Associations between a numerical and a categorical variable for VCE General Mathematics Unit 3 Data analysis

Test each statement against the values.

• A: $5.6 < 7.8$, so false.
• B: $76.5 - 67.2 = 9.3$ years, which is not more than 10, so false.
• C: $Q_3$ for H is 70.8 and $Q_1$ for T is 73.6; $70.8 < 73.6$, so false.
• D: the minimum of Sample T (about 68) is greater than the median of Sample H (about 67.2), so every country in T has a higher life expectancy than the median of H. True.

The answer is D ($82\%$ of students chose it). Statement D is the kind of "all" or "percentage" comparison that can be read straight from the five-number summary: the minimum tells you about all values.

Source: VCAA 2025 General Mathematics Examination 1, Question 3, with the values read from the boxplots as given in the 2025 examination report.

(a) Explanatory: place of residence (city or country town), which is categorical. Response: weekly commuting time, which is numerical. (1 mark) The explanatory variable is the one we think may explain differences in the other.

(b) Parallel boxplots (or a back-to-back stem plot, since there are exactly two groups, or parallel dot plots). (1 mark)

(a) The median for walkers and cyclists (11.8 thousand) is much higher than for car commuters (5.9 thousand). (1 mark)

(b) Yes. Step count differs systematically between the groups: walkers and cyclists tend to take more steps, so step count is associated with commuting mode. (1 mark)

B. Parallel boxplots can show any number of groups side by side. A back-to-back stem plot only works for exactly two groups, a segmented bar chart is for two categorical variables, and a scatterplot is for two numerical variables.

(a) With 13 values, the median is the 7th value and each half has 6 values.

• Year 7: median 52; $Q_1 = \frac{41 + 45}{2} = 43$, $Q_3 = \frac{58 + 61}{2} = 59.5$, IQR $= 16.5$.
• Year 12: median 36; $Q_1 = \frac{28 + 31}{2} = 29.5$, $Q_3 = \frac{41 + 44}{2} = 42.5$, IQR $= 13$.

(2 marks)

(b) Completion time is associated with year level. The median time for Year 12 students (36 seconds) is 16 seconds less than for Year 7 students (52 seconds), and Year 12 times are also slightly less variable (IQR 13 seconds compared with 16.5 seconds). (1 mark)

The Year 7 median is 52 seconds. In the Year 12 data, 12 of the 13 values (all except 55) are less than 52. (1 mark)

$\frac{12}{13} \times 100\% \approx 92\%$ of Year 12 students were faster than the median Year 7 student. (1 mark) Counting leaves in the stem plot gives this exactly; with only a boxplot you could say "more than $75\%$", because $Q_3 = 42.5 < 52$.

There is some association, but only in spread, not in centre. The typical (median) rainfall is about the same at both towns, so knowing the town tells you little about the typical rainfall. (1 mark) However, town B's rainfall is far more variable (IQR 55 mm against 18 mm), so knowing the town does tell you how predictable the rainfall is. (1 mark)

In VCAA questions, an association is usually judged mainly on differences in median; a difference in spread alone should be described carefully.

(a) Car: IQR $= 7.4 - 4.8 = 2.6$; upper fence $= 7.4 + 1.5 \times 2.6 = 11.3$. The maximum, 13.0, is above 11.3, so it is an outlier. (1 mark)

(b) "Daily step count is associated with commuting mode. The median step count increases from 5.9 thousand for car commuters, to 8.2 thousand for public transport users, to 11.8 thousand for walkers and cyclists. The spread also increases slightly: the IQR is 2.6 thousand for car commuters, 2.7 thousand for public transport users and 3.4 thousand for walkers and cyclists." (2 marks: 1 for comparing all three medians with values, 1 for the IQR comparison with values.)

Quoting the actual values is what earns the marks.

(a) Team B's distribution is the same shape as Team A's, shifted up by exactly one goal: median 3 for B against 2 for A, identical ranges (5) and identical IQRs (2). Team B tends to score one more goal per game. (1 mark)

(b) Both distributions are slightly positively skewed (a tail towards the higher scores), so the median is a better measure of the typical score than the mean, which is pulled towards the tail. (1 mark)