Charts for Categorical Data

From raw responses to a frequency table

Categorical data sorts observations into named groups. Examples include favourite fruit, eye colour or method of travel. The categories do not measure an amount: even if they are given number codes, those numbers may only be labels.

A frequency table organises the data by listing every category and recording how many observations belong to it. When collecting data, a tally column is useful: marks are grouped in fives so that they can be counted reliably. The final frequency column contains the numerical totals.

For example, suppose a survey of $40$ people gives these frequencies:

Always check the frequencies against the number surveyed. Here, $14+10+9+7=40$, so every response has been included once.

Showing frequencies with a bar chart

A bar chart makes the frequencies of categories easy to compare. Put the categories on the horizontal axis and frequency on the vertical axis. Choose an even numerical scale that reaches at least the greatest frequency, then draw bars of equal width.

The bars must have equal gaps because the categories are separate rather than positions on a continuous measurement scale. Each bar's height represents its frequency, so a frequency of $14$ must reach $14$ on the vertical scale. Include a descriptive title and label both axes.

To interpret a bar chart, read each height against the scale rather than judging it by eye. The tallest bar identifies the mode. You can add all the bar heights to find the total frequency, or divide a category's frequency by the total to find its fraction of the data.

Turning frequencies into a pie chart

A pie chart shows how the whole data set is divided among categories. A complete circle is $360^\circ$, so a category's sector angle must be in the same proportion as its frequency:

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

For the fruit survey, one person corresponds to

$$360^\circ\div40=9^\circ.$$

The apple sector is therefore $14\times9^\circ=126^\circ$. In the same way, the other angles are $90^\circ$, $81^\circ$ and $63^\circ$. Their sum is $360^\circ$, providing a useful check. Construct the chart by drawing a radius, measuring each angle at the centre with a protractor, and labelling each sector or supplying a key.

The calculation can also be reversed when the total frequency is known:

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

For instance, a $90^\circ$ sector in a chart representing $40$ people contains $\frac{90}{360}\times40=10$ people.

Representing categories with a pictogram

A pictogram replaces numerical bars with repeated pictures or symbols. Its key fixes the value of one complete symbol. If one symbol represents $2$ people, seven symbols represent $14$ people. A half-symbol then represents $1$ person.

Choose a key that allows all the frequencies to be displayed clearly. Symbols must have the same size and be equally spaced; enlarging some symbols would make the data appear more different than it really is. A pictogram without a key cannot be interpreted reliably.

Choosing and interpreting the representation

A frequency table gives exact values in a compact form. A bar chart is particularly helpful when the aim is to compare category frequencies. A pie chart emphasises proportions of one whole, while a pictogram can communicate a small, simple data set visually.

When interpreting any representation, identify what is being counted, read the scale or key, and support conclusions with values. Instead of saying “apple is most popular”, say “apple is the mode, with $14$ responses”.

Pie charts require extra care when different-sized groups are compared. Equal sector angles mean equal proportions, not necessarily equal frequencies. A $90^\circ$ sector is one quarter of its sample: it represents $10$ people in a sample of $40$, but $100$ people in a sample of $400$. Convert each angle to a frequency if the question asks which group contains more people.

You should also check whether a display is fair. Unequal bar widths, uneven numerical scales, missing labels or pictogram symbols of different sizes can distort the impression given by the data.

Frequency tables

List each category once and record how often it occurs. Check that

$$\text{total frequency}=\text{number of observations}.$$

The category with the greatest frequency is the mode.

Bar charts

• Categories on the horizontal axis; frequency on the vertical axis.
• Use an even scale, equal-width bars and equal gaps.
• Add a title and axis labels.
• Read heights from the scale, not by visual guesswork.

Pie charts

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

Sector angles must total $360^\circ$. Sector angles compare proportions; use sample totals to compare actual frequencies across different pie charts.

Pictograms

Use the key to convert symbols into frequencies. A fractional symbol represents the same fraction of the key's value.

Choose appropriately

• Frequency table: exact organised values.
• Bar chart: clear comparison between categories.
• Pie chart: proportions of a whole.
• Pictogram: a small, simple data set shown visually.