Lesson Video: Descending Cumulative Frequency Graphs Mathematics

Video Transcript

In this video, 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. 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 𝑥 can be found by adding all the frequencies less than 𝑥.

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 as follows. The descending cumulative frequency of a value 𝑥 indicates the frequency of values that are greater than or equal to 𝑥.

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 zero 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 zero 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.

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. Complete the descending cumulative frequency table.

Let’s recall that the descending cumulative frequency of a value 𝑥 indicates that frequency of values that are greater than or equal to 𝑥. In the grouped frequency table, we are given groupings such as eight dash, 10 dash, along with their frequencies. For example, the class eight dash indicates ages that are eight months or greater but less than 10 months, the lower boundary of the subsequent class. In the second table, the first column relates the group aged eight 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 two babies started walking at eight months or more up to 10 months. But the babies that started walking at 10 dash months, 12 dash months, and so on up until 18 dash months also started walking at eight months or 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 eight months and more. Hence, our first entry in the descending cumulative frequency row is 150.

To find the second descending cumulative frequency for the group 10 or more, we will now exclude the two 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 150 minus two, which equals 148.

To find the third descending cumulative frequency of the group 12 or more, we subtract both frequencies of the groups eight dash and 10 dash 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. This gives us 133.

To find the fourth descending cumulative frequency, we subtract the frequency of 40 from the third descending cumulative frequency. This gives 133 minus 40, which is equal to 93.

We can then complete the remaining cumulative frequency values in the same way. Notice that the last descending cumulative frequency value is zero. In the first table, the last class is that of 18 dash, 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 less than 20 months. Therefore, the descending cumulative frequency value of zero for the class 20 or more is correct. There were zero 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, seven, and zero.

One of the features that we can observe from the descending cumulative frequency values 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 values to increase, as each value is found by subtracting a 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 𝑥-axis. For example, using the data in the previous example, the first coordinate for the class eight dash, or eight or more, with a descending cumulative frequency of 150 would be eight, 150. In this way, we accurately represent that the descending cumulative frequency of a value 𝑥 indicates the frequency of values that are greater than or equal to 𝑥.

Consider the frequency distribution shown. Which of the following is the descending cumulative frequency diagram that represents this data? Is it graph (A), (B), (C), (D), or (E)?

The descending cumulative frequency of a value 𝑥 indicates the frequency of values that are greater than or equal to 𝑥. 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 zero dash, indicating grade values that are zero 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 zero or greater. The total frequency for this distribution can be calculated by adding all the frequencies, giving eight plus 10 plus six plus six plus two equals 32. The first cumulative frequency is the same as the total frequency of 32, as all 32 students received a grade of zero or greater.

Next, we consider how many students achieved a grade of 10 or more. This will be the total of 32 students excluding the eight students from the first group who did not achieve a grade of 10 or more. This gives us a second descending cumulative frequency of 32 minus eight, which equals 24.

The third descending cumulative frequency representing the grade 20 or more can be found by subtracting the second frequency from the second descending cumulative frequency value. We have 24 minus 10, which equals 14. So 14 students achieved a grade of 20 or more.

We can then complete the remaining descending cumulative frequencies of eight and two in the same way. We commonly finish a descending cumulative frequency with a value of zero. 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 dash. A grade 50 dash was assigned to zero students. Hence, grades of 50 or more would also have a descending cumulative frequency of zero.

To draw this descending cumulative frequency graph, we would plot the grades on the 𝑥-axis and the descending cumulative frequency on the 𝑦-axis. The coordinates of the points would be given as lower boundary of each class, descending cumulative frequency. Hence, the coordinates could be given as zero, 32; 10, 24; 20, 14; 30, eight; 40, two; and 50, zero.

Inspecting the given answer options, we can immediately rule out option (A), as it is increasing, and option (C), as part of the curve is increasing. From the remaining answer options, we can observe that the graph given in option (B) is the correct descending cumulative frequency diagram as it matches the six coordinates. Although the graph in option (E) is very similar, it has an incorrect first coordinate of zero, 30 rather than zero, 32. Graph (D) has an incorrect coordinate at 40, six, which should be at 40, two.

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

Consider the descending cumulative frequency graph shown, which represents the weights of 200 people. How many of the people weigh more than 60 kilograms?

We recall that a descending cumulative frequency of a value 𝑥 indicates the frequency of values that are greater than or equal to 𝑥. Given the curve, we can use this to find the number of people who weigh more than 60 kilograms. We draw a vertical line from 60 kilograms on the 𝑥-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 value on the 𝑦-axis is 150. Hence, we can give the answer that 150 people weigh more than 60 kilograms.

We know that the graph of a descending cumulative frequency for a value 𝑥 gives us the frequency of values that are greater than or equal to 𝑥. We have learned how we can 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 the coordinates that we plot, but also for other values that lie on the curve, as shown. 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.

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.

The descending cumulative frequency of a value 𝑥 gives us the frequency, or number of values, that are equal to or greater than 𝑥. 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 𝑥-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 is greater than 13, we can give the answer that class A has more students achieving a grade of 40 or higher.

We will now summarize the key points from this video. The descending cumulative frequency of a value 𝑥 indicates the frequency of values that are greater than or equal to 𝑥. 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 zero, 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 zero. The coordinates of a descending cumulative frequency diagram will have an 𝑥-coordinate that is the lower boundary of each class. The 𝑦-coordinate will be the corresponding descending cumulative frequency value.