Correlation Analysis

Is there a relationship between the amount McDonalds spends on advertising and its sales? Is there a relationship between smoking and lung cancer? Is there a relationship between violence in TV programs and crime rate? In each of these circumstances there are two variables -- for example advertising amount and sales. Correlation analysis is the study of relationship between such variables. To explain suppose the sales manger of XEROX America, which has a large sales force throughout the United States and Canada, wants to determine whether there is a relationship between the number of sales calls made and the number of copiers sold that month. The manager selects a random sample of 10 sales representatives and determines the number of sales calls each representative made last month and the number of copiers sold. The sample information is shown below

Sales Calls and Copiers Sold for Ten Salespersons
Sales Representative	Number of Sales Calls	Number of Copiers Sold
Edward Jenkins	30	40
William McGee	50	70
Tom Banks	30	50
John Fitzgerald	40	70
Brian Hunt	20	20
Elle Robson	20	30
Dennis Fay	30	45
Lew Azner	25	20
Robert Wilson	30	70
Kelly Arena	20	50

There seems to be some relationship between the number of calls and the number of units sold. We will try Correlation Analysis to arrive at a conclusion. Correlation Analysis measures the strength of the association between two variables. A Scatter diagram is a chat that portrays the relationship between two variables. Following is a scatter diagram for the above data.

Scatter Diagram

The Coefficient of Correlation

Originated by Karl Pearson in early 1900's, the coefficient of correlation describes the strength of the relationship between two sets of interval-scaled or ratio-scaled variables. It is designated by r and is referred to as Pearson's r. It assumes any value between -1.00 to +1.00 inclusive. A correlation coefficient of -1.00 or +1.00 indicates perfect correlation. If coefficient of correlation for our example computed to be +1.00 it would indicate the number of sales calls and the number of copiers sold are perfectly related in positive linear sense. And a computed value -1.00 would suggest that sales calls and number of sales are perfectly related in negative sense. The coefficient of correlation is computed by the following formula.

Coefficient of Correlation

X	Y	XY	X²	Y²
30	40	1200	900	1600
50	70	3500	2500	4900
30	50	1500	900	2500
40	70	2800	1600	4900
20	20	400	400	400
20	30	600	400	900
30	45	1350	900	2025
25	20	500	625	400
30	70	2100	900	4900
20	50	1000	400	2500
295	465	14950	9525	25025

n=10

Using the formula to compute r entails the following

Correlation Example