Displaying and describing distributions: VCE General Mathematics Unit 3 Data analysis
“Types of data (categorical nominal and ordinal, numerical discrete and continuous), frequency tables, bar charts, segmented bar charts, dot plots, stem plots and histograms, and describing and interpreting the distribution of a categorical or numerical variable to answer statistical questions”
Classify each variable as categorical (nominal or ordinal) or numerical (discrete or continuous). Display categorical data with frequency tables and bar charts (segmented bar charts for two categorical variables) and numerical data with dot plots, stem plots or histograms. Describe a numerical distribution by its shape (symmetric, positively or negatively skewed, named after the tail), outliers, centre and spread, answering in context with units.
What this dot point is asking
Every data analysis question starts with two decisions: what kind of variable is this, and what display suits it. VCAA wants you to classify variables as categorical (nominal or ordinal) or numerical (discrete or continuous), to display categorical data with frequency tables and bar charts (including segmented bar charts), to display numerical data with dot plots, stem plots and histograms, and to describe what a display shows (shape, outliers, and roughly where the centre and spread are) so you can answer a statistical question in context.
The study design calls this "investigating data distributions", and it is the first thing tested in Examination 1 and Examination 2 each year. In 2025, of students lost the mark for classifying "type of home" as nominal, which shows that the vocabulary has to be exact.
The answer
Types of data
| Type | Subtype | What it is | Examples |
|---|---|---|---|
| Categorical | Nominal | Categories with no natural order | eye colour, favourite sport, type of home, postcode |
| Categorical | Ordinal | Categories with a natural order | T-shirt size, satisfaction rating, year level |
| Numerical | Discrete | Counts: only certain values, usually whole numbers | number of siblings, goals scored |
| Numerical | Continuous | Measurements: any value in a range | height, time, mass, price (treated as continuous) |
Tricky cases appear regularly. A postcode or a jersey number is written with digits but is categorical (nominal): averaging postcodes is meaningless. Age group (18 to 24, 25 to 34, ...) is categorical ordinal, while age in years is numerical. A rating on a 1 to 5 scale is usually treated as ordinal. If you are asked "numerical, nominal or ordinal", give exactly one of those three words.
Categorical data places each individual into a group (nominal if the groups have no order, ordinal if they do). Numerical data records a quantity: discrete if it is counted, continuous if it is measured.
Displaying categorical data
- A frequency table lists each category with its count (frequency) and, often, its percentage: .
- A bar chart shows each category as a separate bar (bars do not touch). For ordinal data, keep the bars in their natural order.
- A segmented bar chart stacks the categories into one bar; a percentaged segmented bar chart makes each bar 100%. It is the best display for comparing a categorical variable across groups (two categorical variables), as in the 2025 multiple-choice question.
The main summary of a categorical variable is its mode (the most common category) together with the percentages in each category.
Displaying numerical data
- A dot plot places one dot per value above a number line. Best for small data sets with repeated values.
- A stem plot (stem-and-leaf plot) splits each value into a stem (leading digits) and a leaf (last digit). It keeps every value, shows shape, and makes the median easy to find. Always include a key such as 7 | 2 means 72. If most values share one or two stems, split each stem into two rows (leaves 0 to 4 and 5 to 9).
- A histogram groups values into class intervals of equal width and draws touching bars. Use it for larger data sets. An interval such as "30 to 40" contains values from 30 up to but not including 40.
- For data spanning several orders of magnitude, a histogram on a logarithmic (base 10) scale is used (see the log scales page).
Choosing the display
| Data | Good displays |
|---|---|
| One categorical variable | frequency table, bar chart |
| Two categorical variables | two-way table, percentaged segmented (or side-by-side) bar chart |
| One numerical variable, small data set | dot plot, stem plot |
| One numerical variable, larger data set | histogram, boxplot |
| Numerical variable across groups | back-to-back stem plot, parallel dot plots, parallel boxplots |
| Two numerical variables | scatterplot |
Describing shape and outliers
The shape of a numerical distribution is described by comparing its two sides:
- Approximately symmetric: the two sides are roughly mirror images; mean and median are close.
- Positively skewed: the values bunch at the lower end with a long tail to the right (towards larger values); the mean is usually larger than the median.
- Negatively skewed: the values bunch at the upper end with a long tail to the left; the mean is usually smaller than the median.
The direction of the skew is the direction of the tail, not of the peak. Also note any possible outliers: values that sit well apart from the rest. The formal test for an outlier uses the fences and (see the boxplots page).
Classify first: categorical (nominal, ordinal) or numerical (discrete, continuous). Categorical data uses frequency tables and bar charts; numerical data uses dot plots, stem plots and histograms. Describe a numerical distribution by shape (symmetric, positively skewed, negatively skewed), outliers, centre and spread, with units and in context. Skew is named after the direction of the tail.
Reading a display to answer a statistical question
Most exam questions ask you to extract a specific fact:
- A count or percentage in a category or range: add the relevant frequencies, divide by the total.
- The modal category or interval: the tallest bar.
- The median: find its position (th value, or the average of the two middle values when is even) by counting through the ordered display. In a histogram you can only say which interval contains it.
- The range: maximum minus minimum (from a dot plot or stem plot; a histogram only gives an approximate range).
- Comparisons: "What percentage of students have fewer than 2 siblings?" becomes a sum of frequencies.
Then answer in context: " of the workers commute for at least 40 minutes", not just "18".
Describing a distribution in words
A complete description names the shape, any outliers, and gives the centre and spread with values and units. For the commuting data: "The distribution of commuting times is positively skewed, with most workers taking 10 to 30 minutes, a median between 20 and 30 minutes, and a small number of workers taking 50 minutes or more." Calculating and choosing the measures of centre and spread (mean and standard deviation, or median and IQR) is covered on the next page.
Classifying variables
A survey records: shoe size (numbers such as 8, 8.5, 9), favourite subject, number of subjects failed, school year level, and time spent on homework last night. Classify each.
- Shoe size
- Numerical and discrete (only certain sizes exist), though some texts treat it as ordinal; if a question gives only the options numerical, nominal or ordinal, choose numerical.
- Favourite subject
- Categorical, nominal.
- Number of subjects failed
- Numerical, discrete (a count).
- Year level
- Categorical, ordinal.
- Homework time
- Numerical, continuous (a measurement).
Marker's note: the test for numerical is whether arithmetic (such as an average) makes sense for the values.
A categorical frequency table
40 students chose a favourite sport: football 14, netball 10, basketball 8, swimming 5, other 3.
Percentages. Football , netball , basketball , swimming , other .
Describe. Football is the modal category, chosen by ; together football and netball account for of students.
Marker's note: percentages in a frequency table must add to (allowing for rounding).
A stem plot
Resting pulse rates (bpm) of 15 students: 58, 62, 64, 65, 67, 68, 70, 71, 72, 72, 74, 75, 78, 81, 95.
- Construct
- Stems 5 to 9; leaves in order: 5 | 8; 6 | 2 4 5 7 8; 7 | 0 1 2 2 4 5 8; 8 | 1; 9 | 5. Key: 7 | 2 means 72 bpm.
- Median
- 8th value bpm.
- Shape
- Approximately symmetric apart from 95, which is a possible outlier (above the upper fence ).
Marker's note: always write the key. A stem plot without a key cannot be read.
Reading a histogram
The commuting-time histogram above has frequencies 6, 15, 12, 8, 5, 3, 1 for intervals of 10 minutes from 0.
- Modal interval
- 10 to 20 minutes.
- Median interval
- , so the median is between the 25th and 26th values; cumulative frequencies 6, 21, 33 place both in 20 to 30 minutes.
- Percentage at least 40 minutes
- .
- Shape
- Positively skewed.
Marker's note: say "the median lies in the interval 20 to less than 30 minutes"; a single number cannot be read from a histogram.
Calling numbers numerical automatically. Postcodes, phone numbers and jersey numbers are categorical.
Answering "categorical" when asked "numerical, nominal or ordinal". Choose nominal or ordinal.
- Naming skew after the peak
- Skew is named after the tail: a tail to the right is positive skew.
- Drawing gaps in a histogram or joining bars in a bar chart
- Histograms touch; bar charts do not.
- Forgetting the key on a stem plot
- Without it, 7 | 2 could mean 72, 7.2 or 720.
- Picking one of the two middle values as the median
- For even , average the two middle values.
- Using a histogram to state an exact median or range
- A histogram only tells you intervals.
Start every data question by writing the variable type next to each variable. In multiple-choice questions about which display to use, identify the number and types of variables first: that alone eliminates most options. When describing a distribution, use the checklist shape, outliers, centre, spread, and include units and context in every statement.
Before you can make sense of data you need to know what kind it is. Some data sorts people into groups, like favourite sport, and some measures or counts something, like height or number of pets. Groups are shown with bar charts; measurements and counts are shown with dot plots, stem plots or histograms. Then you describe the picture: is it balanced, or does it have a long tail on one side? Are there any odd values far from the rest? And roughly where is the middle, and how spread out is it?
Exam-style questions
Questions in the style of VCAA exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.
2025 VCAA-style2 marksThe prices of 20 homes sold in an inner Melbourne suburb, and the type of each home, are: apartments $350 000, $490 000, $500 000, $620 000, $720 000, $830 000, $875 000, $995 000, $1 100 000, $1 520 000; houses $800 000, $840 000, $920 000, $920 000, $1 010 000, $1 263 000, $1 398 000, $1 460 000, $1 540 000, $1 540 000. (a) Find the median, in dollars, of the variable price. (b) State whether the variable type is numerical, nominal or ordinal.
Show worked answer →
(a) Put all 20 prices in order. With the median is the average of the 10th and 11th values. The ordered list begins 350 000, 490 000, 500 000, 620 000, 720 000, 800 000, 830 000, 840 000, 875 000, and the 10th and 11th values are both 920 000:
(1 mark; of students were correct.)
(b) Type takes the values "apartment" or "house": categories with no natural order, so it is nominal. (1 mark; correct.) "Categorical" alone is not the answer to "numerical, nominal or ordinal".
Source: VCAA 2025 General Mathematics Examination 2, Question 1a and 1b, and the 2025 examination report.
2025 VCAA-style1 markA two-way frequency table records the preferred car colour (black, silver or white) of a sample of female and male car buyers. Which one of the following is the most appropriate way to graphically display the data shown in the table? A. a histogram B. a back-to-back stem plot C. parallel boxplots D. a segmented bar chart
Show worked answer →
Both variables (car colour and sex) are categorical. Histograms and stem plots display numerical variables, and parallel boxplots compare a numerical variable across groups, so A, B and C are ruled out. A (percentaged) segmented bar chart is the standard display for the association between two categorical variables: D. ( of students were correct.)
Source: VCAA 2025 General Mathematics Examination 1, Question 8, and the 2025 examination report.
Practice questions
Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.
foundation3 marksClassify each variable as categorical (nominal or ordinal) or numerical (discrete or continuous): (a) the number of goals scored in a match; (b) T-shirt size (S, M, L, XL); (c) the time taken to run 100 m; (d) a person's postcode; (e) satisfaction rating (very unhappy, unhappy, neutral, happy, very happy); (f) favourite sport.
Show worked solution →
(a) Numerical, discrete (counted). (b) Categorical, ordinal (sizes have a natural order). (c) Numerical, continuous (measured). (d) Categorical, nominal: postcodes are labels, and doing arithmetic with them makes no sense. (e) Categorical, ordinal. (f) Categorical, nominal. (3 marks: half a mark each, rounded down.)
foundation2 marks40 students chose their favourite sport: football 14, netball 10, basketball 8, swimming 5, other 3. (a) Construct a frequency table with percentages. (b) State the modal category and what percentage chose it.
Show worked solution →
(a)
| Sport | Frequency | Percentage |
|---|---|---|
| Football | 14 | 35.0% |
| Netball | 10 | 25.0% |
| Basketball | 8 | 20.0% |
| Swimming | 5 | 12.5% |
| Other | 3 | 7.5% |
| Total | 40 | 100% |
(1 mark)
(b) Football is the modal (most common) category, chosen by of the students. (1 mark)
foundation2 marksThe number of siblings of 20 students is shown in a dot plot: 0 siblings 3 students, 1 sibling 8, 2 siblings 5, 3 siblings 2, 4 siblings 1, 6 siblings 1. (a) Find the median. (b) Describe the shape.
Show worked solution →
(a) With 20 values, the median is the average of the 10th and 11th. Counting up: values 1 to 3 are 0, values 4 to 11 are 1. The 10th and 11th are both 1, so the median is 1 sibling. (1 mark)
(b) Most values are at 0, 1 and 2, with a tail stretching to 3, 4 and 6: the distribution is positively skewed. (1 mark)
core3 marksThe resting pulse rates (beats per minute) of 15 students are 58, 62, 64, 65, 67, 68, 70, 71, 72, 72, 74, 75, 78, 81 and 95. (a) Display them in a stem plot with stems 5 to 9. (b) Find the median. (c) Describe the shape, noting any possible outlier.
Show worked solution →
(a)
| Stem | Leaves |
|---|---|
| 5 | 8 |
| 6 | 2 4 5 7 8 |
| 7 | 0 1 2 2 4 5 8 |
| 8 | 1 |
| 9 | 5 |
Key: 7 | 2 means 72 beats per minute. (1 mark)
(b) The median is the 8th value: 71 beats per minute. (1 mark)
(c) Apart from 95, the distribution is approximately symmetric, centred around 70. The value 95 sits well above the rest and is a possible outlier (the upper fence is ). (1 mark)
core3 marksA histogram shows the commuting times (minutes) of 50 workers: 0 to less than 10: 6; 10 to less than 20: 15; 20 to less than 30: 12; 30 to less than 40: 8; 40 to less than 50: 5; 50 to less than 60: 3; 60 to less than 70: 1. (a) State the modal interval. (b) In which interval does the median lie? (c) What percentage of workers commute for at least 40 minutes?
Show worked solution →
(a) 10 to less than 20 minutes (15 workers). (1 mark)
(b) The median is between the 25th and 26th values. Cumulative counts: 6, then 21, then 33. The 25th and 26th values are in 20 to less than 30 minutes. (1 mark)
(c) workers; . (1 mark)
core2 marksExplain why a histogram has no gaps between its bars but a bar chart does.
Show worked solution →
A histogram displays a numerical variable on a continuous number line: each bar covers an interval that starts exactly where the previous one ends, so the bars touch. (1 mark)
A bar chart displays a categorical variable: the categories are separate and could be listed in any order (or in their natural order if ordinal), so the bars are drawn apart. (1 mark)
exam3 marksFor the 50 commuting times in the histogram above, (a) describe the shape of the distribution, (b) explain whether the mean or the median is likely to be larger, and (c) write a one-sentence description a statistician would accept.
Show worked solution →
(a) Positively skewed: most times are between 10 and 30 minutes, with a tail stretching out to 60 to 70 minutes. (1 mark)
(b) The long tail of large values pulls the mean upward, so the mean is likely to be larger than the median. (1 mark)
(c) "The distribution of commuting times is positively skewed, with a modal interval of 10 to 20 minutes, a median between 20 and 30 minutes, and a few workers commuting for 50 minutes or more." (1 mark)
exam2 marksFor the 20 homes in the 2025 VCAA data, the percentages with prices less than $600 000, from $600 000 to $1 000 000, and more than $1 000 000 are: houses 0%, 40%, 60%; apartments 30%, 50%, 20%. (a) What type of display would best show this? (b) Use the percentages to compare the price distributions of the two types of home.
Show worked solution →
(a) A percentaged segmented bar chart, with one bar for houses and one for apartments, each divided into the three price ranges. (1 mark)
(b) Houses tend to be more expensive: of houses sold for more than $1 000 000 compared with of apartments, and no houses sold for less than $600 000 compared with of apartments. (1 mark)