Bivariate DescriptionBivariate Description

Heibatollah Baghi, and

Mastee Badii

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OBJECTI VESOBJECTI VES

Define bivariate and univariate statistical tests.

Explain when to use correlational techniques to answer research questions.

Understand measure of Pearson Product Moment Correlation Coefficient (Pearson’s r).

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DefinitionsDefinitions

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?

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Typical SituationsTypical Situations

Two nominal variables– Gender and readmission status

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 TabulationCross Tabulation

Describes relationship between two nominal variables

Two dimensional frequency distribution

Male Female TotalReadmitted 15 (30%) 10 (20%) 25Not readmitted 35 (70%) 40 (80%) 75Total 50 50 100

Readmission statusGender

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

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Elements of Cross TabulationElements of Cross Tabulation

Colum

n

Row

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Elements of Cross TabulationElements of Cross Tabulation

Cell countRow %

Column %Total %

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Elements of Cross Tabulation Elements of Cross Tabulation

Marginal

Marginal

Joint distribution

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Group Mean ComparisonGroup Mean Comparison

Describes a nominal variable and an interval/ratio variable

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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)

Linear AssociationLinear Association

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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.

Scatter plotScatter plot

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Scatter Plot of Positively Correlated DataScatter Plot of Positively Correlated Data

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Scatter Plot of Negatively Correlated DataScatter Plot of Negatively Correlated Data

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Scatter Plot of Non Linear DataScatter Plot of Non Linear Data

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Scatter Plot of Uncorrelated DataScatter Plot of Uncorrelated Data

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Covariance FormulaCovariance Formula

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Correlation FormulaCorrelation Formula

Covariance of X & Y

StandardDeviation of

X

StandardDeviation of

Y

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Sum 30.80 6550.0

Mean 2.57 545.80

S.D. 0.54 128.73

GPA SAT

ID Y 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

I 2.4 650

J 3.4 650

K 2.8 700

L 3 750

Example DataExample Data

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STUDENTS Y(GPA) X(SAT)

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

S.D. 0.54 128.73

)X - (X)Y - (Y )X - (X )Y - (Y

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STUDENTS Y(GPA) X(SAT)

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

S.D. 0.54 128.73

)X - (X)Y - (Y )X - (X )Y - (Y

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STUDENTS Y(GPA) X(SAT)

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

S.D. 0.54 128.73

)X - (X)Y - (Y )X - (X )Y - (Y

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STUDENTS Y(GPA) X(SAT)

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

S.D. 0.54 128.73

)X - (X)Y - (Y )X - (X )Y - (Y

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Calculation of Covariance & Calculation of Covariance & CorrelationCorrelation

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Correlations in SPSSCorrelations in SPSS

Limitation of the CovarianceLimitation of the Covariance

It is metric-dependent

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Properties of Pearson rProperties 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?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?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 associationStrength of association

r2 = Coefficient of determination

1 – r2 = Coefficient of non-determination

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Analysis of RelationshipsAnalysis of Relationships

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Take Home LessonsTake 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