Classify data relating to a single random variable as categorical (nominal or ordinal) or numerical (discrete or continuous), and select and use an appropriate graphical display for the data type

Make: categorical (the values are labels). No discrete/continuous sub-branch is required for a categorical variable, though noting "nominal" is a bonus.

Number of seats: numerical and discrete (a whole-number count).

Fuel efficiency: numerical and continuous (measured on a scale, can take any value in a range).

Markers award one mark per correct full classification. A common error is calling the number of seats continuous; it is a count, so it is discrete. Stating only "categorical/numerical" without the discrete/continuous sub-branch on the numerical variables loses the detail marks.

Part (a): Year group is categorical ordinal (the years have a natural order). Preferred meal is categorical nominal (unordered labels). Daily screen time is numerical continuous (measured on a scale).

Part (b): A column (bar) graph or sector graph is appropriate because preferred meal is categorical; a column graph shows one bar per meal with height equal to the frequency, making the most popular meal easy to read. A sector graph is acceptable if the response is justified as showing each meal as a share of the whole.

Markers award a mark for each correct classification (with the ordinal/nominal distinction expected), and marks in part (b) for naming a valid categorical display AND giving a reason tied to the data type. Naming a histogram or dot plot for the meal data loses the mark, since those are for numerical data.

Difference: a discrete variable is counted in whole-number steps (it can only take separated values), while a continuous variable is measured on a scale and can take any value within a range.

Examples (sport-dependent, e.g. cricket): discrete - the number of wickets taken; continuous - a bowler's delivery speed in km/h. Any valid count and any valid measurement earn the marks.

Display: a dot plot is the more natural choice for a discrete variable over a small whole-number range, because each whole-number value gets its own stack of dots, showing every individual value and the most common count clearly; a histogram groups data into intervals, which is better suited to continuous data or a large range.

Markers reward a correct, clearly worded distinction, one valid example of each type, and a justified display choice. Defining the terms the wrong way round, or giving a measured quantity as the "discrete" example, loses marks.

Ask the deciding question: is the value a label or a number you can count or measure?

Part (a) eye colour. The values are labels such as brown, blue and green. They place each student into a named group, so eye colour is a categorical variable.

Part (b) number of siblings. The value is a genuine count, such as $0$, $1$ or $3$, that you can do arithmetic with, so the number of siblings is a numerical variable. (Check: you can find a mean number of siblings, but a mean eye colour makes no sense, which confirms one is numerical and the other categorical.)

Ask the deciding question: is the variable counted in whole steps (discrete) or measured on a scale that can take any value (continuous)?

Part (a) number of cars

You count cars in whole numbers; a household cannot own $2.4$ cars, so this is discrete.

Part (b) mass of a watermelon

Mass is measured and can take any value in a range, such as $3.7$ kg or $3.74$ kg depending on the scales, so this is continuous.

Part (c) number of goals

Goals are counted in whole numbers, so this is discrete. (Check: the two counts can only land on whole numbers, while the mass can sit anywhere between two whole numbers, which is exactly the discrete versus continuous split.)

Ask the deciding question: do the categories have a natural order, or are they just different names?

Part (a) favourite sport. Cricket, netball and soccer are simply different names with no natural ranking, so the variable is nominal.

Part (b) movie rating. Poor, fair, good and excellent have a clear order from worst to best, so the variable is ordinal. (Check: you could sensibly sort the ratings from lowest to highest, but sorting the sports into an order would be meaningless, which is the nominal versus ordinal test.)

Work each variable through the two questions in turn: first categorical or numerical, then the sub-branch.

Postcode

Although it is written with digits, a postcode is a label for an area; adding or averaging postcodes is meaningless. So it is categorical, and the postcodes have no natural order, so it is categorical nominal.

T-shirt size

The values S, M, L, XL are labels, so the variable is categorical, and they run in a clear order from smallest to largest, so it is categorical ordinal.

Height in centimetres

Height is measured and can take any value in a range (such as $172.5$ cm), so it is numerical and continuous.

Number of languages

This is a count in whole numbers, so it is numerical and discrete. (Check: the postcode is the classic trap - digits do not make it numerical, because the deciding test is whether arithmetic on the value is meaningful, and averaging postcodes is not.)

Match the display to the data type: categorical data suits column, sector and divided bar graphs; numerical data suits dot plots, stem-and-leaf plots, histograms and frequency tables.

Part (a) household budget shares

The categories are parts of one whole (the total budget), so a sector (pie) graph is appropriate because it shows each category as a slice of the total $100\%$. A divided bar graph would work equally well.

Part (b) number of pets

This is discrete numerical data over a small range, so a dot plot (or a column/frequency graph) is appropriate: one dot per student stacked above each count shows the distribution and any clustering.

Part (c) favourite cuisine

This is nominal categorical data, so a column (bar) graph is appropriate, with one bar per cuisine and the height showing the frequency. (Check: each chosen graph matches its data type - a slice-of-the-whole for budget shares, stacked dots for a small count, and separated bars for unordered categories.)

Part (a) classify all four. Apply the two questions to each variable.

• Blood type: the values are unordered labels, so it is categorical nominal.
• Pain level: none, mild, moderate and severe are labels with a natural order, so it is categorical ordinal.
• Resting heart rate: measured on a scale and able to take any value in a range, so it is numerical continuous.
• Number of GP visits: a whole-number count, so it is numerical discrete.

Part (b) appropriate displays. Blood type is categorical, so a column graph (one bar per blood type, height equal to the frequency) clearly compares the four groups. Resting heart rate is continuous numerical, so a histogram with the data grouped into class intervals (for example $60$ to $69$, $70$ to $79$, and so on) shows the shape of the distribution.

Part (c) why a sector graph fails for heart rate. A sector graph splits a whole into categories, but resting heart rate is continuous with hundreds of distinct values, so it has no small set of natural slices; a pie of $500$ near-unique values would be unreadable and would hide the shape (centre, spread, skew) that a histogram reveals. (Check: each display matches its type, and the explanation turns on the fact that continuous data has no natural categories to slice, which is the whole reason type drives the display.)

Part (a) classify each. Run each variable through the two questions.

• Suburb: unordered labels, so categorical nominal.
• Satisfaction: ordered labels from very dissatisfied to very satisfied, so categorical ordinal.
• Age in years and months: measured on a continuous scale (age increases smoothly, and "years and months" records part-years), so numerical continuous.
• Number of park visits: a whole-number count, so numerical discrete.

Part (b) display for satisfaction, and why order matters. A column graph with the bars placed in the natural order from very dissatisfied to very satisfied is appropriate, because satisfaction is ordinal: keeping the categories in order lets a reader see the trend from unhappy to happy at a glance. Suburb is nominal, so its bars can sit in any order (for example tallest first) without losing meaning, since there is no underlying ranking to preserve.

Part (c) display for park visits. The visits are discrete numerical data over the small whole-number range $0$ to $12$, so a dot plot (or a column graph of the frequencies) is appropriate: one dot per resident above each count shows the most common number of visits, any gaps, and any unusually high values. (Check: ordering matters only when the categories themselves are ordered, which is precisely what separates ordinal from nominal, and a small whole-number range is exactly where a dot plot is at its clearest.)