Comparing Tax Plans using Averages and Percents

My coworker Mr. Roland recently started his own blog here, which is meant to be his compilation of mathematics teaching resources and miscellaneous commentary. Nice work Mr. Roland! Keep the posts coming.



The above picture is a graphic on this entry, entitled "Every American Should See This". It is particularly compelling in the context of teaching mathematics since it raises some very important concepts, namely how percentages relate to actual quantities and how the scale of a graph, (e.g. how the x-axis or y-axis is partitioned into values), affects how we can interpret a visual representation of information like this one.

Being able to quickly look at graphs and glean useful information and trends is an important skill across professions and people's interests. Graphs really are a case where a picture can be worth a thousand words, since within them is often contained meaning and insight that is difficult to convey in text. Being able to read, interpret and analyze a graph is important not only for professional reasons, but also to gain information and insight from magazines, newspapers and television.

Let's begin analyzing this graphic by looking at how the average tax cut of both of the candidates is calculated. Although most people understand the basic definition of an average from calculating their grades in school, averages can get more complicated when we consider averages using percents, as well as weighted averages.

Use the form at the end of the post to submit your answers to these questions.

As an example, let's say that your grade in a class was formed by averaging your scores on 5 exams, each graded on a scale of 0 to 100.

Q1) If you scored a 73, 89, 83, 100 and 92 on your 5 math exams, what would be your grade in this class?

Q2) If your scores on your 5 exams were x1, x2, x3, x4 and x5, what formula would you use to calculate your grade?

This is an example of calculating a simple arithmetic average by weighing each of our scores equally. Most of us have done it a million times without even noticing it, from splitting a check at a restaurant to calculating our grade in school.

In the graphic above, I was wondering how they calculated the average tax cut since there are a number of ways we can interpret an "average tax cut". And some of those ways will lead to more valid and effective insights than others.

Q3) Using the calculator in the side bar, what do you get for the average tax cut for each of the candidate's tax plans by simple averaging?

Note that this answer is slightly different than the average tax cut value displayed on the graphic. Why can't we just take the arithmetic average of the values we see above to determine the average tax cut of the population in a manner similar to averaging the grades in the example above?

The reason is that when averaging percentages in the example above, particularly in considering how to calculate a new average for the entire population, we must not only consider the actual percentages but also how many people are in each of those groups.

To see this more clearly, let's consider a simpler example.

Let's say that we had a population of 10 people and 7 of them made 35 thousand dollars a year, 2 made 100 thousand dollars a year, and 1 of them made 500 thousand dollars a year.

Q4) What would be the change in the amount of money paid in taxes by this population under each candidate's tax plans?

Now that we know the total change in money paid in taxes, we can determine the average tax cut by dividing this number by the number of people in our population.

Q5) Calculate the average tax cut for each candidate's tax plan in this population. Is this number greater than or equal to the average tax cut that would be calculated by simply taking the average of the percent tax cuts that apply to the categories represented in this problem? Why or why not?

What this population analysis illustrates is that the number of people in each income group affects the average tax cut. You can't calculate the average tax cut just by averaging the percentages.

So, returning to the graphic for the candidate's tax plans, to see what the actual tax cut is for the population we really need to know how many people are in each of these categories.

Q6) Using the Internet or another resource, make an educated estimate for the number of people in each of the income categories listed in the graphic above?

Q7) Use the information you gained in Question 6 to determine the change in the amount of the overall taxes paid, as well as the average tax cut for each of the candidate's tax plans. How does this compare to the values displayed in the graphic?