Video Transcript
The following table shows the
heights of students in a high school. No student is taller than 175
centimeters. Which cumulative frequency graph
correctly represents this data? Options (A), (B), (C), (D), and
(E).
In this question, we are asked to
correctly identify the cumulative frequency graph of the data presented in a grouped
frequency table. To do this, we first recall that
the cumulative frequency is the running total of the frequencies. Since the cumulative frequency is a
running total of the frequencies, it will get larger as we add more frequencies,
this means that the cumulative frequency will never decrease.
In option (D), we see that the
graph curves downwards. This cannot happen in a cumulative
frequency graph. So, we can eliminate option
(D). We then recall that in this
cumulative frequency graph, the 𝑥-coordinates of the points will be the heights of
the students and the corresponding 𝑦-coordinates will be the cumulative frequency
up to the height. We see in the table that all
students are at least 150 centimeters in height. This means that the cumulative
frequency graph will start at the point 150, zero, since there are no students with
height less than this. We can use this to eliminate
options (A) and (E) since these graphs start at the point 150, four.
To determine the correct option
between the final two graphs, we can sketch the cumulative frequency graph. To do this, we need to find the
cumulative frequencies using the table. We need to find the sum of the
previous frequencies in the table. We start by noting the first column
has a cumulative frequency of four. We add this previous cumulative
frequency of four onto the next frequency of 22 to find that the next cumulative
frequency is 26. Adding the previous cumulative
frequency of 26 onto the next frequency of 56 gives us that the next cumulative
frequency is 82. Applying this process two more
times allows us to find the final two cumulative frequencies of 112 and 120.
To plot the cumulative frequency
graph, we note that the 𝑦-coordinates will be the upper bound of each class in the
grouped frequency table. We can add these upper bounds onto
the table where we use the fact that no student is taller than 175 centimeters to
bound the final group.
Let’s now clear some space so we
can sketch the cumulative frequency graph. Using the cumulative frequency
table, we see that we need to plot the points 150, zero; 155, four; 160, 26; 165,
82; 170, 112; and 175, 120. To sketch the graph, we start with
the axes. The 𝑥-axis is the heights, and we
can start this near 150 centimeters, since this is the lowest height in the data
set. The 𝑦-axis is the cumulative
frequency. Next, we plot the coordinates of
the six points we found from the cumulative frequency table as shown. Finally, since our options connect
the points with a smooth curve, we will also connect our plotted points with a
smooth curve. We see that our sketch most closely
resembles the graph in option (B). So, this is the cumulative
frequency graph of the data.
