Lecture note 2 Summarizing Relationships among variables © by Aaed Al- Rabai

4Analyzing a relationship between two (or more) variables is done in many situations. Examples: (i) Relationship between the price of a product and the revenue, (ii) relationship between the promotion cost and revenue, (iii) relationship between the revenue and the rate of price discount. 4We use “Scatter Plot” to represent a relationship between two variables.

Using the following data, we will produce scatter plot that shows the relationships between the number of promotions and the revenues from three different products Product AProduct BProduct C Month Number of promoti ons Sales Number of pro mot ions Sales Number of pro mot ion s Sales April5600, , ,000 May101,000, , ,000 June81,100, , ,000 July9900, , ,000 August101,500, , ,000 September12750, , ,000 October202,200, , ,000 November182,000, , ,000 December171,700, , ,000 Scatter plot example

From the scatter plot we can learn 4There are positive relationships between the number of promotions and the sales from the product A and product B. 4Promotion seems to be most effective for product A. 4There seems to be no relationship between the number of promotions and the sales from product C. (There is a slight downward trend, but this can happen just by chance)

Exercise 4Open the data “Promotion and Sales”, and reproduce the scatter plot.

Numerical measures of summarizing the relationship between two variables 4To think of what numerical measures we would need to represent relationships between variables, see the following three pairs of scatter plots.

Example 1: Relationships between the returns of different stocks Stock B return Stock A return * ** * * * * * * * * * * Stock D return Stock C Return * ** * * * * * * * * * Scatter plot I Scatter Plot II

Example 1 (Continued) 4Scatter Plot I shows a positive relationship while scatter plot II shows negative relationship. 4We need a numerical measure that shows the direction of the relationship. 4 For this purpose, we use “Covariance”

Example 2: Relationships between advertisement spending and revenue Product shows clearer linear relationship between the advertisement spending and revenue than product II. We need to have a numerical measure that shows the strength of linear relationship between two variables. We use “Correlation Coefficient”

Example 3: Number of promotion and sales. Promotion seems to be more effective for Product A than product B in the sense that additional promotion brings greater increase in revenue (i.e., the “slope” is steeper). To measure the effectiveness of the promotion, we use “Regression Analysis”

Numerical measures of summarizing relationships 4In the next lecture note, we will learn 1.Covariance 2.Correlation coefficient 3.Regression Analysis