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Describing Relationships Using Correlations

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Describing Relationships Using Correlations

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.

- 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

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.

- 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

- 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

- 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 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

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.

- 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 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

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

- 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.