# Lesson Video: Cumulative Frequency Graphs Mathematics

In this video, we will learn how to draw a cumulative frequency diagram and how to use it to make estimations about the data.

15:09

### Video Transcript

In this video, we will learn how to draw a cumulative frequency diagram and how to use it to make estimations about the data. Let’s begin by understanding what cumulative frequency is.

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 frequencies. The ascending cumulative frequency of a value 𝑥 can be found by adding all the frequencies less than 𝑥. The use of cumulative frequencies is a statistical method that is typically applied to grouped frequency tables, where data is organized into smaller groups or classes. Let’s look at an example of how we find the cumulative frequency of a set of data that is given in a grouped frequency table.

The table shows the number of hours that 100 students spent revising for an exam. Determine the missing cumulative frequency results.

The grouped frequency table presents the data on the number of hours that students spent studying. The groups or classes have open intervals such that the first group, zero dash, represents values of zero hours or greater but less than two. This is because the next group begins with values greater than or equal to two. We do not have overlapping values in a grouped frequency table.

We are asked to complete the cumulative frequency table based on the frequencies. The cumulative frequency gives the running total of the frequencies. An ascending cumulative frequency will always represent the frequencies of values that are less than a particular value. The first group in the frequency table has a cumulative frequency of zero. This is because we can conclude from the frequency table that there were zero students revising for less than two hours. To find the second cumulative frequency value, we add the frequency of the second group to the previous cumulative frequency. There are 10 students who revised for less than four hours. Hence, the second cumulative frequency is 10 plus zero, which equals 10.

We now need to determine the cumulative frequency of students who revised for less than six hours. The class four dash in the grouped frequency table indicates that 19 students revised four hours or more and less than six hours. However, the 10 students in the previous group also revised for less than six hours. Hence, the cumulative frequency for less than six hours is equal to 19 plus 10, which equals 29. This third cumulative frequency was found by adding the frequency of the third class to the previous cumulative frequency.

We can then continue this process to find each of the cumulative frequency values. It is worth noting that the cumulative frequency of all values will be the same as the total frequency. This is useful for checking whether our values are correct. The total frequencies can be calculated as zero plus 10 plus 19 plus 37 plus 24 plus 10, which equals 100. Since the final cumulative frequency is also 100, then we have confirmed that the missing cumulative frequency values are zero, 10, 29, 66, 90, and 100.

We will now see the most common way in which cumulative frequency is presented, as a cumulative frequency graph. A cumulative frequency graph displays the cumulative frequency of a data set. This can be a cumulative frequency polygon, where straight lines join the points, or a cumulative frequency curve. The cumulative frequency for a value 𝑥 is the total number of data values that are less than 𝑥. Since cumulative frequency is a running total of values, the graph of the cumulative frequency will never descend. It may have horizontal portions where the cumulative frequency remains the same if the frequency of a group is zero.

We will now look at an example where we need to identify the correct representation of a data set as a cumulative frequency graph.

A manufacturer samples the mass, in grams, of 30 pencils from their production line. Their masses are recorded in the table. No pencil has a mass greater than 60 grams. Which cumulative frequency graph correctly shows this information? Is it graph (A), (B), (C), (D), or (E)?

In order to identify the correct graph, we need to calculate the cumulative frequencies for the values in the table. This will give us a running total for values that are less than a given point. The less than value that we will use will be the upper boundary of each class.

We begin by recognizing that the first group in the table represents masses that are 10 grams or greater but less than 20 grams. We can add the cumulative frequency row to our table, which will represent pencils with a mass of less than 20 grams, less than 30 grams, less than 40 grams, and so on. Since no pencil has a mass greater than 60 grams, the last element of the cumulative frequency row represents the number of pencils with masses less than 60 grams.

Recalling that cumulative frequency is the running total, we have values of three, nine, 20, 27, and 30. Note that we calculate these values by adding the frequency in that column to the previous cumulative frequency value. The final cumulative frequency will be the same as the total frequency. In this case, this will be the value 30, since 30 pencils were sampled.

When drawing or identifying the cumulative frequency graph in this context, we have the mass on the 𝑥-axis and cumulative frequency on the 𝑦-axis. The 𝑥-coordinate values will be the less than mass values or the upper boundaries of each class. This allows us to use a cumulative frequency curve to identify values that are less than any particular value. The coordinates that would be plotted can be given as 10, zero; 20, three; 30, nine; 40, 20; 50, 27; and 60, 30. As previously stated, the 𝑥-coordinate is the upper boundary of each group and the 𝑦-coordinate is the corresponding cumulative frequency. The graph that matches these coordinates is that of graph (B). And so this is the cumulative frequency graph for the given information.

Whilst graphs (A), (D), and (E) are cumulative frequency graphs, they do not match the data in the table. Graph (C) is a frequency polygon and is not a cumulative frequency graph. When creating a cumulative frequency diagram, it is preferable to join the points with a smooth curve, rather than with straight lines. This gives us a better approximation for the data and allows us to make more accurate estimations for cumulative frequencies that do not lie on boundaries of classes.

We will now see an example of this, where we are given a cumulative frequency graph and we use it to help us estimate values that are less than, greater than, or equal to particular values.

Mason took a sample of 100 balls from a box. He weighed each ball and recorded its weight in the table. He used the data to draw the cumulative frequency graph shown on the grid. Estimate how many balls had a weight of less than 80 grams. Estimate how many balls had a weight of 130 grams or more.

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 frequencies. The given graph shows the cumulative frequency of the weights of 100 balls. We can see from the graph that the highest cumulative frequency is 100. Any point on the cumulative frequency graph indicates the total number of balls that are less than the given weight.

In order to find an estimate for the number of balls that are less than 80 grams, we can draw a vertical line from 80 on the 𝑥-axis until it meets the curve. We then draw a horizontal line from this point to the 𝑦-axis to allow us to read the corresponding 𝑦-value, the cumulative frequency. Observing that each minor grid line on the 𝑦-axis represents a frequency of two, we can give the answer to the first part of this question. The number of balls less than 80 grams can be estimated as 26 balls.

Although each value on the cumulative frequency curve represents frequencies that are less than a particular value, we can still use the curve to find the values for greater than or equal to values. To estimate the number of balls that are 130 grams or more, we use the same process. We draw a vertical line from 130 on the 𝑥-axis to the curve and then draw a horizontal line from this point to the 𝑦-axis. We can read the cumulative frequency of 78 balls from the 𝑦-axis, which means that 78 balls had a weight less than 130 grams.

In order to find the number of balls that had a weight of 130 grams or more, we subtract this from the total frequency. The total frequency is the total number of balls that have been weighed. Hence, it is 100. Therefore, we have 100 minus 78, which is equal to 22. The answer to the second part of the question is that we estimate that there are 22 balls with a weight of 130 grams or more.

In a grouped frequency table, the groups or classes may be described using different notation. We have seen how a class of 110 dash represents values that are 110 or greater and less than the lower boundary of the subsequent class. We can also use inequalities to represent the boundaries in continuous data sets. For example, data representing heights ℎ may be allocated different intervals written as ℎ is greater than or equal to 110 and less than 120, as shown. Whilst we have not seen an example of this type in this video, we would follow the exact same process when calculating cumulative frequency totals, drawing cumulative frequency graphs, and making estimations about the data.

We will now summarize the key points from this video. 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 frequencies. To draw a cumulative frequency graph, we first determine all the cumulative frequency totals for values that are less than the upper boundary of each class. To plot the coordinates for each cumulative frequency value, we take the upper boundary of a class as the 𝑥-coordinate and the corresponding cumulative frequency as the 𝑦-coordinate. Any point on a cumulative frequency curve represents the cumulative frequency of variables that are less than the corresponding 𝑥-coordinate. To find the frequency of values that are greater than or equal to any 𝑥-coordinate, we subtract the value of the 𝑦-coordinate from the total frequency.