Video Transcript
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.
