1.7: An Index for Your Salary
- Page ID
- 147908
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)INTRODUCTION
In addition to reviewing the concepts of Absolute Change and Relative Change, this collaboration introduces the concept of an Index. Similar to Relative Change, indexes are used to compare two quantities in relative terms. Later in this collaboration we will learn how to use it to keep track of changes over time. Here’s an example that shows how to calculate Absolute Change, Relative Change, salary Ratio, and salary Index:
| 2022 salary | 2023 salary | Absolute Change | Relative Change | |
| Tyler | $78,000 | $88,000 | $88,000 - $78,000 = $10,000 | $10,000 / $78,000 * 100% = 12.82% |
| Amy | $87,000 | $91,000 | $91,000 - $87,000 = $4,000 | $4,000 / $87,000 * 100% = 4.6% |
| Danela | $81,000 | $85,000 | $85,000 - $81,000 = $4,000 | $4,000 / $81,000 * 100% = 4.94% |
| 2022 salary | 2023 salary | Salary Ratio | Salary Index | |
| Tyler | $78,000 | $88,000 | $88,000 / $78,000 = 1.1282 | $88,000 / $78,000 * 100 = 112.82 |
| Amy | $87,000 | $91,000 | $91,000 / $87,000 = 1.0460 | $91,000 / $87,000 * 100 = 104.60 |
| Danela | $81,000 | $85,000 | $85,000 / $81,000 = 1.0494 | $85,000 / $81,000 * 100 = 104.94 |
Based on the calculations shown above, please generalize by matching the concept to the formula that can be used to calculate Absolute Change, Relative Change, Salary Ratio, and Salary Index for any two numbers.
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SPECIFIC OBJECTIVES
By the end of this collaboration, you should understand that
- understanding relative changes in quantities over time can illustrate patterns of growth.
- indexes can be useful ways to compare various changing quantities to each other.
By the end of this collaboration, you should be able to
- create an index of relative changes given raw data.
PROBLEM SITUATION: CAREER PLANNING
As you read in the preparation, Gary and Anne are employees at a retail store for a cell phone provider. Gary is a Sales Associate and Anne is his Sales Supervisor. Their company offers many opportunities for growth in the organization. For example, Anne was promoted to Supervisor from Associate about three years ago. Gary is starting some long-term planning and wants to decide if he should stay with this company and seek a promotion, or if he wants to explore other options with another company, or even try a new career.
Gary has access to the following data from the past seven years about typical salaries for Sales Associates, Sales Supervisors, and Regional Sales Managers. Notice that salaries do not always increase from one year to the next. Gary wants to get a better idea about how the salaries change over time for each of these positions, to gain insight into his earning potential if he stays with his current company.
Salaries of Sales Personnel (2016–2023)
| 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | |
| Sales Associate | $41,500 | $42,600 | $42,800 | $42,000 | $43,100 | $43,600 | $44,700 | $46,300 |
| Sales Supervisor | $56,500 | $57,100 | $58,700 | $57,300 | $60,500 | $59,600 | $60,700 | $62,600 |
| Regional Sales Manager | $85,000 | $89,500 | $87,400 | $82,700 | $87,100 | $89,000 | $86,100 | $91,000 |
(1) What do you notice about the figures in this table? What are some advantages or disadvantages of only using this table to compare salaries to make a decision? Be sure to describe the advantages and disadvantages quantitatively.
(2) (a) Make a quick sketch of a graph or visual display of these data. To make your graph easier to read, consider using different color ink to represent each of the positions.
(b) What are some advantages or disadvantages of using a graph like the one you made above to visualize data over time?
(3) Which position seems to show the most growth in salary over the past seven years? What particular quantities would be valuable for Gary to notice to help make his decision?
(4) The table below shows absolute and relative changes in salaries for each position from 2016 to 2023.
Absolute and Relative Changes from 2016 to 2023
| Position | Absolute Change | Relative Change (rounded to nearest tenth of a percent) |
| Sales Associate | $4,800 | 11.6% |
| Sales Supervisor | $6,100 | 10.8% |
| Regional Sales Manager | $6,000 | 7.1% |
(a) Explain what you observe in the numbers and in the absolute changes and relative changes. For example, you could explain how the Sales Associate can have the lowest absolute increase in salary but the highest relative increase.
(b) Now calculate the absolute and relative changes in salary from 2021 to 2022. Fill in the table below with your answers. Round your answers to the nearest tenth of a percent.
Absolute and Relative Changes from 2021 to 2022
| Position | Absolute Change | Relative Change |
| Sales Associate | ||
| Sales Supervisor | ||
| Regional Sales Manager |
(c) Which position had the greatest absolute increase in salary? Which position had the greatest relative increase in salary? Write an explanation for what you observe in the absolute changes and relative changes.
Salary Index
One way to systematically look at a large set of relative changes over time is to create an index. Making an index requires you to make a lot of calculations similar to what you did in the preparation for this collaboration. A salary index will look like a table of numbers that answer the question, “The salary for this year is what percent of the base year?”
For example, in the preparation, you were given the following information:
In 2022, Gary earned $48,300, but he received a raise in 2023 and earned $53,400.
If Gary meets his supervisor’s benchmark goals in 2023, then he will earn $59,150 in 2024.
| Year | Gary’s Salary |
| 2022 | $48,300 |
| 2023 | $53,400 |
| 2024 | $59,150 |
You then used the given information to answer the following questions:
Gary’s 2023 salary is ______% of his 2022 salary. Round your answer to the nearest percent.
Gary’s 2024 salary is ______% of his 2022 salary. Round your answer to the nearest percent.
In this example, 2022 is the base year or the reference value. Note that both the 2023 and 2024 indices are calculated by dividing the corresponding salaries by the base year salary.
Please review these questions with your group and verify that the correct percentages are rounded to the nearest percent:
- Gary’s 2023 salary is 111% of his 2022 salary
- Gary’s 2024 salary is 122% of his 2022 salary
Gary’s Salary Index
| Year |
Salary Index |
| 2022 | 100 |
| 2023 | 111 |
| 2024 | 122 |
(5) Now, use the salary information from the problem situation to create a salary index for each of the three positions. You will be creating three separate indices, one for Sales Associate, one for Sales Supervisor, and one for Regional Sales Manager. Use 2016 as the base year, and round each answer to the nearest tenth.
Salaries of Sales Personnel (2016–2023)
| 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | |
| Sales Associate | $41,500 | $42,600 | $42,800 | $42,000 | $43,100 | $43,600 | $44,700 | $46,300 |
| Sales Supervisor | $56,500 | $57,100 | $58,700 | $57,300 | $60,500 | $59,600 | $60,700 | $62,600 |
| Regional Sales Manager | $85,000 | $89,500 | $87,400 | $82,700 | $87,100 | $89,000 | $86,100 | $91,000 |
Salary Index (Base Year = 2016)
| Year | Sales Associate | Sales Supervisor | Regional Sales Manager |
| 2016 | 100.0 | 100.0 | 100.0 |
| 2017 | 102.7 | 101.1 | 105.3 |
| 2018 | |||
| 2019 | |||
| 2020 | |||
| 2021 | |||
| 2022 | |||
| 2023 |
(6) Use quantities in your index table to explain your observations and ideas about what Gary should know about how the salaries change over time.
(7) Make a graph of the indices that you created. What are the advantages and disadvantages of using this graph to interpret how the salaries changed over time?
MAKING CONNECTIONS
Record the important mathematical ideas from the discussion.