Analyzing Bivariate Data Doing Statistics for Business Data, Inference, and Decision Making Marilyn K. Pelosi Theresa M. Sandifer Chapter 5 Analyzing Bivariate Data

Doing Statistics for Business Chapter 5 Objectives Bivariate Qualitative Data: Contingency Tables Clustered & Stacked Bar Charts Identifying Quantitative Relationships: Scatter Plots The Least-squares Line

Doing Statistics for Business A Contingency Table is a table whose rows represent the possible values of one variable and whose columns represent the possible values for a second variable. The entries in the table are the number of times that each pair of values occurs.

Doing Statistics for Business TRY IT NOW! Quality Problems Creating a Contingency Table A company that manufactures cardboard boxes is trying to understand some of their quality problems. They have analyzed some data and determined that their major defects are Printing, Color and Skewness (how square the box is). Further attempts to pinpoint the problems have resulted in many opinions and finger pointing about responsibility. They decide to collect some additional data on defect type and the shift during which production occurred.

Doing Statistics for Business TRY IT NOW! Quality Problems Creating a Contingency Table (con’t) The data (sorted by defect type) are as follows:

Doing Statistics for Business TRY IT NOW! Quality Problems Creating a Contingency Table (con’t) Create a relative frequency contingency table for the data. What percentage of the defects were color? Does there appear to be any credence tot he claim that the majority of the defects occur on the third shift? Ans.

Doing Statistics for Business Figure 5.1 Two bar charts representing a contingency table

Doing Statistics for Business In a Clustered Bar Chart, the bars for one variable are grouped according to the values of the other qualitative variables.

Doing Statistics for Business Figure 5.2 Example of a Clustered Bar Chart

Doing Statistics for Business TRY IT NOW! Quality Problems Creating a Clustered Bar Chart The management of the company that manufactures cardboard boxes decide to have a meeting with the crews about the quality problems. The managers decide that a graphical display of the data is the best way to convey the information they have obtained and that they want to display the crew data for each of the quality defects.

Doing Statistics for Business TRY IT NOW! Quality Problems Creating a Clustered Bar Chart (con’t) Create a clustered bar chart that display the percent defects for each shift using the quality problems as the categories for the x axis.

Doing Statistics for Business TRY IT NOW! Quality Problems Creating a Clustered Bar Chart (con’t) Which shift contributes the largest percentage of color defects? The least percentage? What do you notice about the skewness defects? Ans. Crew 1, Crew 3, Only made by Crew 2

Doing Statistics for Business In a Stacked Bar Chart the data for the selected variable are represented as a percentage of the total for each category of the second variable. Each value of the selected variable is represented in a different way and the bars are “stacked” to total 100%.

Doing Statistics for Business Figure 5.3 Stacked Bar Chart

Doing Statistics for Business TRY IT NOW! Quality Problems Creating a Stacked Bar Chart The company that manufactures cardboard boxes decides to create a graphical display of the data to show to the employees. What the company wants to do is to display type of defect as a percentage of total defects for each shift. It is hoped that these data will help each shift concentrate on its own priorities. The contingency table for the defect is shown on the following slide.

Doing Statistics for Business TRY IT NOW! Quality Problems Creating a Stacked Bar Chart (con’t) Modify the contingency table to display defect types as a percentage of total defects for each shift.

Doing Statistics for Business TRY IT NOW! Quality Problems Creating a Stacked Bar Chart (con’t) Create a stacked bar chart for the data. Which defect type should each crew concentrate on? Why? Ans. Crew 1 color; Crew 2 printing; Crew 3 color

Doing Statistics for Business In a Scatter Plot an axis is used to represent each of the variables and the data are plotted as points on the graph. Typically, the independent variable is plotted on the x axis and the dependent variable is plotted on the y axis.

Doing Statistics for Business Figure 5.4 Scatter Plot

Doing Statistics for Business TRY IT NOW! Airline Fares Creating a Scatter Plot A travel agency was interested in knowing how airline fares were related to the length of the flight in miles. The agency hypothesized that the longer the flight, the more the airfare. The following data were collected:

Doing Statistics for Business TRY IT NOW! Airline Fares Creating a Scatter Plot (con’t) Which variable is the dependent variable and which is the independent variable? Make a scatter plot of the data. From your plot, do you think that the travel agency’s hypothesis was correct? Why or why not? Ans: Length = Independent Fare = Dependent Yes

Doing Statistics for Business Figure 5.6 Scatter Plot Indicating no Relationship Between Two Variables

Doing Statistics for Business TRY IT NOW! Starting Salaries & Math Courses Finding the Least-Squares Line The Career Planning office of a large university is interested in knowing whether there is a relationship between the starting salary of graduates and the number of mathematics courses the graduates had taken as students. The Career Planning office goes through the records for the last year and finds the following data.

Doing Statistics for Business TRY IT NOW! Starting Salaries & Math Courses Finding the Least-Squares Line (con’t) Which variable is the independent variable? The dependent variable?

Doing Statistics for Business TRY IT NOW! Starting Salaries & Math Courses Finding the Least-Squares Line (con’t) Use a grid to create a scatter plot of the data. Do you think that there is a linear relationship between the number of math courses taken and the starting salary? If so, describe the relationship. Ans. Dependent - salary, Independent number of Math Courses

Doing Statistics for Business Discovery Exercise 5.1 Discovering Relationships 1. The data below represent data taken on the number of weeks that a student was enrolled in a speed reading program and the speed gain in words per minute that the student has experienced:

Doing Statistics for Business Discovery Exercise 5.1 Discovering Relationships (con’t) 2. Plot the data on graph paper. Be sure to label your axes and to construct the graph so that it uses most of the paper. 3. Draw a straight line through the points that you think best represent the relationship between x and y. What criteria did you use for drawing the line you selected?

Doing Statistics for Business The Least Squares technique finds the equation of the line that minimizes the sum of the squared errors between the actual data points and the line.

Doing Statistics for Business Figure 5.7 Deviations Between the Data Points and the Line

Doing Statistics for Business The distance between the predicted value of Y, called the deviation or error. The deviation, e, is equal to - y.

Doing Statistics for Business TRY IT NOW! Starting Salaries and Math Courses Finding the Least-Squares Line The Career Planning office at the university wants to use least squares to find the equation that relates the number of math courses taken and starting salary. The data are: Number of math courses Starting Salary ($) 1 26,284 1 25,470 2 26,777 3 27,269 4 28,553 6 30,054

Doing Statistics for Business TRY IT NOW! Starting Salaries and Math Courses Finding the Least-Squares Line (con’t) (a) Find the equation for the least-squares line for the data. (b) Explain what the least-squares line tells the Career Planning Office about math courses and starting salary. Ans: = 25,020 + 840.6 The Base Salary is $25,020 and every math class increases it by $840.60.

Doing Statistics for Business TRY IT NOW! Starting Salaries and Math Courses Finding the Least-Squares Line Once the least-squares equation is obtained the Career Planning Office wants to see whether the equation does a good job of predicting starting salary for a given number of math courses. Plot the data and the least-squares line on the same graph.

Doing Statistics for Business TRY IT NOW! Starting Salaries and Math Courses Finding the Least-Squares Line Use the equation you just found to predict the starting salary of a person who takes 2 mathematics courses. Do the same thing for the starting salary of a person who takes 5 mathematics courses. Do you think the predictions from the least-squares line are useful? Why or why not? Is what you just did interpolation or extrapolation?

Doing Statistics for Business TRY IT NOW! Starting Salaries and Math Courses Finding the Least-Squares Line (con’t) Now use the equation of the least-squares line to predict the starting salary of a person who takes no math courses. Do the same thing for a person who takes 10 math courses. Do you think these predictions are valid? Why or why not? Ans. $26,701; $29,223 Interpolation $25,020, $33,426 No. You are extrapolating.

Doing Statistics for Business Creating Contingency Tables in Excel 1. From the Data menu, select PivotTable and Pivot Chart Report . 2. Since you have already highlighted the range that contains your data, you can just proceed by clicking Next>. 3. Indicate where you want the Pivot table to appear and click the button labeled Layout.

Doing Statistics for Business Creating Contingency Tables in Excel 4. Drag the field box for Rank to the area marked Row; and the field box for School to the area marked Column. Now drag the field box for either Rank or School into the area marked Data. 5. Click on Next and tell Excel where you want the table to appear; either in the current worksheet or in a new worksheet. 6. Finally, click on Finish.

Doing Statistics for Business Figure 5.10 Contingency Table of Rank and School

Doing Statistics for Business The basics of creating a chart in Excel, using the Chart Wizard. 1. Highlight the data (Frequency table) that you want to graph. 2. Invoke the Chart Wizard by clicking on the icon on the toolbar. 3. Follow the directions and hints from the Chart Wizard. 4. Edit the graph to include any other features or changes you want.

Doing Statistics for Business Creating Clustered Bar Charts in in Excel 1. Highlight just a portion of the pivot table by copying it to another location in the worksheet using the Paste Special command. To do this click on the lower right hand corner of the table and highlight it. Select Copy from the Edit menu. Then, click on the location for the copy and select Paste Special. In the Paste section, click on the radio button for Values and in the Operation section, select None. Click OK.

Doing Statistics for Business Creating Clustered Bar Charts in in Excel (con’t) 2. Highlight the copy of the table, being careful not to highlight the Grand Total row or column. Then start the Chart Wizard and select Column as the type of chart and clustered chart as the sub-type. 3. Click Next twice to get to the Options step; add titles and make any desired formatting changes. 4. Click Next to indicate where you want the chart to be located and then click Finish.

Doing Statistics for Business Figure 5.14 Clustered Bar Chart

Doing Statistics for Business Creating Stacked Bar Charts in in Excel 1. Highlight the rows of the pivot table, being careful not to include the Grand Total row and column and start the Chart Wizard. 2. Select Column as the chart type and 100% Stacked Column as the sub-type. 3. Click Next twice to get to the Options step; put in titles and any other formatting changes that you want. 4. Click Next, enter the location for the chart, and click Finish to display the chart.

Doing Statistics for Business Figure 5.17 Stacked Bar Chart

Doing Statistics for Business Analyzing Quantitative Bivariate Data in Excel 1. Highlight the range of the data and start the Chart Wizard. Select XY (Scatter) as the chart type and Scatter no lines) for the subtype. 2. Select Next> twice to get to the Chart Options dialog box; put in titles and any other formatting changes that you want. 3. Select Next> to tell Excel where to locate the chart and click Finish to display the chart.

Doing Statistics for Business Figure 5.20 Finished Scatter Plot

Doing Statistics for Business Adding the Least-Squares Line to Your Scatter Plot 1. Click on any one of the points in the scatter plot to highlight them all. 2. From the Chart menu, select Add Trendline. The Add Trendline dialog box will open. 3. In the Trend/Regression section, highlight Linear. 4. Click on the Options tab and click the box labeled Display equation on chart. 5. Click OK.

Doing Statistics for Business Figure 5.24 Scatter Plot with Least-Squares Line

Doing Statistics for Business Chapter 5 Summary In this chapter you have learned: Some methods for looking at how two variables are related. There are graphical methods, such as stacked bar charts, clustered bar charts, and scatter plots. There are quantitative methods, such as least-squares analysis.