The following table shows the
heights of players in a basketball game. Note that no player is taller than
200 centimeters. Which of the following is the
descending cumulative frequency graph? Is it graph (A), (B), (C), (D), or
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 graph represents the given data, we can first
calculate these values. We note that the first group in the
frequency table is that of 140 dash, indicating a height of 140 centimeters or
greater, up to but not including 150 centimeters, since this is the lower boundary
of the next group. Therefore, the first descending
cumulative frequency will be for heights that are 140 centimeters or greater.
We see from the table that there
were a total of 30 players. And as all 30 have a height of 140
centimeters or greater, the first descending cumulative frequency is 30. Next, we consider how many players
had a height of 150 centimeters or greater. This will be the total of 30
players minus those players from the first group, those with a height greater than
or equal to 140 centimeters, but less than 150 centimeters. 30 minus four equals 26. So, our second descending
cumulative frequency is 26.
We then repeat this process by
subtracting six from 26. The third descending cumulative
frequency, which represents those players with a height greater than or equal to 160
centimeters, is therefore equal to 20. Continuing this process, we have
three further descending cumulative frequency values of 11, eight, and one.
At this stage, we note that we
typically 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. In this question, this would be 200
dash, which corresponds to those players with a height of 200 centimeters or
greater. This group has a descending
cumulative frequency value of zero.
To draw the descending cumulative
frequency graph, we plot the height on the 𝑥-axis and the descending cumulative
frequency on the 𝑦-axis. The coordinates are given by lower
boundary of each class, descending cumulative frequency. Hence, the first coordinate is 140,
30. Next, we have 150, 26, followed by
160, 20; 170, 11; 180, eight; 190, one; and 200, zero.
Inspecting the given answer
options, we can observe that the given graph in option (C) is the correct descending
cumulative frequency graph. Whilst options (B) and (E) are very
similar, we note that they have incorrect coordinates at 150, 30 and 200, one,
We can therefore conclude that the
correct answer is option (C).