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Correlation

Correlation. Measurement of linear relationship between variables.

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Correlation

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  1. Correlation Measurement of linear relationship between variables

  2. Consider the following scatterplot of fire damage to houses (in thousands of dollars) during house fires and distance to the nearest fire station (in miles). The data are plotted and one additional point, , is shown, as well, using a square symbol.

  3. As we begin looking at the relationship between the explanatory and response variables, we are sometimes distracted by the scale of the data. To eliminate the effect of scale, we can standardize each value. In place of each (x,y), we now have (zx,zy).Whenwe standardize, the graph will be centered around the origin with becoming (0,0).

  4. Compare the two scatterplots, and note that with standardization, the shape of the plot remains the same, but the axes are shifted. Note that the majority of the points now fall in the first and third quadrants. There are few points in the second and fourth quadrants.

  5. Recall that in the first and third quadrants both x and y have the same algebraic sign, while in the second and fourth quadrants the signs are different. If we multiply the z-scores for each ordered pair, the products from quadrants one and three will be positive, while the products from quadrants two and four will be negative.

  6. If there is a positive association between x and y, then there are more points in quadrants one and three than in quadrants two and four. When there is a negative association between x and y, just the opposite is true. The sum of these products gives an indication of the strength of the association. To adjust for sample size, we will divide this sum by the degrees of freedom, as we have done in other circumstances.

  7. This brings us to a formula: We call this quantity r, the correlation coefficient, or the Pearson Product-Moment Correlation Coefficient. It measures the direction and strength of a linear relationship between two variables. The correlation coefficient is an entity that is often used, and occasionally misused, in the practice of statistics.

  8. The correlation coefficient, r, can vary between -1 and 1. The closer it is to 1 or -1, the stronger the association. The closer it is to zero, the weaker the association. If data is perfectly linear, then r = 1 or -1, as in the examples shown below. r=1 r=-1

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