It is the numerator of the formula for the variance of Y.įor the illustrative data set, "y bar" = 30.883 and ss yy = 3159.68.įinally, let us define the sum of the crossproducts ( ss xy ) as: This statistics measures the spread of dependent variable Y. The sum of squares around the mean of the Y variable ( ss yy ) as: įor the illustrative data set, x = 30.833 and ss xx = 7855.67. This statistic measures the spread of independent variable X, and is the numerator of formula for the variance of X. The sum of squares around the mean of the X variable ( ss xx ) as: There are three types of sums of squares. To calculate correlation coefficients, we need to calculate various sums of products and cross-products. We may judge the strength of the correlation in qualitative terms: the closer r is to-1 or +1, the stronger is the correlation.Īlthough there are no firm cutoffs for strength, let us say that absolute correlations | r | greater than or equal to 0.7 are "strong." Absolute correlations less than 0.3 are "weak." Absolute correlations between 0.3 and 0.7 are moderate. Perfect positive and negative correlations, however, are seldom encountered, with most correlations coefficients falling short of these extremes. When all points fall on a downward slope, r = -1. When all points fall on a trend line with an upward slope, r = +1. The closer the correlation coefficient is to +1 or -1, the better the two variables "keep in step." This can be visualized by the degree to which the scatter cloud adheres to an imaginary trend line through the data. Pearson's correlation coefficient ( r ) is a statistic that quantifies the relationship between X and Y in unit-free terms. For insights into how to address outliers, please see Correlation Pearson's correlation coefficient ( r ) However, they should never be entirely ignored. In some instances, outliers should be excluded before analyzing the data and in other instances they should remain present during analysis. Identifying and dealing with outliers is an important statistical undertaking. Observations that do not fit the general data pattern are called outliers. (Suggestion: Enter the illustrative data set into an SPSS file and produce this scatter plot.) Outliers SPSS: To draw a scatter plot with SPSS, click on Graphs | Simple | Scatter, and then select the variables you wish to plot. Negative correlation (high values of X associated with low values of Y),.Positive correlation (high values of X associated with high values of Y),.Thereby, a negative correlation is said to exist. That is, as the number of children receiving reduced-fee meals at school increases, the bicycle helmet use rate decreases. Notice that this graph reveals that high X values are associated with low values of Y. The scatter plot of the illustrative data set is shown below: This type of graph shows ( x i, y i ) values for each observation on a grid. The basis of both correlation and regression lies in bivariate ("two variable") scatter plots. X represents the percentage of children receiving free or reduced-fee meals at school. Y represents as the percentage of bicycle riders in the neighborhood wearing helmets. Data come from a study of bicycle helmet use ( Y ) and socioeconomic status (X). To illustrate both methods, let us use the data set called BICYCLE.SAV. In general, the dependent (outcome) is referred to as Y and the independent (predictor) variable is called X. This is used to analyze the relationship between two continuous variables. We will just address the tip of the iceberg for this topic, by basic linear correlation and regression techniques. 11: Correlation and Regression 11: Linear Correlation & RegressionĬorrelation and regression are complex and powerful statistical techniques that have wide application in data analysis.
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