## What you need to know

In short, a frequency table is a concise way of displaying a bunch data that’s been collected. The frequency column tells us how many times a particular value (or group of values) appeared when we were collecting data. Cumulative frequency is the number of times that anything up to and including (or if you prefer, less than or equal to) that value (or group of values) appeared. So, a frequency table might tell you the number of people who had size 6 feet in the survey, whereas a cumulative frequency table would tell you the number of people whose feet were any size up to and including size 6. To work out this value, all we have to do is add up all the values before it in the table.

Example: Below is a frequency table of data compiled on a group of college students’ heights. Construct a cumulative frequency table for this data.

The first cumulative frequency value is easy, since the number of people who were up to 160cm tall was just 13. Then, for the next cumulative frequency value, we need to include both the first and second category, since all people in either of those categories were all less than or equal to 170 cm. So, we get

$13 + 33 = 46$

people were 170cm tall or shorter. Now we just have to keep adding on the next frequency to the current running total (46) to get the next cumulative frequency.

Continuing this, we get that

$46 + 35 = 81$

people were 180cm of shorter, and

$80 + 11 = 92$

people were 190cm or shorter.

Note that all of the lower bounds of the groups, highlighted in blue, are now all 150cm, since the cumulative frequency values include all the lower values too.

From a cumulative frequency table, you are expected to know how to plot a cumulative frequency diagram/graph. To do this, you need to remember two things:

– The points plotted on your graph should be plotted at the end of each class, i.e. the point which has cumulative frequency 13 should be plotted at 160 on the height axis, and so on.

– You should join up the plotted points with a smooth curve (technically called an ogive). It should end up looking like an elongated ‘S’ shape.

Example: From the table above, plot a cumulative frequency diagram.

Once you’ve plotted the points and joined them together, the result should look like the graph to the right.

As you can see, the points are plotted at the end of each class interval with an extra point at $(150, 0)$, given that we know there are no values smaller than that.

Additionally, the points have been joined with a curved line forming a nice-looking Ogive.

Now that we’ve constructed our cumulative frequency diagram, we can put it to good use: we can use to form a boxplot. If you don’t remember what they are, then click here (https://mathsmadeeasy.co.uk/gcse-maths-revision/box-plots-gcse-revision-and-worksheets/).

Example: From the cumulative frequency diagram above, construct a boxplot representing the data on students’ heights.

To draw a boxplot, we need 5 things: smallest value, largest value, lower quartile, upper quartile, and median. Because this diagram was based on grouped data, these will be estimates.

Firstly, we estimate the smallest and largest values to be the lower bound and upper bound of the whole data set. Here, that is 150 and 190.

There are 92 people in total, so the lower quartile, median, and upper quartile will be the 23rd person, 46th person, and 69th person. So, we find these points on the y-axis, and then draw a line across to graph to find the corresponding heights on the x-axis.

Here, we get

$Q_1 = 163,\hspace{5mm} \text{ median } = 170,\hspace{5mm} Q_3 = 176$.

Now we have all the information we need, and the resulting boxplot looks like this:

## Example Questions

#### 1) Below is a frequency table of data on people’s time spent on a particular website in one day. Complete the cumulative frequency column in the table below and use it to plot a cumulative frequency diagram.

To find the cumulative frequency, we add up the frequencies as we move downwards in the table. The result should look like

Then, we need to plot each of the cumulative frequency at the point corresponding to the end of their class intervals (so plot (20, 16), (30, 40) and so on), and connect them with a smooth curve. The outcome is

#### 2) Shown below is a cumulative frequency diagram of data collected on people’s weekly hours spent exercising. Use it to construct a boxplot.

Firstly, we can see that the minimum number of hours spent exercising was zero and the maximum was 12.

Then, reading from the diagram, there are 72 people in total in this data set. Therefore,

$Q_1 \text{ is the } \dfrac{1}{4} \times 72 = 18\text{th term}$

$Q_2 \text{ is the } \dfrac{1}{2} \times 72 = 36\text{th term}$

$Q_3 \text{ is the } \dfrac{3}{4} \times 72 = 54\text{th term}$

So, now we know where the two quartiles and the median are, we can draw across from these values on the y-axis to find the corresponding numbers of hours on the x-axis. This looks like

Considering that since each big square is worth 2, each small square must be worth 0.4. Reading off the graph, we get that the lower quartile $Q_1 = 1.6$ hours, the median $Q_2 = 3.2$ hours, and the upper quartile $Q_3 = 5$ hours.

Using this information, the resulting boxplot is

## Cumulative Frequency Revision and Worksheets

Cumulative Frequency
Level 6-7
Cumulative Frequency 2
Level 6-7

Good GCSE Maths revision materials can be difficult to find. At Maths Made Easy we have collected some of he best Cumulative Frequency Curve questions and resources to make revision easier. Our Cumulative Frequency Cure worksheets and resources can also be used by teachers and tutors. You may want to set one of the worksheets for homework or use some of the questions as a starter. For more fantastic GCSE Maths revision resources go to the Maths Made Easy homepage.