Chapter 3D Chapter 3, part D Fall 2000

IV. Measures of Association Between 2 Variables The world of business and economics is a complicated one. Many interacting variables create interesting relationships that, if they can be described, can help the decision maker.

A. Covariance The covariance is a measure of association that can tell us if 2 variables are positively, negatively, or not related. Each xi is paired with a corresponding yi. Sample covariance: Population covariance:

B. Interpretation of the covariance Suppose that a sample covariance between the amount of advertising spending (x) and sales (y) is calculated to be 150.75. What can be said about this figure? Not much. Only that it appears that there exists a positive linear relationship between ad spending and sales. Be careful, don’t assume that a larger variance means a stronger relationship!

C. Correlation Coefficient This statistic does give us an idea of the strength of the relationship between x and y. Sample correlation: Population correlation:

D. Interpretation of the correlation coefficient The correlation ranges from -1 to 0 for negative relationships and 0 to +1 for positive relationships. Thus the closer the value gets to 1, in absolute value, the stronger the relationship between x and y. If the coefficient is approximately 0, there is apparently no significant relationship between x and y.

V. Grouped Data These are techniques that are used when data are only available in a frequency distribution, but you don’t have the actual data. These will only be approximations of the actual mean, variance and standard deviation.

Back to our sample frequency distribution of beef prices

A. The mean Treat the midpoint of each class (i=1...k) as the mean for each class (Mi). Grouped mean: Our example: = 106.9

B. Variance and Standard Deviation Again, because we don’t have the actual data, we use the midpoint as an approximation. The standard deviation is the square root of the variance.

Our Example Calculate the variance and standard deviation for our grouped data. The sample grouped variance should be = (1969.8/19) = 103.67. Thus the sample grouped standard deviation should be = 10.18.