Year 11 General Univariate Data Analysis

Displaying And Describing Categorical Data Distributions

20 practice questions 1 video lesson Theory + worked examples

Theory

Categorical data is summarised with frequency tables and bar charts. Convert raw counts to percentages using $percentage = \frac{frequency}{total} \times 100 %$ . The mode is the most common category. Bar charts have gaps between bars to mark distinct categories; histograms (for numerical data) do not.

For categorical data, the goal is to summarise how often each category occurs. The two main displays are frequency tables and bar charts.

A frequency table lists each category and how many times it occurs. A percentage column is often added:

$percentage = \frac{frequency}{total} \times 100 %$

A bar chart shows categories on the horizontal axis and frequency (or percentage) on the vertical axis. The bars are separated by gaps to indicate the categories are distinct, not continuous.

The mode is the most common category — the one with the tallest bar (or the largest frequency in the table).

A segmented (stacked) bar chart is a single bar split into coloured sections, with each section representing a category's share. It's useful for percentage breakdowns or for comparing two groups side-by-side.

The first diagram is the favourite snacks bar chart from the worked example, with the mode (Chips) highlighted. The second compares a bar chart (categorical, with gaps) to a histogram (numerical, no gaps).

Five separated bars for five categories. Chips is the mode (tallest bar at 36).

Bar charts have gaps (distinct categories). Histograms don't (continuous numerical values).

The only real formula is the percentage conversion.

Percentage from frequency

$percentage = \frac{frequency}{total} \times 100 %$

% = \frac{frequency}{total} \times 100 %

Frequency from percentage

$frequency = \frac{percentage}{100} \times total$

Quick reference

Quantity	How to find it
Mode	category with the largest frequency
Missing frequency	total minus the sum of the others
Combined %	add the relevant percentages, or add their frequencies then ÷ total × 100%

Watch the totals. A "45%" in one survey isn't directly comparable to "45%" in another unless the totals are the same. Always compute percentages within each group before comparing.

Building a frequency table

List every category in the data — including any with zero count.
Count (tally) how many times each category appears.
Add a percentage column using the formula above.
Check that all frequencies add to the total and all percentages add to $100 %$ (allowing small rounding error).

Drawing a bar chart

Put categories on the horizontal axis, frequency (or percentage) on the vertical.
Make all bars the same width, separated by equal-sized gaps.
Start the vertical axis at $0$ (no truncation).
Label both axes and give the chart a title.

Describing a categorical distribution

State the mode — the most common category.
State the least popular category.
Describe the general pattern: "one category dominates", "frequencies are roughly even", or similar.

EXAMPLE 1 — FREQUENCY FROM PERCENTAGE

32 %

250

Year 11 students chose blue as their favourite colour. How many students chose blue?

SOLUTION

Convert the percentage to a decimal and multiply by the total.

Number	$=$	$0.32 \times 250$
	$=$	$80$

Answer: $80$ students chose blue.

number = 80

EXAMPLE 2 — PERCENTAGE FROM FREQUENCY

A survey of

180

pizza orders had

76

Medium pizzas. What percentage were Medium?

SOLUTION

Use $percentage = \frac{frequency}{total} \times 100 %$ .

$%$	$=$	$\frac{76}{180} \times 100 %$
	$\approx$	$42.2 %$

Answer: about $42.2 %$ of pizzas were Medium.

% \approx 42.2 %

EXAMPLE 3 — FIND A MISSING FREQUENCY

A pet survey of

100

Brisbane families: Dog

45

, Cat

?

, Bird

12

, Fish

8

. Find the missing Cat count.

SOLUTION

The total of all categories equals $100$ . Subtract the known categories.

Cat	$=$	$100 - 45 - 12 - 8$
	$=$	$35$

Answer: $35$ families chose Cat.

Cat = 35

EXAMPLE 4 — COMBINED PERCENTAGE

For the snack data above (Chips

36

, Choc

24

, total

120

), what percentage of students chose Chips or Chocolate?

SOLUTION

Either add the individual percentages, or add the frequencies then divide by the total.

Combined	$=$	$30 % + 20 %$
	$=$	$50 %$

Check directly: $\frac{36 + 24}{120} \times 100 % = \frac{60}{120} \times 100 % = 50 %$ . ✓

Answer: $50 %$ of students chose Chips or Chocolate.

combined % = 50 %

Common pitfalls

Bar charts need gaps; histograms don't. The bars in a bar chart (categorical data) should be separated by gaps. Histograms (numerical, continuous data) have bars touching. Drawing them the wrong way around is a classic mark-losing mistake.

Truncated axes mislead. If the vertical axis doesn't start at

0

, small differences between bars look much bigger than they are. Always check the axis — and always start your own bar chart at

0

Percentages need their totals. "

45 %

of one survey" isn't directly comparable with "

45 %

of another" unless the totals are the same. For comparing groups of different sizes, compute percentages within each group.

The mode is the category, not the frequency. If Chips has frequency

36

and is the most common, the mode is "Chips" — not "

36

". A common slip.

Don't drop categories with zero frequency. A frequency of

0

is still a valid data point — it tells you nobody chose that category. Keep the row in the table.

Frequently asked questions

What is a frequency table?

A frequency table lists each category in one column and how many times it occurs in another. A percentage column is often added: percentage equals frequency divided by total, times 100.

What is the difference between a bar chart and a histogram?

Bar charts are for categorical data and have GAPS between the bars to show categories are distinct. Histograms are for numerical data and have bars TOUCHING because values run continuously into each other.

What is the mode of a categorical dataset?

The mode is the most common category — the one with the tallest bar or largest frequency. Note that the mode is the CATEGORY itself (like 'Chips') and not the frequency number (like 36).

What is a segmented bar chart?

A segmented (stacked) bar chart is a single bar divided into coloured sections, with each section's length showing one category's share. It's useful for visualising a percentage breakdown or comparing two groups side-by-side.

What is a truncated axis and why is it misleading?

A truncated axis is one that doesn't start at zero. This makes small differences between bars look much larger than they really are. It's a classic technique used in misleading graphs — always check whether the vertical axis starts at zero.

How do I compare two groups of different sizes?

Convert raw frequencies to percentages within each group first. If 30 out of 60 Year 7s prefer soccer and 10 out of 60 Year 11s do, the percentages are 50 percent versus about 17 percent. Comparing raw numbers when totals differ is misleading.

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Interpreting And Describing Frequency Tables And Column Charts

Displaying And Describing Categorical Data Distributions

📖 Theory

Percentage from frequency

Frequency from percentage

Quick reference

Building a frequency table

Drawing a bar chart

Describing a categorical distribution

Common pitfalls

Frequently asked questions

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Theory

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