Charts for Categorical and Discrete Data

Begin by identifying the type of data

A chart is a representation of data, so the type of data determines which representations make sense.

Categorical data sort observations into named groups. Examples include favourite fruit, eye colour and method of travelling to school. Even if a category is given a code such as 1, 2 or 3, those codes do not become measured numerical values.

Discrete numerical data consist of separate numerical values, usually obtained by counting. The number of books read might be 0, 1, 2 or 3, but not 1.6 books. Because these values have a numerical order and meaningful spacing, they should be placed on a numerical horizontal scale.

For categorical data, a frequency table, bar chart, pie chart or pictogram may be appropriate. For ungrouped discrete numerical data, a frequency table or vertical line chart can display the frequency at each exact value. “Ungrouped” means that the values have not been combined into intervals.

Constructing a frequency table

Suppose 12 pupils name their favourite lunchtime fruit. Their results can be organised as follows:

A tally is a counting aid; the frequency is the completed numerical count. Groups of five tally marks make larger counts easier to check. Each response must be placed in exactly one category, and the frequencies should add to the number of observations.

A frequency table can also be interpreted directly. Apple is the most frequent category, while pear is the least frequent. There are $6+3=9$ pupils who chose either apple or banana. The table is also the starting point for constructing the other charts.

Representing categories with a bar chart

In a bar chart, place the categories on one axis and frequency on the other. Draw bars of equal width, with a consistent frequency scale and a gap between neighbouring bars. The gaps emphasise that the categories are separate groups rather than adjoining numerical intervals.

The height of each bar gives its frequency. A suitable chart must therefore have:

• a descriptive title;
• labelled axes;
• a clear, evenly spaced frequency scale;
• bars whose heights accurately match the frequencies.

The order of categories may be arbitrary, alphabetical or otherwise logical. What matters is that every bar is correctly labelled. When reading a bar chart, check the scale before reading a height: one grid square does not necessarily represent one observation.

Showing proportions with a pie chart

A pie chart uses the full circle to represent the whole data set. Since a full turn is $360^\circ$, the angle for a category is

$$\text{sector angle}=\frac{\text{category frequency}}{\text{total frequency}}\times360^\circ.$$

For the fruit data, the apple sector has angle

$$\frac{6}{12}\times360^\circ=180^\circ.$$

The banana, orange and pear sectors have angles $90^\circ$, $60^\circ$ and $30^\circ$ respectively. These total $360^\circ$, providing a useful check. Construct the chart by marking the angles from the centre with a protractor, then label the sectors or provide an unambiguous key.

To work backwards from a sector, use

$$\text{frequency}=\frac{\text{sector angle}}{360^\circ}\times\text{total frequency}.$$

A pie chart mainly communicates proportion. A semicircle represents one half of the data, whatever the sample size. Therefore, without knowing the total frequency, an angle alone does not reveal an exact number of observations. Similarly, equally sized sectors in two pie charts can represent different frequencies if the totals are different.

Using a pictogram and its key

A pictogram replaces numbers with repeated symbols. Its key is part of the mathematics, not decoration. If one fruit symbol represents 2 pupils, apple requires 3 whole symbols, orange requires 1 whole symbol, and banana requires $1\frac12$ symbols.

A fraction of a symbol represents the same fraction of the key value. With a key of one symbol representing 2 pupils, half a symbol represents 1 pupil. Symbols must be identical in size and equally spaced; making a symbol larger would exaggerate the frequency.

When constructing a pictogram, choose a key that keeps the diagram readable. When interpreting one, calculate the value represented by each whole or partial symbol before finding the total.

Constructing a vertical line chart

Now consider discrete numerical data about the number of books read by 12 pupils:

On a vertical line chart, the horizontal axis is a numerical scale showing the number of books. At each value, draw a thin vertical line up to its frequency: the line at 0 reaches 2, the line at 1 reaches 5, and so on. A cross or point may be marked at the top of each line.

Unlike the categories of a bar chart, the horizontal positions here have numerical meaning: 2 lies twice as far from 0 as 1 does. However, the separate vertical lines and the gaps between them show that only the displayed discrete values are being counted. Do not join their tops, as that could suggest meaningful intermediate values.

From the finished chart, you can read that 1 book is the most frequent result, $2+5+4+1=12$ pupils were surveyed, and $4+1=5$ pupils read at least 2 books.

Choosing and interpreting a representation

Frequency tables preserve the exact counts and make totals easy to check. Bar charts make category frequencies easy to compare. Pie charts emphasise each category’s share of the whole. Pictograms can make small, simple data sets accessible, although their keys and partial symbols need careful reading. Vertical line charts show both the order of discrete numerical values and their frequencies.

Interpretation should connect the display to its context. State what the numbers mean, compare relevant categories or values, and support every conclusion with a frequency, angle, fraction or proportion from the representation.

Choose the representation

• Categorical data: frequency table, bar chart, pie chart or pictogram.
• Ungrouped discrete numerical data: frequency table or vertical line chart.

Construction checks

• Frequency table: categories do not overlap; frequencies give the correct total.
• Bar chart: labelled axes, even frequency scale, equal-width bars and gaps.
• Pie chart: labelled sectors and angles totalling $360^\circ$.
• Pictogram: state and apply the key, including for partial symbols.
• Vertical line chart: numerical horizontal scale; one separate vertical line at each discrete value.

Pie-chart equations

$$\text{sector angle}=\frac{\text{frequency}}{\text{total}}\times360^\circ$$ $$\text{frequency}=\frac{\text{sector angle}}{360^\circ}\times\text{total}$$

Interpret accurately

Read the scale or key first. Use frequencies, proportions or angles to support comparisons, and remember that a pie sector shows a proportion rather than an exact frequency unless the total is known.