The Definition of Correlation
14: Correlation
Introduction | Scatter Plot | The Correlational Coefficient | Hypothesis Test | Assumptions | An Additional Example
Introduction
Correlation quantifies the extent to which two quantitative variables, X and Y, “go together.” W hen high values of X are associated with high values of Y, a positive correlation exists. W hen high values of X are associated with low values of Y, a negative correlation exists. Illustrative data set. W e use the data set bicycle.sav to illustrate correlational methods. In this cross-sectional data set, each observation represents a neighborhood. The X variable is socioeconomic status measured as the percentage of children in a neighborhood receiving free or reduced-fee lunches at school. The Y variable is bicycle helmet use measured as the percentage of bicycle riders in the neighborhood wearing helmets. Twelve neighborhoods are considered: X Neighborhood Fair Oaks Strandwood W alnut Acres Discov. Bay Belshaw Kennedy Cassell Miner Sedgewick Sakamoto Toyon Lietz Three are twelve observations (n = 12). Overall, (% receiving reduced-fee lunch) 50 11 2 19 26 73 81 51 11 2 19 25 = 30.83 and Y (% wearing bicycle helmets) 22.1 35.9 57.9 22.2 42.4 5.8 3.6 21.4 55.2 33.3 32.4 38.4 = 30.883. W e want to explore the relation
between socioeconomic status and the use of bicycle helmets. It should be noted that an outlier (84, 46.6) has been removed from this data set so that we may quantify the linear relation between X and Y.
Page 14.1 (C:\data\StatPrimer\correlation.wpd)
Scatter Plot
The first step is create a scatter plot of the data. “There is no excuse for failing to plot and look.” 1 In general, scatter plots may reveal a • • • positive correlation (high values of X associated with high values of Y) negative correlation (high values of X associated with low values of Y) no correlation (values of X are not at all predictive of values of Y).
These patterns are demonstrated in the figure to the right.
Illustrative example. A
Introduction | Scatter Plot | The Correlational Coefficient | Hypothesis Test | Assumptions | An Additional Example
Introduction
Correlation quantifies the extent to which two quantitative variables, X and Y, “go together.” W hen high values of X are associated with high values of Y, a positive correlation exists. W hen high values of X are associated with low values of Y, a negative correlation exists. Illustrative data set. W e use the data set bicycle.sav to illustrate correlational methods. In this cross-sectional data set, each observation represents a neighborhood. The X variable is socioeconomic status measured as the percentage of children in a neighborhood receiving free or reduced-fee lunches at school. The Y variable is bicycle helmet use measured as the percentage of bicycle riders in the neighborhood wearing helmets. Twelve neighborhoods are considered: X Neighborhood Fair Oaks Strandwood W alnut Acres Discov. Bay Belshaw Kennedy Cassell Miner Sedgewick Sakamoto Toyon Lietz Three are twelve observations (n = 12). Overall, (% receiving reduced-fee lunch) 50 11 2 19 26 73 81 51 11 2 19 25 = 30.83 and Y (% wearing bicycle helmets) 22.1 35.9 57.9 22.2 42.4 5.8 3.6 21.4 55.2 33.3 32.4 38.4 = 30.883. W e want to explore the relation
between socioeconomic status and the use of bicycle helmets. It should be noted that an outlier (84, 46.6) has been removed from this data set so that we may quantify the linear relation between X and Y.
Page 14.1 (C:\data\StatPrimer\correlation.wpd)
Scatter Plot
The first step is create a scatter plot of the data. “There is no excuse for failing to plot and look.” 1 In general, scatter plots may reveal a • • • positive correlation (high values of X associated with high values of Y) negative correlation (high values of X associated with low values of Y) no correlation (values of X are not at all predictive of values of Y).
These patterns are demonstrated in the figure to the right.
Illustrative example. A