# Drawing Histograms with Unequal Class Widths

Higher GCSE maths students are required to plot and interpret histograms with unequal class widths.  Drawing histograms with unequal class widths are very common in GCSE maths papers.

## Histograms versus bar charts

Histograms look like bar charts but have important differences.

• Histograms use a continuous horizontal scale which means the bars touch so the difference between them is zero.
• The frequency of the data is measured by area not height.

## How to calculate the height of the bar (Frequency Density)

When grouping continuous data, it may be necessary to have different class widths if the data are not equally spread out.  When class widths are not equal frequency density becomes the vertical axes.

## Working with grouped data

I start the lesson by asking students to find a probability from a set of data in a grouped frequency table.  I use this example to help students recap inequality notation and to discuss whether this is an exact or estimated probability.  It is important to remind everyone that grouping helps to organise large samples of data but there is a trade off with accuracy when interpreting the results.

## Drawing histograms with unequal class widths

As we move on to the main part of the lesson I ask the class to think about how the table in the second slide is different to that in the starter.  Students quickly pick up the class widths are no longer equal.

## Representing data with unequal class widths

We discuss it may be necessary to use unequal class widths for data that are unevenly distributed and when we do so frequency is measured as the product of frequency density (vertical scale) and class width (horizontal scale).

When drawing histograms for Higher GCSE maths students are provided with the class widths as part of the question and asked to find the frequency density.

I work through the first example with the class plotting the histogram as we complete the table.

## Completing a table and histogram with unequal class intervals

In the plenary students are challenged to complete a table and histogram by working out the scale of the frequency density axis.  This task frequently appears in exam papers.  I provide the students with a print-out of this slide, so no time is wasted copying it into books.  Examiner reports state that those who understand frequency as measured by the area of a bar often go on to achieve full marks.

Here is an extract of an Edexcel examiner report for a similar question.

‘In general candidates appear not to be aware that the area of the bars of a histogram are the frequencies, evidenced by a lack of frequency density calculations.’

To ensure all students have enough time to complete the plenary I ask those who have finished first to additionally estimate the number of flowers that grew to between 5 and 8 cm tall.   The most able students calculate this as a compound area of the bars on the histogram.

## Interpreting histograms with unequal class widths

In the next lesson students practice finding the frequencies from histograms to calculate an estimate of the mean.  As learning progresses we move on to using interpolation to estimate the median average from a histogram.

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