Baseball Salaries - Dynamic Math



Baseball Salaries

Why do owners of major league baseball teams pay their players such high salaries? In this Fathom activity, we’ll explore this question and learn some basic Fathom features.

1. Before we start using Fathom, let’s look at three possible reasons the owners might pay huge salaries to their players. These three reasons, or attributes, are listed below. Discuss each one with your partner. Next to each attribute, write your prediction as to whether our data will show a relationship between a team’s total payroll costs and each attribute and why or why not.

|Owners pay high salaries because: |Agree? |Why or why not? |

|Attendance will increase at ballgames | | |

|Their team’s batting average will increase because they attract | | |

|baseball’s top batters to their team | | |

|Their team will win more games and possibly be a World Series | | |

|contender | | |

STOP HERE SO WE CAN DISCUSS AS A CLASS.

Linear Correlations

2. Let’s use Fathom to determine if there is a relationship between salary costs and each of our three attributes. Open the file “BaseballSalary1” and click on the collection box named “baseball teams.”

3. Drag down a new case table from the tool bar. Adjust the table so you can see all of the teams and all of the table headings. Double click on the collection box and click on the Comments tab for information about each column of data. What do you notice about the units used for total payroll and attendance?

4. Reduce the case table, but make sure you can still see all of the teams. This will give you more room on your screen.

5. First we’ll explore the relationship between total salary costs and annual attendance at ballgames. Drag a new graph from the tool bar. Drag the heading PayrollMillions from the case table onto the x-axis. Drag the heading AttendanceThousands from the case table onto the y-axis. You should have a scatter plot of the data.

6. What general pattern do you notice on the scatter plot? Does it look like there is a relationship between a team’s total salary costs and attendance at ballgames?

7. By highlighting points on the graph and using the case table, which team has the highest attendance? ______________ Which team has the lowest attendance? __________________ Highlight Minnesota’s row on the case table and locate it’s point on the graph. If you were the owner of this team and saw this graph, what could you say about your payroll costs/attendance compared to other teams?

8. Highlight SanFrancisco’s row on the case table and locate its point on the graph. It appears that this team’s payroll is slightly below the average of all the teams, but attendance at games is much higher than the average of all teams. Why might this be the case?

9. Now let’s see if there is a strong linear relationship between total payroll costs and attendance at games. Highlight the graph, right-click on your mouse and select Least-Squares Line. Fathom has drawn the regression line that best fits all 30 data points. Are the data points closely clustered around this line? What does this indicate about using a linear model to describe the relationship between these two variables?

10. We can quantify how well the least-squares line explains the change in attendance as payroll costs change by looking at the r^2 value shown in red below the graph. This value can range from 0 to 1; a value of 1 means all of the variation in attendance is explained by the linear relationship with payroll costs and a value of 0 means none of the variance is explained. What is the r^2 value and does it support your answer to 9 above?

11. The equation of the least-squares line is also shown in red below the graph in the form y = mx + b. What is the slope of this line? _______ Describe in words what this slope means for this particular graph.

12. Based on your analysis above, if you were a team owner and wanted to increase attendance, would you consider hiring high-salary players for your team? Why or why not?

13. Now let’s look at the relationship between payroll costs and increasing the team’s batting average. Create a scatter plot of these two variables using the process in Step 5 above. What general pattern do you notice on the scatter plot? Does it look like there is a relationship between a team’s total salary costs and the team’s batting average?

14. Plot the least-squares line and find the r^2 value using Steps 9 and 10 above. How does the line and the r^2 value support your answer to Step 13 above?

15. Pitchers usually have very low batting averages. How does this affect the scatter plot? What could you do to make this a more meaningful scatter plot?

16. Lastly, let’s look at the relationship between payroll costs and the number of games a team wins. Create a scatter plot of these two variables. What general pattern do you notice on the scatter plot? Does it look like there is a relationship between a team’s total salary costs and the number of games won?

17. Plot the least-squares line and find the r^2 value. How does the line and the r^2 value support your answer to Step 16 above?

STOP HERE SO WE CAN DISCUSS OUR FINDINGS UP TO THIS POINT.

Using Formulas in Fathom

Let’s look more closely at the relationship between total payroll costs and attendance. Most owners of major league baseball teams are savvy businesspeople who want to increase attendance and make a profit. Remember that Profit = Revenue – Expenses. In this case, we’ll look at a team’s revenue as the average price fans pay for a ticket and as expenses as payroll costs per ticket. We’ll then determine if any teams earn a profit on each ticket sold after paying out salaries.

18. So far we’ve looked at a team’s total payroll costs compared to total attendance. Now let’s look at the payroll cost per each fan that attended a ballgame. Highlight the open case table and scroll over to the first empty column. Change the name by double clicking on the column heading and typing CostperAtt. This column will represent the payroll cost per each fan that attended a game.

19. Right-click on this heading and select Edit Formula. You should see a blank formula box. We want to divide total payroll costs by attendance using the attributes or columns in our case box. Type PayrollMillions / AttendanceThousands. Each of these attributes should turn blue to indicate Fathom recognizes which data to use in this calculation. Be careful – this formula is not yet correct! What else do we have to add to the formula?

20. Let’s graph each team’s payroll cost per fan. Create a new graph that shows CostperAtt on the x-axis. Change the dot plot to a histogram using the box on the upper-right corner of the graph. Look at the differences in each team’s data. Do most teams pay the same per fan or is there a large spread?

Click the right-most bar on the histogram. Which team is this? Where does this team’s data fall on the three graphs you created above? Is this team’s owner seeing a benefit from spending the most per fan on salaries?

21. Now add a new column in the cast study box called ProfitTick to represent the profit from each ticket sold. Enter the formula to calculate this profit per ticket using the process from Step 19. Make a new histogram that shows ProfitTick on the x-axis. Do all teams earn a profit after paying out salaries? Where does the team identified in Step 20 fall on this graph? What should the owner consider doing?

Discuss with your partner the other revenue and costs of owning a baseball team and whether you think the teams earn a total profit or loss. Summarize your thoughts below.

22. FOR HOMEWORK: Based on all of your analysis above, if you were the owner of a major league baseball team, would you pay high salaries to attract the better known or more talented players? Why or why not?

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