Z-scores and Correlations

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• Z-scores and Correlations
• Lecture 6, Psych 350 - R. Chris
  Fraleyhttp//www.yourpersonality.net/psych350/fal
  l2012/

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Announcements

• No lab on Wed this week
• No lecture next week
• Email TA about zero-acquaintance data working
  with it on Friday

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Answering Descriptive Questions in Multivariate
Research

• When we are studying more than one variable, we
  are typically asking one (or more) of the
  following two questions
• How does a persons score on the first variable
  compare to his or her score on a second variable?
• How do scores on one variable vary as a function
  of scores on a second variable?

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Making Sense of Scores

• Lets work with this first issue for a moment.
• Lets assume we have Marcs scores on his first
  two Psych 350 exams.
• Marc has a score of 50 on his first exam and a
  score of 50 on his second exam.
• On which exam did Marc do best?

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Example 1

• In one case, Marcs exam score is 10 points above
  the mean
• In the other case, Marcs exam score is 10 points
  below the mean
• In an important sense, we must interpret Marcs
  grade relative to the average performance of the
  class

Exam1
Exam2
Mean Exam2 60
Mean Exam1 40
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Example 2

• Both distributions have the same mean (40), but
  different standard deviations (10 vs. 20).
• In one case, Marc is performing better than
  almost 95 of the class. In the other, he is
  performing better than approximately 68 of the
  class.
• Thus, how we evaluate Marcs performance depends
  on how much spread or variability there is in the
  exam scores.

Exam1
Exam2
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Standard Scores

• In short, what we would like to do is express
  Marcs score for any one exam with respect to (a)
  how far he is from the average score in the class
  and (b) the variability of the exam scores.
• how far a person is from the mean
• (X M)
• variability in scores
• SD

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Standard Scores

• Standardized scores, or z-scores, provide a way
  to express how far a person is from the mean,
  relative to the variation of the scores.
• (1) Subtract the persons score from the mean.
  (2) Divide that difference by the standard
  deviation.
• This tells us how far a person is from the
  mean, in the metric of standard deviation units

Z (X M)/SD
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Example 1
Marcs z-score on Exam1 z (50 - 40)/10
1 (one SD above the mean) Marcs z-score on
Exam2 z (50 - 60)/10 -1 (one SD below the
mean)
Exam1
Exam2
Mean Exam2 60 SD 10
Mean Exam1 40 SD 10
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Example 2
An example where the means are identical, but the
two sets of scores have different spreads Marcs
Exam1 Z-score Z (50-40)/5 2 Marcs Exam2
Z-score Z (50-40)/20 .5
Exam1 SD 5
Exam2 SD 20
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Some Useful Properties of Standard Scores

• (1) The mean of a set of z-scores is always zero
• Why? If we subtract a constant, C, from each
  score, the mean of the scores will be off by that
  amount (M C). If we subtract the mean from
  each score, then mean will be off by an amount
  equal to the mean (M M 0).

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(2) The SD of a set of standardized scores is
always 1 Why? SD/SD 1
if x 60,
M 50 SD 10
50
60
70
80
40
30
20
x
0
1
2
3
-1
-2
-3
z
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(3) The distribution of a set of standardized
scores has the same shape as the unstandardized
(raw) scores
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A Normal Distribution
The normalization (mis)interpretation
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Some Useful Properties of Standard Scores

• (4) Standard scores can be used to compute easily
  centile scores the proportion of people with
  scores less than or equal to a particular score.
  

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The area under a normal curve
50
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34
14
14
2
2
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Some Useful Properties of Standard Scores

• (5) Z-scores provide a way to standardize
  different metrics (i.e., metrics that differ in
  variation or meaning). Different variables
  expressed as z-scores can be interpreted on the
  same metric (the z-score metric). (Each score
  comes from a distribution with the same mean
  zero and the same standard deviation 1.)

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Person Heart Rate Complaints Z-score (Heart Rate) Z-score (Complaints) Average
A 80 2 (80-100)/20 -1 (2-2.5)/.5 -1 -1
B 80 3 (80-100)/20 -1 (3-2.5)/.5 1 0
C 120 2 (120-100)/20 1 (2-2.5)/.5 -1 0
D 120 3 (120-100)/20 1 (3-2.5)/.5 1 1
Average 100 2.5 0 0 0
SD 20 .5 1 1 1
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Correlations in Personality Research

• Many research questions that are addressed in
  personality psychology are concerned with the
  relationship between two or more variables.

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Some examples

• How does dating/marital satisfaction vary as a
  function of personality traits, such as emotional
  stability?
• Are people who are relatively sociable as
  children also likely to be relatively sociable as
  adults?
• What is the relationship between individual
  differences in violent video game playing and
  aggressive behavior in adolescents?

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Graphic presentation

• Many of the relationships well focus on in this
  course are of the linear variety.
• The relationship between two variables can be
  represented as a line.

aggressive behavior
violent video game playing
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• Linear relationships can be negative or positive.

aggressive behavior
aggressive behavior
violent game playing
violent game playing
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• How do we determine whether there is a positive
  or negative relationship between two variables?

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Scatter plots
One way of determining the form of the
relationship between two variables is to create a
scatter plot or a scatter graph. The form of the
relationship (i.e., whether it is positive or
negative) can often be seen by inspecting the
graph.
aggressive behavior
violent game playing
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How to create a scatter plot
Use one variable as the x-axis (the horizontal
axis) and the other as the y-axis (the vertical
axis). Plot each person in this two dimensional
space as a set of (x, y) coordinates.
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How to create a scatter plot in SPSS
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How to create a scatter plot in SPSS

• Select the two variables of interest.
• Click the ok button.

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positive relationship
negative relationship
no relationship
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Quantifying the relationship

• How can we quantify the linear relationship
  between two variables?
• One way to do so is with a commonly used
  statistic called the correlation coefficient
  (often denoted as r).

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Some useful properties of the correlation
coefficient

• Correlation coefficients range between 1 and
  1.
• Note In this respect, r is useful in the same
  way that z-scores are useful they both use a
  standardized metric.

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Some useful properties of the correlation
coefficient

• (2) The value of the correlation conveys
  information about the form of the relationship
  between the two variables.
• When r gt 0, the relationship between the two
  variables is positive.
• When r lt 0, the relationship between the two
  variables is negative--an inverse relationship
  (higher scores on x correspond to lower scores on
  y).
• When r 0, there is no relationship between the
  two variables.

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r .80
r -.80
r 0
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Some useful properties of the correlation
coefficient

• (3) The correlation coefficient can be
  interpreted as the slope of the line that maps
  the relationship between two standardized
  variables.
• slope as rise over run

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r .50
takes you up .5 on y
rise
run
moving from 0 to 1 on x
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How do you compute a correlation coefficient?

• First, transform each variable to a standardized
  form (i.e., z-scores).
• Multiply each persons z-scores together.
• Finally, average those products across people.

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Example
Person Violent game playing (z-scores) Zx Aggressive behavior (z-scores) Zy
Adair 1 1 1
Antoine 1 1 1
Colby -1 -1 1
Trotter -1 -1 1
Average 0 0 1
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Why products? Important Note on 2 x 2
Matching z-scores via products
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Important Note on 2 x 2
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Computing Correlations in SPSS

• Go to the Analyze menu.
• Select Correlate
• Select Bivariate

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Computing Correlations in SPSS

• Select the variables you want to correlate
• Shoot them over to the right-most window
• Click on the Ok button.

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Magnitude of correlations

• When is a correlation big versus small?
• Cohens standards
• .1 small
• .3 medium
• gt .5 large

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What are typical correlations in personality
psychology?
Typical sample sizes and effect sizes in studies
conducted in personality psychology.

Mdn M SD Range
N 120 179 159 15 508
r .21 .24 .17 0 .96
Note. The absolute value of r was used in the
calculations reported here. Data are based on
articles published in the 2004 volumes of
JPSPPPID and JP.
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A selection of effect sizes from various domains
of research

Variables r
Effect of sugar consumption on the behavior and cognitive process of children .00
Chemotherapy and surviving breast cancer .03
Coronary artery bypass surgery for stable heart disease and survival at 5 years .08
Combat exposure in Vietnam and subsequent PTSD within 18 years .11
Self-disclosure and likeability .14
Post-high school grades and job performance .16
Psychotherapy and subsequent well-being .32
Social conformity under the Asch line judgment task .42
Attachment security of parent and quality of offspring attachment .47
Gender and height for U.S. Adults .67

Note. Table adapted from Table 1 of Meyer et al.
(2001).
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Magnitude of correlations

• real world correlations are rarely get larger
  than .30.
• Why is this the case?
• Any one variable can be influenced by a hundred
  other variables. To the degree to which a
  variable is multi-determined, the correlation
  between it and any one variable must be small.

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Qualify

• For the purposes of this class, I want you to
  describe the correlation What is it numerically?
  And, qualitatively speaking, is it zero or close
  to zero (lt .1), small (.1 to .29), medium
  (.30 to .49), or large (gt .50).