores to Standardize a
Distribution
► When
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 zero;
 The s.d. is transformed into a value of 1.
Standardized Distribution
►A
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.
Using z-scores to Make Comparisons
► Example:
pg. 112-113
► Psychology score = 60
 m = 50 and s = 10
► Biology
score = 56
 m = 48 and s = 4
► z=
X – m = 60-50 = 10 = + 1.0
s
► z=
10
10
X – m = 56-48 = 8 = +2.0
s
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