Frequency Graph

Here we will learn about frequency graphs, including what they are and how to draw them.

There are also frequency graph worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is a frequency graph?

A frequency graph is a way of representing a set of data (a frequency distribution).

To represent data using a frequency graph we need to know which type of graph would be preferable for the data we are representing (and be able to justify why this type of frequency graph has been chosen), and how to draw each type of frequency graph.

There are two types of data that can be represented using a frequency graph.

• Categorical data – data which is in the form of words rather than numbers. For example, colours, makes of cars, or types of music.
• Numerical data – data which is in the form of numbers. There are two types of numerical data.
  • Discrete – a count that involves integers (such as frequency)
  • Continuous – an uncountable number of values within a range (such as height)

What is a frequency graph?

How to use a frequency graph

We can use different types of frequency graphs to display data in a variety of ways.

● Pie charts

Pie charts can be used for categorical data. Each sector of the pie chart (or “slice”) is proportional to the frequency of the item in the data set.

To calculate the angle for each sector, we use the formula

where T is the total frequency and F is the category frequency.

Below is an example of drawing a pie chart to represent the favourite colours of a sample of 2 year old children.

Step-by-step guide: Pie chart

● Bar charts (and vertical line graphs)

Bar charts can also be used for categorical data. The horizontal axis ( x -axis) is for the data values. The vertical axis ( y -axis) is labelled frequency.

To draw a bar chart, we need to know the frequency for each category. Below is an example of a standard bar chart.

There are different types of bar charts including comparative and compound bar charts.

Step-by-step guide: Bar chart

Bar charts and vertical line graphs can be used for discrete numerical data.

For example,

Here is an example of a frequency diagram for grouped numerical data. This data is continuous and so there are no gaps between the bars.

Step-by-step guide: Frequency diagram

● Frequency polygons / line graphs

Grouped data can also be displayed in a frequency polygon. This is a type of line graph. The frequencies can be joined together using straight line segments because the horizontal axis is a continuous scale.

Here we can see the use of a break on the vertical axis as there are no data values between 0 and 14. We cannot use a break when drawing a bar chart.

Step-by-step guide: Frequency polygons

● Cumulative frequency graphs

A frequency graph represents individual frequencies of each category; a cumulative frequency shows the frequencies of each category accumulated together. This allows us to analyse the distribution of the data in more detail than if we used a frequency polygon and calculate statistics.

Below is an example of a cumulative frequency graph along with the data set.

The horizontal axis of a cumulative frequency graph is a continuous scale and the vertical axis represents the cumulative frequency. To calculate a cumulative frequency, the frequency of values in the next category is added to the frequency of all the previous values in the data set. This means that the data must be in order.

We can order data that is in numerical groups (see table 1 below).

We cannot add the frequencies of discrete categories such as favourite pet because there is no order to which pet is named first, or second, etc (see table 2 ).

A cumulative frequency graph can help us calculate statistics such as the median, the upper and lower quartiles, the interquartile range (IQR) and percentiles.

● Histograms

A histogram is different to a bar chart in multiple ways. The horizontal ( x -axis) is a continuous scale with no gaps between intervals and the vertical ( y -axis) is labelled frequency density instead of frequency.

For any histogram, the area of the bar represents the frequency of the class; not the height of the bar. The larger the area of the bar, the higher the frequency within that interval.

Step-by-step guide: Histogram

Related lessons on representing data

Frequency graph is part of our series of lessons to support revision on representing data. You may find it helpful to start with the main representing data lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Representing data
• Line graph
• Scatter graphs
• Two way tables
• x and y axis
• Time series graph
• Tally chart
• Stem and leaf diagram
• Pictograph

Frequency graph examples

Example 1: constructing a pie chart

The table below shows the frequency of purchases from an online football kit store.

Draw a pie chart for this set of data.

1. Calculate the angle for each category.

To determine the angle for each sector of the pie chart, we can use the formula

or A=\frac{360}{T}\times{F}

where

• A is the angle of the sector,
• T is the total frequency,
• and F is the category frequency.

2Draw a circle, mark the centre and draw a radius.

3Measure and draw the angle for the first category.

The first category has an angle of 135^{\circ}. Drawing this angle from the radius in a clockwise direction, we have

We can label this category ‘Red and navy’.

4Measure and draw the angle for each further category, in order.

The next sector has an angle of 60^{\circ}. Using the previous line drawn from the sector before and measuring 60^{\circ} in a clockwise direction, we get

Labelling this sector, we have

Continuing the pie chart by drawing the next sector line which is 75^{\circ} clockwise around the centre of the circle, we have

We can then fill in the information for the final sector as the remaining angle should be exactly the same value as in the table.

5Add data labels / complete a key.

Adding the data label to the final category completes the pie chart.

Example 2: constructing a bar chart

A farmer was keeping track of the animals on the farm. Draw a bar chart for this set of data.

Draw the axes with a ruler and label them.

Draw a pair of axes and label them with frequency on the vertical axis ( y -axis) and animals on the horizontal axis ( x -axis).

Use a ruler to draw each bar with the correct height. 

Draw the heights of the bars depending on its frequency. Draw each bar the same width. There should be gaps between the bars as the data is discrete.

Give the chart a title.

The last step is to add a title to the bar chart so that you can quickly see what the bar chart is representing. Here, ‘The frequency of animals on a farm’.

Example 3: constructing a vertical line graph

An estate agent was researching the number of bedrooms each house has in a small hamlet. Draw a vertical line graph for this set of data.

Draw the axes with a ruler and label them.

Draw a pair of axes and label them with frequency on the vertical axis ( y -axis) and bedrooms on the horizontal axis ( x -axis).

Use a ruler to draw each bar with the correct height.

Draw the heights of the vertical lines depending on its frequency.

Give the chart a title.

Example 4: constructing a frequency diagram for grouped data

21 students participated in a reaction time study where they were to catch a falling ruler. The measurements where they caught the ruler were recorded in the grouped frequency table below.

Construct a frequency diagram to represent the data in the table.

Draw the axes with a ruler and label them.

Draw a pair of axes and label them with ‘Frequency’ on the vertical axis ( y -axis) and ‘Measurement’ on the horizontal axis ( x -axis).

Use a ruler to draw each bar with the correct height. 

Draw the heights of the bars depending on its frequency. As the scale is continuous for the scores, we do not need gaps between the bars.

The horizontal axis has measurements from 0 to 80cm, and the highest frequency is 8 and so the vertical axis should be labelled from 0 to at least 8. Here, we have drawn the vertical axis to 9 so that we can clearly see the top of the highest bar.

Give the chart a title.

Example 5: constructing a frequency polygon

The test scores for a sample of people were recorded in a grouped frequency table.

Construct a frequency polygon to show this data.

Label the axes and add an axis title.

Draw a pair of axes and label them with ‘Frequency’ on the vertical axis ( y -axis) and ‘Score, x ’ on the horizontal axis ( x -axis).

The range of values for the frequency is 0 to 7 minimum. Increase in steps of 1.

The range of values for scores is 0 to 40. Increase in steps of 5.

Plot each data point accurately.

We need to find the midpoints of the class intervals. These can be found by adding the lower class boundary to the upper class boundary and dividing by 2.

Connect each pair of consecutive points with a straight line.

The plotted points are then joined by straight line segments.

Example 6: constructing a cumulative frequency graph

The number of hours students play games per week was recorded into the grouped frequency table below.

Draw a cumulative frequency graph to represent this information.

Calculate the cumulative frequency values for the data set.

To calculate the cumulative frequency of the data, we add the current frequency value to all of the frequency values before the class interval.

Draw a set of axes with suitable labels.

The greatest number of hours is 10 and so we need a horizontal axis labelled in equal steps from 0 to at least 10.

The total frequency of the data is 70 and so we need to start the vertical axis at 0 (no break allowed) and increase in equal steps up to 70.

Plot each value at the end of the interval.

Remember: we can only know the location of all the data at the end of the class interval. This is why the data in a cumulative frequency graph is an estimated distribution of the data set because we do not know the exact values of all 70 items.

Plotting the endpoint and the cumulative frequency for the class intervals, we have

Join the points with a smooth curved line.

The line must be a single smooth curve.

Add a title to the cumulative frequency graph.

Here, a suitable title would be,

Example 7: constructing a histogram

The word count for 30 books was recorded into a grouped frequency table. 

Draw a histogram to represent this information.

Calculate the frequency density for each class interval.

Using the frequency density formula D=\frac{F}{W}, we substitute the information from each row to calculate the frequency density. Remember to calculate the class width for each class as well.

Use the frequency density and class intervals to create suitable vertical and horizontal axes.

The maximum frequency density is 3.5 and the horizontal scale needs to go from 0 to 20.

Draw bars for each class interval using the frequency density as the height of the bar.

Drawing each bar one after the other with no gaps, we have the completed histogram.

Common misconceptions

• Gaps between bars of a bar chart

There should be gaps between the bars of a bar chart when given discrete data.

• Inconsistent labelling on each axis of a bar chart / vertical line graph

Each axis must be labelled in equal steps. This will help keep the bars/lines at an equal width, and the height of each bar/line remains consistent.

• Not labelling axis

The horizontal and vertical axes need to have data labels.

• Total of the angles is \bf{360^{\circ}} for a pie chart

The total of the angles in a pie chart is 360^{\circ} . This is a good check. However, if you are drawing a pie chart using real life data, the total might not be quite 360^{\circ} as we may need to round values, which loses the accuracy of the data.

• The angle of the sector is equal to \bf{360} degrees, divided by the frequency

The angle of each sector is proportional to the frequency of that category. This is misinterpreted to mean that you divide 360^{\circ} by the frequency of each group to get the angle. Instead, we need to find the fraction of the frequency out of the total frequency, of 360^{\circ}.

• The frequency is multiplied by the total frequency to get the angle of the sector

Similarly to the previous misconception, the angle of the sector is incorrectly calculated to be the frequency, multiplied by the total frequency. Instead, we need to find the fraction of the frequency out of the total frequency, of 360^{\circ}.

• Pie charts and doughnut charts (donut charts)

Pie charts and donut charts are very similar. A pie chart displaces individual categories as slices of the circle, whereas donut graphs have a hole in the centre and displays the categories as arcs.

• Frequency vs frequency density (histograms)

A very common error that occurs in histogram questions is that the frequency is used instead of the frequency density. Frequency density must be found because the groups provided in the frequency table are usually not equal width.

• The height of the bar is the frequency, instead of the area

Similar to the previous misconception, the height of the bar for a histogram is the frequency whereas it is the area of the bar that represents the frequency for a histogram. The height of the bar is the frequency for a bar chart.

The next lessons are

• Cumulative frequency
• Frequency table
• Mean, median, mode

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