Topic 1.4 Notes – Graphical Representations for One Categorical Variable

What a Categorical Distribution Shows

A categorical variable puts each observational unit into exactly one category. If the variable is “usual transportation,” each student goes into one category like car, bus, or walk.

The distribution tells you two things:

• what the categories are
• how often each category occurs

You can express that three equivalent ways:

• Frequency = count in each category
• Relative frequency = proportion in each category
• Percentage = relative frequency $\times 100\%$

The main calculation here is

$$\text{Relative frequency}=\frac{\text{category count}}{\text{total count}}$$

If 48 out of 120 students usually come by car, the relative frequency is

$$\frac{48}{120}=0.40=40\%$$

Always check totals:

• frequencies add to $n$
• relative frequencies add to $1$
• percentages add to $100\%$ apart from rounding

One easy test mistake is reading a graph as counts when it actually shows proportions. Check the axis or labels every time.

Also, categorical distributions are described by which categories are common or uncommon and how categories compare. You do not describe them with shape, center, spread, or outliers. Those are for quantitative data.

Bar Charts and Pie Charts

Both graphs display one categorical variable. For one data set, the category with the largest count also has the largest proportion, tallest bar, and biggest slice. Here, the transportation categories are shown as both a bar chart and a pie chart, so you can see how the same distribution looks in each format.

Bar charts

A bar chart has one bar per category.

• bar height or length shows frequency or relative frequency
• bars have equal width
• bars have gaps between them
• categories go on one axis
• counts, relative frequencies, or percents go on the other axis
• the numerical scale should begin at zero

In the example, the bar chart uses percentages on the vertical axis and starts at zero, which is exactly what you want.

Category order can be:

• logical
• alphabetical
• decreasing by frequency

A common confusion is bar chart vs histogram.

• Bar chart = categorical data, bars have gaps
• Histogram = quantitative data grouped into intervals, bars touch

Pie charts

A pie chart shows the whole data set as one circle.

• each slice is one category
• slice area shows relative frequency
• all slices together make $100\%$

The transportation pie chart labels each slice by percent, which makes the part-to-whole relationship easy to scan.

The slice angle comes from

$$360^\circ \times p$$

where $p$ is the relative frequency. If $p=0.25$, the angle is $360^\circ(0.25)=90^\circ$.

Pie charts only make sense when:

• categories are mutually exclusive
• categories make up the whole data set

They get weak when there are lots of categories or the categories are close in size. Bar charts are usually easier to compare.

Making and Choosing the Right Graph

Here’s the flow.

1. List the categories and counts.
2. If needed, convert counts to relative frequencies by dividing by total $n$.
3. Build the graph that matches what you want to show.

For a bar chart:

• label categories clearly
• decide whether the axis shows counts or proportions
• use equal-width bars, gaps, and a zero baseline

For a pie chart:

• convert each category to a proportion or percent
• if drawing by hand, find each slice angle
• label slices clearly

Use frequency when actual numbers matter. Use relative frequency when you’re making percent/proportion claims or comparing groups of different sizes. A pie chart is always a relative-frequency display, even if counts are written on it.

Reading the Graph and Justifying Claims

Your claim has to name the variable and the category in context. Say “Car was the most common transportation method among these 120 students,” not “the tallest bar was car.”

You should be able to justify claims like these:

• most or least common category
• one category is more common than another
• a category is above $50\%$ or above one-third
• combined categories make a majority

Use numbers from the graph or table. If you estimate from the picture, say about or approximately.

Graph evidence is descriptive. It does not prove causation, and by itself it does not justify broad generalizations.

Comparing Multiple Data Sets and Common Mistakes

To compare groups, you need the same categorical variable and the same category definitions. Keep the same category order and compatible scales.

If sample sizes are equal, compare frequencies directly. If sample sizes differ, compare relative frequencies.

The two bar charts below show both situations. On the left, the groups each have $n=100$, so counts can be compared directly. On the right, the totals are different, so the bars have the same pattern in percent even though the counts are much larger for one group.

Comparing counts and relative frequencies

Example. Group A has 35% choosing fruit and Group B has 30%. You’d say fruit is more common in Group A by 5 percentage points, since $35\%-30\%=5$ percentage points.

Common mistakes:

• ignoring whether the graph shows count or percent
• comparing raw counts when sample sizes differ
• using a truncated bar-chart axis
• forgetting gaps in bar charts
• making a pie chart when categories overlap or do not make up the whole
• making vague claims with no numerical support