Math Homework Problems

Task

Over a two week period, Jenna had the following number of math homework problems given each day:$$20,\,0,\,7,\,10,\,1,\,11,\,0,\,25,\,15,\,1.$$

1. What is the mean number of homework problems Jenna had?
2. What is the Mean Absolute Deviation for the number of homework problems?
3. What do the mean and Mean Absolute Deviation tell you about the number of homework problems Jenna had over these two weeks?

IM Commentary

The goal of this task is to calculate and interpret the Mean Absolute Deviation in a context. It is intended to be an introductory task but can readily be adapted for a more in depth study. The teacher may wish to have students make a dot plot of the data or analyze the quartiles and create a box plot. The data set is peculiar in that it roughly divides into three separate parts: the days (4) when there is little or no homework, the days (3) when there is a moderate number of homework problems, and the days (3) when the assignment is relatively large. Teachers may wish to have students discuss why homework assignments might have this structure: for example, the assignments with only one question are likely challenging problems, perhaps involving modeling or work outside of class. Or there may be an upcoming exam so students are given more time to prepare. The large assignments are likely more routine problems while the days when no homework is assigned the class is probably wrapping up previous material. 

Teachers may prefer to substitute actual data from their own class. Alternatively, they could examine and compare data from different math classes. A follow-up challenging question would be: how much flexibility is there in the data if the mean is 8 and the Mean Absolute Deviation is 7.2? In other words, how much information do these statistics tell us about the actual data?

Solution

1. To calculate the mean, we first find the total number of homework problems assigned: $$20 + 0 + 7 + 10 + 1 + 11 + 0 + 25 + 15 + 1 = 90.$$ There were 10 homework assignments so the mean number of homework problems is 90 $\div$ 10 = 9.
2. To calculate the Mean Absolute Deviation, we first need to find the difference between the number of homework problems assigned each day and the mean number of homework problems. This is shown in the table below: 

   Homework Problems	Mean	|Homework Problems - Mean|
   20	9	|20-9| = 11
   0	9	|0-9| = 9
   8	9	|7-9| = 2
   10	9	|10-9| = 1
   1	9	|1-9| = 8
   10	9	|11-9| = 2
   0	9	|0-9| = 9
   25	9	|25-9| = 16
   15	9	|15-9| = 6
   1	9	|1-9| = 8

   If we add the numbers in the last column we get 72. There are 10 assignments so the Mean Absolute Deviation is 72 $\div$ 10 = 7.2.

3. The mean is 9 and the Mean Absolute Deviation is 7.2. The mean tells us the average number of homework problems Jenna had each day. The Mean Absolute Deviation is almost as large as the mean so this tells us that there is substantial variation in the number of homework problems assigned each day: some days there are a lot more than the mean of 9 and some days there are a lot less.