describing relationships using correlations. 2 more statistical notation correlational analysis...
TRANSCRIPT
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Describing Relationships Using Correlations
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More Statistical Notation
Correlational analysis requires scores from two
variables.
X stands for the scores on one variable.
Y stands for the scores on the other variable.
Usually, each pair of XY scores is from the same
participant.
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• As before, indicates the sum of the X scores, indicates the sum of the squared X scores, and indicates the square of the sum of the X scores
• Similarly, indicates the sum of the Y scores, indicates the sum of the squared Y scores, and indicates the square of the sum of the Y scores
X 2X2)( X
Y 2Y2)( Y
More Statistical Notation
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Now, indicates the the sum of the X
scores times the sum of the Y scores and
indicates that you are to multiply each X score
times its associated Y score and then sum the
products.
))(( YX XY
More Statistical Notation
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Correlation Coefficient
• A correlation coefficient is the statistic that in a single number quantifies the pattern in a relationship
• It does so by simultaneously examining all pairs of X and Y scores
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Understanding Correlational Research
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Drawing Conclusions
• The term correlation is synonymous with relationship
• However, the fact that there is a relationship between two variables does not mean that changes in one variable cause the changes in the other variable
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Plotting Correlational Data
• A scatterplot is a graph that shows the location of each data point formed by a air of X-Y scores
• When a relationship exists, a particular value of Y tends to be paired with one value of X and a different value of Y tends to be paired with a different value of X
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A Scatterplot Showing the Existence of a Relationship Between the Two
Variables
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Scatterplots Showing No Relationship Between the Two
Variables
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Types of Relationships
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Linear Relationships
• A linear relationship forms a pattern that fits a straight line
• In a positive linear relationship, as the scores on the X variable increase, the scores on the Y variable also tend to increase
• In a negative linear relationship, as the scores on the X variable increase, the scores on the Y variable tend to decrease
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A Scatterplot of a Positive Linear Relationship
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A Scatterplot of a Negative Linear Relationship
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Nonlinear Relationships
In a nonlinear, or curvilinear, relationship, as
the X scores change, the Y scores do not tend
to only increase or only decrease: at some
point, the Y scores change their direction of
change.
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A Scatterplot of a Nonlinear Relationship
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Strength of the Relationship
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Strength
• The strength of a relationship is the extent to which one value of Y is consistently paired with one and only one value of X
• The larger the absolute value of the correlation coefficient, the stronger the relationship
• The sign of the correlation coefficient indicates the direction of a linear relationship
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Correlation Coefficients
• Correlation coefficients may range between -1 and +1. The closer to 1 (-1 or +1) the coefficient is, the stronger the relationship; the closer to 0 the coefficient is, the weaker the relationship.
• As the variability in the Y scores at each X becomes larger, the relationship becomes weaker
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Computing the Correlation Coefficient
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])()([])()([
))(()(2222 YYNXXN
YXXYNr
Pearson Correlation Coefficient
• r used to describe a linear relationship between two scale variables
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• describes the linear relationship between two variables measured using ranked scores. The formula is
where N is the number of pairs of ranks and D is the difference between the two ranks in each pair.
)2(
)(61
2
2
NN
Drs
Spearman Rank-Order Correlation Coefficient