 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Session 2, Part C:
Frequency Tables

In This Part: Making a Table | Cumulative Frequencies | Another Method
Intervals and Ranges

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.
Note 7

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:  Number of Raisins in 17 Half-Ounce Boxes   29 27 27 28 31 26 28 28 30 29 26 27 29 29 25 28 28  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:  Position Raisin Count  1 25 2 26 3 26 4 27 5 27 6 27 7 28 8 28 9 28 10 28 11 28 12 29 13 29 14 29 15 29 16 30 17 31 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.   Position Cumulative Frequency  25 1 26 3 27 28 29 30 31    Position Cumulative Frequency  25 1 26 3 27 6 28 11 29 15 30 16 31 17  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.   Session 2: Index | Notes | Solutions | Video