# Bivariate Description

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```					Bivariate Description

Heibatollah Baghi, and
OBJECTI VES
 Define bivariate and univariate
statistical tests.
 Explain when to use correlational
questions.
 Understand measure of Pearson
Product Moment Correlation
Coefficient (Pearson’s r).

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Definitions
 Univariate: examination of variable’s
frequency distribution, central tendency,
and variability.
 Bivariate: examination of two variables
simultaneously.
– Is SES related to intelligence?
– Do SAT scores have anything to do with how
well one does in college?
• The question is: do these variables correlate or
covary?

3
Typical Situations
   Two nominal variables
   A nominal and interval/ratio variables
– Delivery type and weight of child
– Bed rest and weight gain during pregnancy
   Two interval ratio variables
– Respiratory function and extent of anxiety

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Cross Tabulation
 Describes relationship between two
nominal variables
 Two dimensional frequency distribution
Gender
Readmitted         15 (30%) 10 (20%) 25
Not readmitted     35 (70%) 40 (80%) 75
Total                    50       50 100

are
Also appropriate if either or both variables 5
ordinal-level with a small number of categories
Elements of Cross Tabulation

Column
Row
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Elements of Cross Tabulation

Cell count
Row %
Column %
Total %

7
Elements of Cross Tabulation

Marginal
Joint
distribution

Marginal
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Group Mean Comparison
   Describes a nominal variable and an
interval/ratio variable

9
Linear Association

   The correlation coefficient is a bivariate
statistic that measures the degree of
linear association between two
interval/ratio level variables. (Pearson
Product Moment Correlation
Coefficient)

10
Scatter plot

 Reveals   the presence of association
between two variables. The stronger
the relationship, the more the data
points cluster along an imaginary line.
 Indicates the direction of the
relationship.
 Reveals the presence of outliers.

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Scatter Plot of Positively Correlated Data

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Scatter Plot of Negatively Correlated Data

13
Scatter Plot of Non Linear Data

14
Scatter Plot of Uncorrelated Data

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Covariance Formula

Cov. & Correlation

(X - X)(Y - Y)
SXY    
n -1

SXY
rXY   
(SX )(SY )         16
Cov. & Correlation
Correlation Formula
(X - X)(Y - Y)
SXY             of
Covariancen - 1
X&Y

SXY
Standard      rXY                    Standard
Deviation of           (SX )(SY )     Deviation of
X                                     Y

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Example Data           ID
GPA
Y
SAT
X
A     1.6     400
B         2   350
C     2.2     500
D     2.8     400
E     2.8     450
F     2.6     550
G     3.2     550
H         2   600
Sum    30.80   6550.0
I     2.4     650
Mean   2.57    545.80
J     3.4     650
S.D.   0.54    128.73
K     2.8     700
L         3   750

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STUDENTS   Y(GPA)   X(SAT)   (Y - Y ) (X - X )   (Y - Y )(X - X )
A          1.6      400
B          2.0      350
C          2.2      500
D          2.8      400
E          2.8      450
F          2.6      550
G          3.2      550
H          2.0      600
I          2.4      650
J          3.4      650
K          2.8      700
L          3.0      750
Sum        30.80    6550.0
Mean       2.57     545.80
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S.D.       0.54     128.73
STUDENTS   Y(GPA)   X(SAT)   (Y - Y ) (X - X )   (Y - Y )(X - X )
A          1.6      400      -0.97
B          2.0      350      -0.57
C          2.2      500      -0.37
D          2.8      400      0.23
E          2.8      450      0.23
F          2.6      550      0.03
G          3.2      550      0.63
H          2.0      600      -0.57
I          2.4      650      -0.17
J          3.4      650      0.83
K          2.8      700      0.23
L          3.0      750      0.43
Sum        30.80    6550.0
Mean       2.57     545.80
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S.D.       0.54     128.73
STUDENTS   Y(GPA)   X(SAT)   (Y - Y ) (X - X )   (Y - Y )(X - X )
A          1.6      400      -0.97    -145.80
B          2.0      350      -0.57    -195.80
C          2.2      500      -0.37    -45.80
D          2.8      400      0.23     -145.80
E          2.8      450      0.23     -95.80
F          2.6      550      0.03     4.20
G          3.2      550      0.63     4.20
H          2.0      600      -0.57    54.20
I          2.4      650      -0.17    104.20
J          3.4      650      0.83     104.20
K          2.8      700      0.23     154.20
L          3.0      750      0.43     204.20
Sum        30.80    6550.0
Mean       2.57     545.80
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S.D.       0.54     128.73
STUDENTS   Y(GPA)   X(SAT)   (Y - Y ) (X - X )   (Y - Y )(X - X )
A          1.6      400      -0.97    -145.80        141.43
B          2.0      350      -0.57    -195.80        111.61
C          2.2      500      -0.37    -45.80         16.95
D          2.8      400      0.23     -145.80        -33.53
E          2.8      450      0.23     -95.80         -22.03
F          2.6      550      0.03     4.20           0.13
G          3.2      550      0.63     4.20           2.65
H          2.0      600      -0.57    54.20          -30.89
I          2.4      650      -0.17    104.20         -17.71
J          3.4      650      0.83     104.20         86.49
K          2.8      700      0.23     154.20         35.47
L          3.0      750      0.43     204.20         87.81
Sum        30.80    6550.0                           378.33
Mean       2.57     545.80
22
S.D.       0.54     128.73
Calculation of Covariance &
Correlation
Covariance & Correlation

(X - X)(Y - Y) 378.33
SXY                            34.39
n -1        11

SXY          34.39
rXY                              0.50
(SX )(SY ) (0.54)(128.73)

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Correlations in SPSS

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Limitation of the Covariance

It is metric-dependent
Properties of Pearson r
   r is metric-independent
   r reflects the direction of the
relationship
   r reflects the magnitude of the
relationship

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What does positive correlation mean?
 Scores above the mean on X tend to be
associated with scores above the mean on Y
 Scores below the mean on X tend to be
accompanied by scores below the mean of Y
 Note for this reason deviation score is an
important part of Covariance

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What does negative correlation mean?

 Scores above the mean on X tend to be
associated with scores below the mean on Y
 Scores below the mean on X tend to be
accompanied by scores above the mean of Y.

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Strength of association

r2 = Coefficient of determination
1 – r2 = Coefficient of non-determination

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Analysis of Relationships

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Take Home Lessons
   Always make a scatter plot
 See the data first
   Examining the scatter plot is not enough
   A single number can represent the degree
and direction of the linear relation between
two variables

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 views: 20 posted: 2/21/2012 language: English pages: 31