Describing relationships using correlations
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Describing Relationships Using Correlations. 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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Describing Relationships Using Correlations

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Describing relationships using correlations

Describing Relationships Using Correlations


More statistical notation

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.


More statistical notation1

More Statistical Notation

  • 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


More statistical notation2

More Statistical Notation

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.


Correlation coefficient

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

Understanding Correlational Research


Drawing conclusions

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


Plotting correlational data

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


A scatterplot showing the existence of a relationship between the two variables

A Scatterplot Showing the Existence of a Relationship Between the Two Variables


Scatterplots showing no relationship between the two variables

Scatterplots Showing No Relationship Between the Two Variables


Types of relationships

Types of Relationships


Linear relationships

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


A scatterplot of a positive linear relationship

A Scatterplot of a Positive Linear Relationship


A scatterplot of a negative linear relationship

A Scatterplot of a Negative Linear Relationship


Nonlinear relationships

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.


A scatterplot of a nonlinear relationship

A Scatterplot of a Nonlinear Relationship


Strength of the relationship

Strength of the Relationship


Strength

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


Correlation coefficients

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

Computing the Correlation Coefficient


Pearson correlation coefficient

Pearson Correlation Coefficient

  • r used to describe a linear relationship between two scale variables


Spearman rank order correlation coefficient

Spearman Rank-Order Correlation Coefficient

  • 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.


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