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Lesson Explainer: Descending Cumulative Frequency Graphs Mathematics

In this explainer, we will learn how to draw a descending cumulative frequency diagram and use it to make estimations about the data.

Usually when we consider cumulative frequency, we mean an ascending cumulative frequency. This gives us a running total of the frequency, with the final frequency being the total frequency of the data set. We can recap this as follows.

Definition: Cumulative Frequency (Ascending)

Cumulative frequency is the sum of all the previous frequencies up to the current point. It is often referred to as the running total of the frequencies.

The ascending cumulative frequency of a value π‘₯ can be found by adding all the frequencies less than π‘₯.

A descending cumulative frequency is also useful for analyzing data. A descending cumulative frequency always has a starting value that is equal to the total frequency. A set of (ascending) cumulative frequency values is always increasing, whereas a set of descending cumulative frequency values is always decreasing. Each successive descending cumulative frequency is less than or equal to the previous value.

We can define descending cumulative frequency below.

Definition: Descending Cumulative Frequency

The descending cumulative frequency of a value π‘₯ indicates the frequency of values that are greater than or equal to π‘₯.

Calculating a descending cumulative frequency can feel less intuitive than calculating an ascending cumulative frequency. It can be helpful to think about a problem in its context. For example, if we are considering the grades that students achieved in an examination, where the greatest grade is 100, then all the students (the total frequency) would have achieved a grade of 0 or more. The first descending cumulative frequency would be equal to the total frequency, that is, the total number of students. If the following group achieved a grade of 10 or more, this group has a descending cumulative frequency of all the students who achieved a grade of 10 or more, excluding the students who achieved a grade of 0 up to less than a grade of 10.

Let’s see an example of how we calculate the descending cumulative frequency values for a data set.

Example 1: Completing a Descending Cumulative Frequency Table from a Grouped Frequency Table

As part of a childcare course, students recorded the age, in months, at which a group of babies and toddlers began to walk. The data is recorded in the grouped frequency table below.

Age (Months)8–10–12–14–16–18–Total
Frequency2154051357150

Complete the descending cumulative frequency table.

Age (Months)8 or more10 or more12 or more 14 or more16 or more18 or more20 or more
Descending Cumulative Frequency

Complete the descending cumulative frequency table.

Answer

Let’s recall that the descending cumulative frequency of a value π‘₯ indicates the frequency of values that are greater than or equal to π‘₯. In the grouped frequency table, we are given groupings such as 8–, 10–, along with their frequencies. For example, the class 8– indicates ages that are 8 months or greater, but less than 10 months (the lower boundary of the subsequent class). In the second table, the first row relates the group aged 8 months or more and a descending cumulative frequency.

We know from the total frequency that this study involves 150 babies and toddlers. From the table, we can observe that 2 babies started walking at 8 months or more, up to 10 months. But the babies that started walking at 10– months, 12– months, and so on up until 18– months also started walking at 8 months and more. Therefore, the first cumulative frequency will be the total frequency of 150. All 150 babies and toddlers recorded were walking at an age of 8 months and more. Hence, our first entry in the descending cumulative frequency column is 150.

To find the second descending cumulative frequency for the group β€œ10 and more,” we will now exclude the 2 babies from the first class who walked at an age less than 10 months. In other words, to find the second descending cumulative frequency, we subtract the first frequency from the previous descending cumulative frequency. This gives us 150βˆ’148=2.

To find the third descending cumulative frequency of the group β€œ12 and more,” we subtract both frequencies of the groups 8– and 10– months from the total frequency. Alternatively, we can consider this as subtracting the second frequency of 15 from the second descending cumulative frequency of 148.

To find the fourth descending cumulative frequency, we subtract the frequency of 40 from the third descending cumulative frequency. This gives 133βˆ’40=93.

We can then complete the remaining descending cumulative frequency values in the same way.

Notice that the last descending cumulative frequency value is 0. In the first table, the last class is that of 18–, meaning a time period of 18 months or more. We can assume that this group has the same class width as the previous group. Hence, this group is babies or toddlers who walked at less than 20 months. Therefore, the descending cumulative frequency value of 0 for the class β€œ20 or more” is correct. There were 0 toddlers recorded as having started walking at 20 months or more.

We can give the answer that the missing descending cumulative frequency values are 150,148,133,93,42,7,0.and

One of the features that we can observe from the descending cumulative frequency values in the previous example is that they are all decreasing. Successive descending cumulative frequency values are always less than or equal to the previous value. It is not possible for successive descending cumulative frequency to increase, as each value is found by subtracting a (positive) frequency from the previous descending cumulative frequency.

We will now see how we can create a descending cumulative frequency graph. To do this, we plot descending cumulative frequency on the 𝑦-axis and the variable under study on the π‘₯-axis. The π‘₯-coordinate for each descending cumulative frequency will be the lower boundary of the class. For example, using the data in the previous example, the first coordinate for the class β€œ8–” or β€œ8 or more” with a descending cumulative frequency of 150 would be (8,150). In this way, we accurately represent that the descending cumulative frequency of a value π‘₯ indicates the frequency of values that are greater than or equal to π‘₯.

Example 2: Identifying a Descending Cumulative Frequency Diagram for a Grouped Data Set

Consider the frequency distribution shown.

Grade (𝑔)0–10–20–30–40–
Number of Students810662

Which of the following is the descending cumulative frequency diagram that represents these data?

Answer

The descending cumulative frequency of a value π‘₯ indicates the frequency of values that are greater than or equal to π‘₯. In order to identify which descending cumulative frequency diagram represents the given data, we can first calculate these values.

We note that the first group in the frequency table is that of 0–, indicating grade values that are 0 or greater, up to a value of 10 (the lower boundary of the subsequent class). Therefore, the first descending cumulative frequency will be for grades that are 0 or greater.

The total frequency of this distribution can be calculated by adding all the frequencies, giving totalfrequency=8+10+6+6+2=32.

The first cumulative frequency is the same as the total frequency of 32, as all 32 students received a grade 0 or greater. We can add a row to the table to record our results.

Next, we consider how many students achieved a grade of 10 or more. This will be the total of the 32 students, excluding the 8 students from the first group (who did not achieve a grade of 10 or more). This gives us a second descending cumulative frequency of 32βˆ’8=24.

The third descending cumulative frequency representing the grades 20 or more can be found by subtracting the second frequency (10) from the second descending cumulative frequency value (24). We have 24βˆ’10=14. 14 students achieved a grade of 20 or more.

We can then complete the remaining descending cumulative frequencies of 8 and 2 in the same way.

We commonly finish a descending cumulative frequency with a value of 0. To do this, we consider the last class to have the same class width as the others and define an additional class in the distribution; here this would be 50–. A grade of 50– was assigned to 0 students. Hence, grades of 50 or more would also have a descending cumulative frequency of 0. We can add an additional column to the table to record this result.

To draw this descending cumulative frequency graph, we would plot the grade on the π‘₯-axis and the descending cumulative frequency on the 𝑦-axis. The coordinates of the points would be given as ().lowerboundaryofeachclass,descendingcumulativefrequency

Hence, the coordinates could be given as (0,32),(10,24),(20,14),(30,8),(40,2),(50,0).and

Inspecting the given answer options, we can observe that the graph given in option B is the correct descending cumulative frequency diagram. Although the graph in option E is very similar, it has an incorrect first coordinate of (0,30) rather than (0,32).

In the following example, we will draw the descending cumulative frequency graph. We plot the variable under study on the π‘₯-axis, and the descending cumulative frequency on the 𝑦-axis. Note, we must ensure that our 𝑦-axis extends to the highest descending cumulative frequency, that is, the total frequency, and not simply the highest frequency in the table. It is always good practice to determine the descending cumulative frequencies and the associated coordinates to plot, before drawing the axes of our graph. Coordinates should be joined with a smooth curve.

Example 3: Drawing a Descending Cumulative Frequency Graph for a Data Set

The times, in seconds, for adults to complete a puzzle were recorded. The results are given in the table below.

Time (s)0–30–60–90–120–
Frequency31735432

Draw a descending cumulative frequency diagram to represent this data.

Answer

The frequency table gives the times in groups. For example, 0– represents a time of 0 seconds or more, up to, but not including, 30 seconds (the lower boundary of the next class). The descending cumulative frequency for any value π‘₯ gives us the frequency of the values that are greater than or equal to π‘₯. In order to draw a descending cumulative frequency diagram, or curve, we first need to determine the descending cumulative frequency values. We can add a column to the table to help us do this.

The first descending cumulative frequency is always the total frequency. We can calculate this as totalfrequency=3+17+35+43+2=100.

In the context of the problem, this means that 100 adults took a time of 0 seconds or more to complete the puzzle.

The second descending cumulative frequency represents the number of adults who took 30 seconds or more to complete the puzzle. This is 100βˆ’3=97 since we do not count the 3 adults (in the first group) who took less than 30 seconds to complete it.

The third descending cumulative frequency is calculated by subtracting the second frequency from the second descending cumulative frequency value. This gives 97βˆ’17=80. 80 adults took 60 seconds or more to complete the puzzle.

We can then complete the rest of the descending cumulative frequencies in the same way. It is common to include a final descending cumulative frequency of 0. To do this, we recognize that there are 0 adults recorded as taking a time of 150 seconds or more. When we add this to the table, we have a final descending cumulative frequency of 0.

We now need to plot these values as coordinates on a grid. The π‘₯-axis will represent the time, in seconds, and the 𝑦-axis will represent the descending cumulative frequency. Each set of coordinates will have the lower boundary of each class as the π‘₯-coordinate and the corresponding descending cumulative frequency as the 𝑦-coordinate.

The correct descending cumulative frequency diagram is shown.

We will now see an example of how we can interpret a descending cumulative frequency graph.

Example 4: Interpreting a Descending Cumulative Frequency Graph

Consider the descending cumulative frequency graph shown, which represents the weights of 200 people.

How many people weigh more than 60 kg?

Answer

We recall that a descending cumulative frequency of a value π‘₯ indicates the frequency of values that are greater than or equal to π‘₯. Given the curve, we can use this to find the number of people who weigh more than 60 kg or more.

We draw a vertical line from 60 kg on the π‘₯-axis until it meets the curve, and then we draw a horizontal line from this point to the 𝑦-axis.

We can observe that the point on the 𝑦-axis is 150. Hence, we can give the answer that 150 people weigh more than 60 kg.

We know that the graph of a descending cumulative frequency for a value π‘₯ gives us the frequency of values that are greater than or equal to π‘₯. We have learned how we plot a descending cumulative frequency curve to show a distribution. However, we can also use a descending cumulative frequency diagram to give us information about a data set, not only for the values of coordinates that we plot, but also for other values that lie on the curve.

In the next example, we will see how we can combine the skills of drawing and interpreting a descending cumulative frequency graph.

Example 5: Drawing and Then Interpreting a Descending Cumulative Frequency Graph

The table shows the times of runners who completed a 5 km race.

Time (Minutes)15–17–19–21–23–
Number of Runners52118124

  1. Draw a descending cumulative frequency diagram to represent this data.
  2. How many runners had a race time of 21 minutes or more?

Answer

Part 1

We recall that a descending cumulative frequency of a value π‘₯ indicates the frequency of values, or number of values in a data set, that are greater than or equal to π‘₯. In order to draw a descending cumulative frequency diagram, we first need to establish the descending cumulative frequency values for each class.

Since descending cumulative frequency means values that are greater than or equal to a certain value, we take the lower boundary of each class and calculate the number of values that are greater than or equal to this value. The first descending cumulative frequency, for the first class, will be equal to the total frequency. In this example, this will be the total number of runners with race times recorded in the table. Hence, we have totalfrequency=5+21+18+12+4=60.

The first descending cumulative frequency is 60. It is helpful to add an additional row to the table to record these values.

The next descending cumulative frequency value indicates the number of runners who had a race time of 17 minutes or more. This can be found by subtracting the number of runners who had a race time of less than 17 minutes (the class before) from the previous descending cumulative frequency of 60. This gives us a second cumulative frequency of 60βˆ’5=55.

In this way, each descending cumulative frequency is found by subtracting the frequency of the class before from the previous descending cumulative frequency. We can fill in the remaining descending cumulative frequency values in the table as shown.

It is common to have a final descending cumulative frequency value of 0. To achieve this, we assume that the final class has the same class width as the others and create an additional class with a frequency of 0. Here, we have an additional class of 25– representing race times of 25 minutes or greater. The descending cumulative frequency value of 0 would represent that 0 runners had a race time of 25 minutes or more.

In order to represent this data graphically, we plot time (in minutes) on the π‘₯-axis and descending cumulative frequency on the 𝑦-axis. The π‘₯-coordinate of each point is the lower boundary of each group (the greater-than-or-equal-to values) and the 𝑦-coordinate is the corresponding descending cumulative frequency. The coordinates can be given as (15,60),(17,55),(19,34),(21,16),(23,4),(25,0).and

The correct graph is shown below.

Part 2

To find the number of runners with a race time of 21 minutes or more, we can use the graph. We draw a vertical line from 21 on the π‘₯-axis to the curve and then draw a horizontal line to the 𝑦-axis. We observe that this is at the descending cumulative frequency of 16.

As β€œ21 minutes or more” also appeared in our table, we could alternatively have used the list of descending cumulative frequencies to determine the value.

We can give the answer that 16 runners had a race time of 21 minutes or more.

Using our knowledge of how to interpret a descending cumulative frequency graph, we will now see how we can compare two data sets using their descending cumulative frequency graphs.

Example 6: Comparing Two Data Sets Using Descending Cumulative Frequency Graphs

Descending cumulative frequency diagrams have been drawn to represent the grades achieved in a test by students in two classes, class A and class B.

Determine which class has more students achieving a grade of 40 or higher.

Answer

The descending cumulative frequency of a value π‘₯ gives us the frequency, or number of values that are equal to or greater than π‘₯. Given the descending cumulative frequency graphs for classes A and B, we can use these to determine the number of students in each class who achieved a grade of 40 or more.

We draw a vertical line on each graph from 40 on the π‘₯-axis until it meets the curve and then a horizontal line from this point to the 𝑦-axis.

We observe that in the graph for class A, the line meets the 𝑦-axis at 17. In the graph for class B, the line meets the 𝑦-axis at 13. This means that class A has 17 students achieving a grade of 40 or more and that class B has 13 students achieving 40 or more.

As 17>14, we can give the answer that class A has more students achieving a grade of 40 or higher.

We now summarize the key points.

Key Points

  • The descending cumulative frequency of a value π‘₯ indicates the frequency of values that are greater than or equal to π‘₯.
  • The first descending cumulative frequency value of a data set is equal to the total frequency of the data set.
  • Subsequent descending cumulative frequency values can be calculated by subtracting the frequency of the previous class from the previous descending cumulative frequency.
  • It is common to end with a descending cumulative frequency of 0, particularly when creating a descending cumulative frequency diagram. To achieve this, we assume the final class has a class width equal to the others and create a new class that has a lower boundary equal to the upper boundary of the final class. This new class will have a frequency of 0.
  • The coordinates of a descending cumulative frequency diagram will have an π‘₯-coordinate that is the lower boundary of each class. The 𝑦-coordinate will be the corresponding descending cumulative frequency value.

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