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.

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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 scatter plot of the illustrative data is shown to the right. The plot reveals that high values of X are associated with low values of Y. That is to say, as the number of children receiving reduced-fee meals at school increases, bicycle helmet use rates decrease’ a negative correlation exists. In addition, there is an aberrant observation (“outlier”) in the upper-right quadrant. Outliers should not be ignored— it is important to say something about aberrant observations. 2 W hat should be said exactly depends on what can be learned and what is known. It is possible the lesson learned from the outlier is more important than the main object of the study. In the illustrative data, for instance, we have a low SES school with an envious safety record. W hat gives? Figure 2

1

Tukey, J. W . (1977). EDA. Reading, Mass.: Addison-W esley, p. 43. Kruskal, W . H. (1959). Some Remarks on W ild Observations. http://www.tufts.edu/~gdallal/out.htm. Page 14.2 (C:\data\StatPrimer\correlation.wpd)

2

Correlation Coefficient
The General Idea Correlation coefficients (denoted r) are statistics that quantify the relation between X and Y in unit-free terms. W hen all points of a scatter plot fall directly on a line with an upward incline, r = +1; W hen all points fall directly on a downward incline, r = !1. Such perfect correlation is seldom encountered. W e still need to measure correlational strength, –defined as the degree to which data point adhere to an imaginary trend line passing through the “scatter cloud.” Strong correlations are associated with scatter clouds that adhere closely to the imaginary trend line. W eak correlations are associated with scatter clouds that adhere marginally to the trend line. The closer r is to +1, the stronger the positive correlation. The closer r is to !1, the stronger the negative correlation. Examples of strong and weak correlations are shown below. Note: Correlational strength can not be quantified visually. It is too...
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