4 Measuring inequality: An Exercise

This project is designed to do two things.

1. Get you to think about was that you could actually quantify the level of income inequality in an entity like an organization or a country.
2. Introduce or review some basic statistical terminology.

To do this you need a spreadsheet you can copy from here:

https://docs.google.com/spreadsheets/d/1OkwpWNUS8TkWjPit_JbmONP7uxkbwtXicvmvRbfpvbY/edit?usp=sharing

(Click on File > Make a Copy or File >  Download)

Here is a copy of this chapter that you can  copy or download or make a copy of and then answer the questions within the document.

https://docs.google.com/document/d/19zfE62OmrzK3yz-PTG6eNprndb6b8J-P4GzRyyQkKsM/edit?usp=sharing

Q1.  Suppose you are starting a new company in which you are the CEO (Chief Executive Officer). Including yourself there will be 10 employees doing various jobs.  You have a budget of one million dollars to use on salaries. How would you distribute those salaries?

Q2 Write the story of your budget choices here. Why did you choose the salaries you did? Feel free to give your employees job titles and descriptions to justify their salaries.

Some ideas for employees: CFO, COO, VPs, Mid-level Managers, Project Leads, Sales Associates, Assistants, Interns, Cleaning People.

Q3

1. Copy your data to the first tab of the spreadsheet.
2. Use the second tab of the spreadsheet to create a new salary structure for your company where there is, in your opinion, the most possible inequality. (But everyone gets something)
3. Use the third tab of the spreadsheet create a salary structure which is the most equal. (Everyone gets something)

Use the mean function in the spreadsheet for each salary distribution.   How do the means compare?

Q4  Why did you get the results for the mean that you got? What is it about the mean that caused you to get that result?

Q5 Do you think that the mean is a good measure of inequality?

Q6 Brainstorm some ideas about how you could measure how much salaries vary in an organization? (There are many good answers!) Explain one of your ideas.

Q7 One way to summarize variation is the range (highest value – lowest value).

1. Calculate the range for your three salary distributions. What values for the range did you get?
2. How is the range as a measure of variation?

Q8  Another way to think about variation is to calculate the distance of each observation from the overall mean.   For your three salary structures calculate how far each person’s salary is from the mean salary.   That is, take the difference:  Person’s salary – Mean.

When you do this make sure that you don’t keep retyping values, use the spreadsheet to do the work.

Q9 What are some things you notice about these differences?

Q10 What happens if you add up the differences?   What does this tell you about the mean? Is the sum of the differences from the mean a good measure of how much inequality there is in a data set?

Q11 What would be another way you could use the differences to measure variation? (there are many possible answers).

Q12 One way might be to simply turn all of the differences into positive numbers (which is removing the information about whether a salary is above or below the mean).   In math there is a function called the absolute value that does this.  The absolute value of -4 is 4. The absolute value of 4 is 4.

In a new column, get the absolute values for the salaries in your three salary structures and add them up. Use the ABS() function in your spreadsheets.

12 A What do you get when you add up the absolute values in each structure? Why are some higher than others?

12 B How is the sum of the absolute values as a measure of variation?

Q13 Now take the mean of the absolute values.   How is that as a measure of variation?

Q14  This measure is called the mean absolute deviation or MAD. Why do you think statisticians and researchers would usually use the mean rather than the sum?

Q15 Based on MAD, which of your salary structures is the most equal?

Q16 MAD is a measure of variation, while the mean is a measure of central tendency.  Why is it that that we would not use the mean salary as a way to measure inequality in a company? Why is a measure of variation what we need?

Looking at inequality Part 2

In the first part of this project you created several possible salary distributions for your new company.  We looked at the mean of these distributions and at the Mean Absolute Deviation of these.

Start by opening up your spreadsheet.  (If you can’t find it and you already uploaded it to Blackboard you can go look at the assignment where you uploaded it and download the file. If you totally don’t have it you can download the blank spreadsheet and quickly make some distributions.)

Q1 Describe in words what each of your three salary distributions and how they differ from each other.

Q2 A second way that statisticians measure how much variation there is in a set of data is by squaring the differences rather than taking the absolute value.  Start a new column in your spreadsheet and put the squared differences.  Look at the spreadsheet.

1. Looking at the numbers, how is squaring the differences similar to using the absolute value?
2. Looking at the numbers, how is squaring the differences different from the absolute value?

Q3  Use the spreadsheet to calculate the sum of the squared differences and to get the mean of the differences.

What are your results for the three distributions?

Q4. Which one has the highest and which has the lowest mean of the sum of squared differences? Is the ranking different than for the MAD?

Q5 The mean of the squared differences from the mean is called the variance.  The square root of the variance is called the standard deviation.  Use the sqrt() function to find the standard deviations. What are your results?

Q6  Make a new tab in your spreadsheet.  In that tab make a company (10 employees and a total budget of one million dollars) with the smallest possible variance.  To find this you will want to try different arrangements. You must pay everyone something!  (It’s okay if this repeats a structure you already have.)  What was the smallest variance you got?  Describe that salary distribution in words.

Q7 Make another new tab.  This time try to find the highest possible variance.  Experiment with different possibilities until you are satisfied that you got the maximum.  What was the maximum variance you got? Describe that distribution in words.

Q8  In your opinion, which  worked better for understanding the distribution of salaries, the MAD or the variance?

Q9 Another way people talk about the distribution of salaries is to compare the salary of the CEO to the mean salary of the other employees.    For each of your distributions calculate the mean salary of everyone but the CEO (you).   What are your results?

Q10 Now divide the CEO salary by the mean of the other salaries and multiply that by 100.  This will let you say “the CEO’s salary is __ times the salary of the mean other employee.”  Write that for each of your distributions.

Q11 If a bigger number in Q10 represents more inequality how did your distributions rank?  Is the ranking different than it was for the variance or the MAD?

Q12. Another measure of inequality that is used by social scientists is called the Gini Coefficient. This ranges from 0 (perfect equality) to 1 (maximum inequality).

Use this app to calculate the Gini Coefficient for your distributions.

https://elinw.shinyapps.io/gini/

How do the rankings compare to the other approaches?

Q13 What do you think about using either of these statistics instead of the variance or the MAD?

Q14 Overall, if you were really creating a company and had to assign salaries, how much variation or inequality would you have? What is your thinking on this?