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.
