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

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

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

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

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