Who doesn't love a frequency table?

What is a frequency table.  It's a table that shows you frequency: how often something happens.  This is the sort of frequency table and question you might see on the SAT.

The students in a class took a survey about the number of pets that they have.  The table below shows the results of the survey:

Number of Pets	Frequency
0	8
1	12
2	4
3	2
4	2
5	2
6	0
7	1

 Which of the following statements is true?

1. The median number of pets is greater than the mean number of pets.
2. The mean number of pets is greater than the median number of pets.
3. The mode of the number of pets is greater than the median number of pets.
4. The range of the number of pets is less than the mean number of pets. 

In a question like this, the first thing that students tend to struggle with is how to read the frequency table. Some are perplexed entirely.  Others aren't sure if 0 people have 8 pets or if 8 people have 0 pets.  The first thing we do is ask the students to slow down.  What does the table say? Read the title (or description) and the column headings.

The left column says "Number of pets."  The right column says "Frequency." So the right column is showing how often the choice noted on the right (that particular number of pets) was chosen by a student.  So, 8 students have 0 pets. 

Helping students read the table right is the first step.  Then, they need to think through how to use the table to calculating statistics. 

If you ask a student what the median of the data is, many students will say 4.  Many others will say 2. The median is actually 1.  Why do so many students get that wrong?  Students remember that the median is the middle number, so they choose the number in the middle of the table (4 if you look at the middle entry in the "Number of Pets" column and 2 if you look at the middle entry in the "Frequency" column.)

Students know that in order to find a median, you have to line the data up in order, but it takes a minute for them to realize what the individual data points in this table are.  Once they start talking about it, they can tell you that 8 students have no pets, 12 students have 1 pet, 4 students have 2 pets, etc.  We then ask them to write those datapoints out so that they can see all of the data (not compressed in a frequency table) and calculate the median (obviously, most will not do this step on the SAT, but it's helping as they are learning how to work with frequency tables):

$0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,4,5,5,7$

So, we find the center datapoint: that's the median.

 $\require{cancel}\cancel{0},\cancel{0},\cancel{0},\cancel{0},\cancel{0},\cancel{0},\cancel{0},\cancel{0},\cancel{1},\cancel{1},\cancel{1},\cancel{1},\cancel{1},\cancel{1},\cancel{1},\boxed{1},\cancel{1},\cancel{1},\cancel{1},\cancel{1},\cancel{2},\cancel{2},\cancel{2},\cancel{2},\cancel{3},\cancel{3},\cancel{4},\cancel{4},\cancel{5},\cancel{5},\cancel{7}$

The median is 1. 

Once students get good at this, they can skip writing everything out.  There are 31 datapoints.  They just need to find the 16th datapoint to find the middle datapoint.  But, before they can make that leap they need to understand that the numbers in the frequency column represent the number of times that datapoint shows up in the data. 

This information is also helpful for finding the mean.  When finding the mean, students' first instinct is often to add the numbers in the frequency column and then divide by 9 (the number of rows in the table).  In fact, they need to add up all of those datapoints to find the sum and then divide by 31 (the total number of students in the class -- which you can find by adding the frequencies).  As students get more savvy, they learn that they do not have to write out all of the datapoints and add.  Instead, they can multiply the frequencies by the "Number of Pets" they represent, then divide by the total number of frequencies. 

So the mean (or average) is 1.65.   Again, not hard, as long as you understand the table.  But, for many students, it takes a few tries to wrap their heads around the table. 

What is the mode?  The mode is the most common answer.  Here again, students sort of like to choose 2.  Why?  Because 2 is the only number that appears multiple times in the "Frequency" column!  But, if you ask students the most common number of pets, they can easily tell you that it's 1.  That's the mode: the most frequent datapoint (when you write out all of the datapoints), which, in normal English is just the most common answer. Sometimes thinking in math messes the kids up. 

What is the range?  The maximum value (7) minus the minimum value (0). $7-0=7$  The range is $7$.

So, which of those answers is true?

1. 1. The median number of pets is greater than the mean number of pets. No.
   2. The mean number of pets is greater than the median number of pets. Yes. 
   3. The mode of the number of pets is greater than the median number of pets. No (they are equal!)
   4. The range of the number of pets is less than the mean number of pets. No. 

If you said B from the start, you probably totally understand frequency tables.  If not, review.  It's all there. It's just a matter of getting used to data presented this way.