## Learning Math: Data Analysis, Statistics, and Probability

# Data Organization and Representation Part C: Frequency Tables: Making a Table (40 minutes)

**In This Par****t: Making a Table **

As you saw in Part A, a line plot is a graphical representation of data. For the raisin-count data, it showed how many times each raisin count occurred among the 17 boxes of Brand X raisins. You can also describe the same data using a frequency table, which shows the number of times each value occurs. The frequency table contains the same information as the line plot, but in tabular rather than graphical form. See Note 6 below.

**Problem C1**

Use the line plot to complete the frequency table for the Brand X raisin counts. The first column lists each of the values that occurred in the raisin counts. The corresponding cell in the second column indicates the frequency — the number of times that that value occurred. For instance, only one box contained 25 raisins, so the frequency of 25 is 1.

**Problem C2**

Use the frequency table to answer the following questions:

a. What is the minimum (smallest) raisin count for a box of Brand X raisins?

b. What is the maximum (largest) raisin count for a box?

c. How many boxes have between 26 and 28 raisins, inclusively (i.e., including 26 and 28)?

d. How many boxes have between 25 and 31 raisins, inclusively (i.e., including 25 and 31)?

e. Which raisin count occurred most frequently?

f. How many boxes contain more than 29 raisins?

g. How many boxes contain 29 or fewer raisins?

h. How many boxes contain fewer than 26 raisins?

i. How many boxes contain 25 or fewer raisins?

j. How many boxes contain between 26 and 29 raisins, inclusively?

**Problem C3**

Which questions in Problem C2 were easier to answer with a frequency table than with a line plot? Which were harder?

**In This Part: Cumulative Frequencies**

Let’s look at another useful way to describe variation in numerical data: A cumulative frequency specifies how many data values are of a particular number or smaller. See Note 7 below.

To obtain cumulative frequencies, it is first useful to obtain an ordered list of the data. Let’s do this now with our original Brand X raisin data.

You may recall that the original listing of the raisin counts was not in order:

We can obtain an ordered list from the line plot we created:

- The smallest raisin count is 25. Therefore, the ordered list begins with 25. As there is only one box of count 25, we look to the next count to find the next number in the sequence
- The next-smallest raisin count is 26. There are two boxes of size 26. The ordered list is now 25, 26, 26.
- The next-smallest raisin count is 27. There are three boxes of count 27. The ordered list is now 25, 26, 26, 27, 27, 27.

This table shows the final ordered list of Brand X raisin counts:

In this problem, the cumulative frequency specifies how many boxes have raisin counts *of a particular count or smaller.* Reading this table in terms of cumulative frequency tells us, for example, that there are 11 values that are 28 or smaller and 15 values that are 29 or smaller.

A cumulative frequency table lists the cumulative frequency for each value in the data set. To construct a cumulative frequency table, start with the smallest raisin count in the data. According to the ordered list, there is only one box with 25 raisins or fewer, so we record this in the cumulative frequency column. Moving on to the next count in the ordered list, we see that there are three boxes with 26 or fewer raisins (see chart below).

**Problem C4**

Use the ordered list of raisin counts given earlier to complete the cumulative frequency table.

**Problem C5**

Use the cumulative frequency table to answer the following questions:

a. What is the minimum (smallest) raisin count for a box of Brand X raisins?

b. What is the maximum (largest) raisin count for a box?

c. How many boxes have between 26 and 28 raisins, inclusively (i.e., including 26 and 28)?

d. How many boxes have between 25 and 31 raisins, inclusively (i.e., including 25 and 31)?

e. Which raisin count occurred most frequently?

f. How many boxes contain more than 29 raisins?

g. How many boxes contain 29 or fewer raisins?

h. How many boxes contain fewer than 26 raisins?

i. How many boxes contain 25 or fewer raisins?

j. How many boxes contain between 26 and 29 raisins, inclusively?

**Problem C6**

Which of the questions in Problem C5 were easier to answer with a cumulative frequency table? Which were more difficult?

The cumulative frequency table becomes more important in data sets with a wide spread of values. For example, it may not be that useful to know that 1.7% of students scored exactly 510 on a standardized test, but it is much more useful to know that 53.6% of students scored no higher than 510 on the same test. In this way, a cumulative frequency table can be used to calculate percentiles.

**In This Part:**** Another Method**

Here is another way that you could use either the line plot or the frequency table to obtain the cumulative frequencies. Look at the number of boxes with 25 or fewer raisins, which are highlighted in red on both the line plot and the frequency table:

There is one box with 25 or fewer raisins.

Now look at the number of boxes with 26 or fewer raisins, highlighted in green on the line plot and the frequency table:

There are three boxes with 26 or fewer raisins.

**Problem C7**

You can add a third column to the table to indicate cumulative frequencies. Use the line plot or the frequency table to complete the cumulative frequency values.

A quicker way to find the cumulative frequency is to add the previous cumulative frequency to the frequency of the new value. So for 28 or fewer, add the previous cumulative frequency (6) to the frequency of the new value (5) to get the new cumulative frequency (11).

**In This Part: Intervals and Ranges**

We use cumulative frequencies to describe intervals and ranges of data. For example, consider the boxes with between 27 and 29 raisins, inclusively, which are represented in blue on the line plot:

The number of boxes with between 27 and 29 raisins (12) is easy to determine from this line plot, but in problems with very large data sets, this might not be the case.

Here’s how a cumulative frequency table can be used to answer the question of how many boxes have between 27 and 29 raisins, inclusively. First, look at the number of boxes with counts of 29 or smaller. There are 15 of these, represented in red on the frequency table and line plot:

Remove the boxes that have fewer than 27 raisins. There are three of these, highlighted in green on the frequency table and line plot:

The number of remaining boxes is:

15 – 3 = 12

Therefore, there are 12 boxes that contain between 27 and 29 raisins.

**Problem C8**

Use the method described above to find the following:

**a.** The number of boxes that contain between 26 and 30 raisins, inclusively

**b.** The number of boxes that contain between 27 and 31 raisins, inclusively

**c.** The number of boxes that contain more than 28 raisins

### Notes

Note 6

The frequency table that corresponds to the line plot contains the same information, but it’s in another form (or representation). Part of the lesson here is to understand that there can be more than one way to represent data and to see the connection between two representations. You will be asked to consider the same 10 questions as before. Though you already know the answers, by solving the problems again, you will learn how to use the table, and you can compare the experience of using the table with that of using the line plot.

Keep in mind that different people see ideas in different ways. Some prefer graphical representations, and some prefer tabular representations. Luckily, in statistics we use both.

Note 7

The frequency table gave us the number of times a specific value occurred. We have also seen how an interval is used to describe how many raisins are in a box; for example, most of the boxes (14/17) contain between 26 and 29 raisins. We were able to determine how many of the counts fall in the interval 26-29 by adding the individual frequencies in this range.

The cumulative frequency function simplifies this process and gives us a more convenient device for obtaining frequencies for an interval of data; the cumulative frequency function has already done the adding! Keep in mind that with large data sets this would be an even greater advantage. If you have the cumulative frequencies, then the computation of the frequency within an interval is simply the difference between two numbers. This idea may not be obvious at first, but you’ll see as you get to play with it a bit in this part of the session.

If you are working with actual raisins, make frequency and cumulative frequency tables with your own data.

### Solutions

**Problem C1**

**Problem C2**

a. The minimum raisin count is 25 raisins.

b. The maximum raisin count is 31 raisins.

c. Adding the frequencies results in a total of 10 boxes containing between 26 and 28 raisins.

d. All 17 boxes do.

e. The count of 28 raisins has the highest frequency, which is five.

f. Two boxes contain more than 29 raisins: one has 30 and one has 31.

g. Fifteen boxes contain 29 or fewer raisins.

h. One box contains fewer than 26 raisins.

i. One box contains 25 or fewer raisins.

j. Fourteen boxes contain between 26 and 29 raisins.

**Problem C3**

Answers vary. Questions about individual raisin counts tend to be easier to answer with a frequency table (as there is no counting required), but questions about ranges of values are often easier with a line plot.

**Problem C4
**

**Problem C5**

The answers are identical to those in Problem C2.

a. The minimum raisin count is 25 raisins.

b. The maximum raisin count is 31 raisins.

c. Adding the frequencies results in a total of 10 boxes containing between 26 and 28 raisins.

d. All 17 boxes do.

e. The count of 28 raisins has the highest frequency, which is five.

f. Two boxes contain more than 29 raisins: one has 30 and one has 31.

g. Fifteen boxes contain 29 or fewer raisins.

h. One box contains fewer than 26 raisins.

i. One box contains 25 or fewer raisins.

j. Fourteen boxes contain between 26 and 29 raisins.

**Problem C6
**Questions about ranges may be easier with the cumulative frequency table. C5 (d) and (g) in particular are easier, since their answers can be read directly from the table. Questions about individual frequencies are more difficult because they require subtraction to go from the cumulative frequency table to the frequency table.

**Problem C7**

**Problem C8
**a. The cumulative frequency for 30 raisins is 16, and the cumulative frequency for 25 raisins is 1, so the number of boxes containing between 26 and 30 raisins is 15. Note that we use the frequency for 25 raisins as the lower boundary if we want to include 26 in the count.

b. The cumulative frequency for 31 raisins is 17, and the cumulative frequency for 26 is 3, so the answer is 14 boxes.

c. The cumulative frequency for 28 raisins is 11, so all six other boxes (17-11) must have more than 28 raisins.